H. M. Srivastava

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All published works (1388)

Carlitz's q$$ q $$‐Operators for the Generalized Homogeneous Hahn Polynomials

2025-05-26

ABSTRACT In this paper, motivated by Carlitz's ‐operators and Liu's generalized homogeneous Hahn polynomials, we show how to construct Carlitz's ‐operators of the generalized homogeneous Hahn polynomials. We derive several … ABSTRACT In this paper, motivated by Carlitz's ‐operators and Liu's generalized homogeneous Hahn polynomials, we show how to construct Carlitz's ‐operators of the generalized homogeneous Hahn polynomials. We derive several generating functions for the generalized homogeneous Hahn polynomials by applying the method of exponential operator decomposition. In addition, we deduce Rogers type bilinear, trilinear, and mixed‐type generating functions as well as the Srivastava–Agarwal‐type generating functions for the generalized homogeneous Hahn polynomials, which provide extensions of the results of Carlitz, Abdlhusein, and Saad and Abdlhusein.

Solving Time-Fractional Fisher Models by Non-Polynomial Splines in Terms of Logarithmic Derivatives

2025-04-23

H. M. Srivastava , Majeed A. Yousif , Pshtiwan Othman Mohammed +3 more

Sharp Starlike and Convex Radii for the Differential Operator of Analytic Functions

2025-04-23

Zhenyong Hu , H. M. Srivastava , Ying Zhang

Solutions comparative for a novel model of fractional delta difference type

2025-04-14

Pshtiwan Othman Mohammed , H. M. Srivastava , Nejmeddine Chorfi +3 more

Korovkin-Type Theorems for Positive Linear Operators Based on the Statistical Derivative of Deferred Cesàro Summability

2025-04-11

H. M. Srivastava , Bidu Bhusan Jena , Susanta Kumar Paikray +1 more

In this paper, we introduce and investigate the concept of statistical derivatives within the framework of the deferred Cesàro summability technique, supported by illustrative examples. Using this approach, we establish … In this paper, we introduce and investigate the concept of statistical derivatives within the framework of the deferred Cesàro summability technique, supported by illustrative examples. Using this approach, we establish a novel Korovkin-type theorem for a specific set of exponential test functions, namely 1, e−υ and e−2υ, which are defined on the Banach space C[0,∞). Our results significantly extend several well-known Korovkin-type theorems. Additionally, we analyze the rate of convergence associated with the statistical derivatives under deferred Cesàro summability. To support our theoretical findings, we provide compelling numerical examples, followed by graphical representations generated using MATLAB software, to visually illustrate and enhance the understanding of the convergence behavior of the operators.

The LDG Finite-Element Method for Multi-Order FDEs: Applications to Circuit Equations

2025-04-05

Mohammad Izadi , H. M. Srivastava , Mahdi Kamandar

The current research study presents a comprehensive analysis of the local discontinuous Galerkin (LDG) method for solving multi-order fractional differential equations (FDEs), with an emphasis on circuit modeling applications. We … The current research study presents a comprehensive analysis of the local discontinuous Galerkin (LDG) method for solving multi-order fractional differential equations (FDEs), with an emphasis on circuit modeling applications. We investigated the existence, uniqueness, and numerical stability of LDG-based discretized formulation, leveraging the Liouville–Caputo fractional derivative and upwind numerical fluxes to discretize governing equations while preserving stability. The method was validated through benchmark test cases, including comparisons with analytical solutions and established numerical techniques (e.g., Gegenbauer wavelets and Dickson collocation). The results demonstrate that the LDG method achieves high-accuracy solutions (e.g., with a relatively large time step size) and reduced computational costs, which are attributed to its element-wise formulation. These findings position LDG as a promising tool for complex scientific and engineering applications, particularly in modeling fractional-order systems such as RL, RLC circuits, and other electrical circuit equations.

Investigations of a Class of Liouville-Caputo Fractional Order Pennes Bioheat Flow Partial Differential Equations through Orthogonal Polynomials on Collocation Points

2025-04-01

Vijay Saw , Pratibhamoy Das , H. M. Srivastava

Boundedness properties and Parseval-type relations for the index Whittaker transforms over Lebesgue spaces

2025-03-28

H. M. Srivastava , Jeetendrasingh Maan , E. R. Negrín

A New Class of Korovkin-Type Theorems on Double Sequences

2025-03-17

H. M. Srivastava , Bidu Bhusan Jena , Susanta Kumar Paikray

Asymptotic analysis of solutions of delay difference equations

2025-03-16

Qin Diao , Yong-Guo Shi , H. M. Srivastava +2 more

Properties of solutions of the $$\alpha$$-harmonic equation in the unit disk

2025-02-24

Z. Y. Hu , J. H. Fan , H. M. Srivastava

Modular identities for some special cases of Ramanujan’s general continued fraction

2025-02-03

Shruthi C. Bhat , H. M. Srivastava , B. R. Srivatsa Kumar

Mathematical tools and techniques in the applied scientific disciplines

2025-01-01

H. M. Srivastava

Effect of Viscosity Models on Diatomic Shock Structure Using Multi-temperature Approach

2025-01-01

H. M. Srivastava , Tapan K. Mankodi , R.S. Myong

Some new results associated with Ramanujan’s numbers <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:mi>τ</mml:mi><mml:mo class="MathClass-open" stretchy="false">(</mml:mo><mml:mi>n</mml:mi><mml:mo class="MathClass-close" stretchy="false">)</mml:mo></mml:mrow></mml:math>

2025-01-01

Aleksandar Petojević , H. M. Srivastava , Marijana Gorjanac Ranitović

In this article, the authors derive more precise limits in the recursion formula of D. H. Lehmer for Ramanujan’s numbers <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:mi>τ</mml:mi><mml:mo class="MathClass-open" stretchy="false">(</mml:mo><mml:mi>n</mml:mi><mml:mo class="MathClass-close" stretchy="false">)</mml:mo></mml:mrow></mml:math>, which are generated … In this article, the authors derive more precise limits in the recursion formula of D. H. Lehmer for Ramanujan’s numbers <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:mi>τ</mml:mi><mml:mo class="MathClass-open" stretchy="false">(</mml:mo><mml:mi>n</mml:mi><mml:mo class="MathClass-close" stretchy="false">)</mml:mo></mml:mrow></mml:math>, which are generated by <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" class="equation" display="block"><mml:mtable class="equation-star"><mml:mtr><mml:mtd> <mml:mrow> <mml:munderover accent="false" accentunder="false"><mml:mrow><mml:mo>∑</mml:mo> </mml:mrow><mml:mrow><mml:mi>n</mml:mi><mml:mo class="MathClass-rel" stretchy="false">=</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mi>∞</mml:mi></mml:mrow></mml:munderover><mml:mi>τ</mml:mi><mml:mo class="MathClass-open" stretchy="false">(</mml:mo><mml:mi>n</mml:mi><mml:mo class="MathClass-close" stretchy="false">)</mml:mo><mml:msup><mml:mrow><mml:mi>z</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msup> <mml:mo class="MathClass-rel" stretchy="false">=</mml:mo> <mml:mi>z</mml:mi><mml:munderover accent="false" accentunder="false"><mml:mrow><mml:mo>∏</mml:mo> </mml:mrow><mml:mrow><mml:mi>n</mml:mi><mml:mo class="MathClass-rel" stretchy="false">=</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mi>∞</mml:mi></mml:mrow></mml:munderover><mml:msup><mml:mrow><mml:mo class="MathClass-open" stretchy="false">(</mml:mo><mml:mn>1</mml:mn> <mml:mo class="MathClass-bin" stretchy="false">−</mml:mo> <mml:msup><mml:mrow><mml:mi>z</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msup><mml:mo class="MathClass-close" stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mn>24</mml:mn></mml:mrow></mml:msup><mml:mo class="MathClass-punc" stretchy="false">,</mml:mo><mml:mspace class="qquad" width="2em"/><mml:mo class="MathClass-open" stretchy="false">(</mml:mo><mml:mo class="MathClass-rel" stretchy="false">|</mml:mo><mml:mi>z</mml:mi><mml:mo class="MathClass-rel" stretchy="false">|</mml:mo> <mml:mo class="MathClass-rel" stretchy="false"><</mml:mo> <mml:mn>1</mml:mn><mml:mo class="MathClass-close" stretchy="false">)</mml:mo><mml:mo class="MathClass-punc" stretchy="false">.</mml:mo> </mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:math> A presumably new recursion formula for Ramanujan’s numbers <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:mi>τ</mml:mi><mml:mo class="MathClass-open" stretchy="false">(</mml:mo><mml:mi>n</mml:mi><mml:mo class="MathClass-close" stretchy="false">)</mml:mo></mml:mrow></mml:math> is also presented.

The second Hankel determinant and the Fekete-Szegö functional for a subclass of analytic functions by using the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.svg" display="inline" id="d1e24"><mml:mi>q</mml:mi></mml:math>-Sălăgean derivative operator

2024-12-27

Ekram E. Ali , H. M. Srivastava , Wafaa Y. Kota +2 more

EXISTENCE RESULT FOR A COUPLED SYSTEM OF HYBRID FRACTIONAL DIFFERENTIAL EQUATIONS IN A BANACH ALGEBRA

2024-12-13

H. M. Srivastava , Benaouda Hedia

In this work, we investigate the existence result for a coupled system of hybrid fractional differential equations in a Banach algebra. Our main result is based on a generalization of … In this work, we investigate the existence result for a coupled system of hybrid fractional differential equations in a Banach algebra. Our main result is based on a generalization of Darbo’s fixed point theorem in Banach algebra. We apply in our approach the technique of measure of non-compactness, we prove that the Kuratowski measure of noncompactness satisfies a condition (m) which will be useful in our considerations. An example is given to illustrate the feasibility of our main result. An example is provided to illustrate our result.

Positivity and uniqueness of solutions for Riemann–Liouville fractional problem of delta types

2024-11-29

H. M. Srivastava , Pshtiwan Othman Mohammed , Dumitru Băleanu +3 more

Dynamics and numerical analysis of a fractional-order toxoplasmosis model incorporating human and cat populations

2024-11-14

Waleed Adel , H. M. Srivastava , Mohammad Izadi +2 more

Toxoplasmosis is a significant zoonotic disease that poses risks to public health and animal health, making the understanding of its transmission dynamics crucial. In this study, we present a novel … Toxoplasmosis is a significant zoonotic disease that poses risks to public health and animal health, making the understanding of its transmission dynamics crucial. In this study, we present a novel fractional-order model that captures complex interactions among human, cat, and mouse populations, providing deeper insights into the disease spread and control. We utilize mathematical techniques to analyze the model properties, including the existence, uniqueness, positivity, and boundedness of solutions, along with stability analysis of the equilibrium states. The basic reproduction number $R_{0}$ is derived, revealing the threshold for potential outbreaks. Our findings indicate that key parameters significantly influence the dynamics of toxoplasmosis, with implications for targeted intervention strategies. We propose the QLM-FONP numerical scheme for efficient resolution of the model and provide a comprehensive convergence analysis, demonstrating the reliability of the numerical solutions. The results confirm the effectiveness of our approach, illustrating that the proposed model not only offers accurate predictions but also extends beyond previous efforts in the literature by incorporating fractional-order dynamics, which better reflect real-world transmission processes. Overall, this study enhances the understanding of toxoplasmosis transmission and informs future research and control efforts.

Sharp inequalities for a class of novel convex functions associated with Gregory polynomials

2024-11-08

H. M. Srivastava , Nak Eun Cho , A. A. Alderremy +3 more

This paper explores the class $\mathcal{C}_{G}$ , consisting of functions g that satisfy a specific subordination relationship with Gregory coefficients in the open unit disk E. By applying certain conditions … This paper explores the class $\mathcal{C}_{G}$ , consisting of functions g that satisfy a specific subordination relationship with Gregory coefficients in the open unit disk E. By applying certain conditions to related coefficient functionals, we establish sharp estimates for the first five coefficients of these functions. Additionally, we derive bounds for the second and third Hankel determinants of functions in $\mathcal{C}_{G}$ , providing further insight into the class's properties. Our study also investigates the logarithmic coefficients of $\log \left ( \frac{g(t)}{t}\right ) $ and the inverse coefficients of the inverse functions $(g^{-1})$ within the same class.

On the Lanczos Method for Computing Some Matrix Functions

2024-11-04

Ying Gu , H. M. Srivastava , Xiaolan Liu

The study of matrix functions is highly significant and has important applications in control theory, quantum mechanics, signal processing, and machine learning. Previous work has mainly focused on how to … The study of matrix functions is highly significant and has important applications in control theory, quantum mechanics, signal processing, and machine learning. Previous work has mainly focused on how to use the Krylov-type method to efficiently calculate matrix functions f(A)β and βTf(A)β when A is symmetric. In this paper, we mainly illustrate the convergence using the polynomial approximation theory for the case where A is symmetric positive definite. Numerical results illustrate the effectiveness of our theoretical results.

Sharp starlike and convex radius for the differential operator of analytic functions

2024-10-31

Zhenyong Hu , H. M. Srivastava , Ying Zhang

UDC 517.5 Under given coefficient conditions for analytic functions <mml:math> <mml:mrow> <mml:mi>f</mml:mi> </mml:mrow> </mml:math> in the unit disk <mml:math> <mml:mrow> <mml:mi mathvariant="bold">D</mml:mi> </mml:mrow> </mml:math>, we first obtain the starlike and … UDC 517.5 Under given coefficient conditions for analytic functions <mml:math> <mml:mrow> <mml:mi>f</mml:mi> </mml:mrow> </mml:math> in the unit disk <mml:math> <mml:mrow> <mml:mi mathvariant="bold">D</mml:mi> </mml:mrow> </mml:math>, we first obtain the starlike and convex radius for the linear combination of the differential operator <mml:math> <mml:mrow> <mml:mi>z</mml:mi> <mml:mi>f</mml:mi> <mml:mo>'</mml:mo> </mml:mrow> </mml:math> of analytic functions <mml:math> <mml:mrow> <mml:mi>f</mml:mi> </mml:mrow> </mml:math> in <mml:math> <mml:mrow> <mml:mi mathvariant="bold">D</mml:mi> <mml:mo>.</mml:mo> </mml:mrow> </mml:math> Then we obtain the starlike and convex radius for the differential operator <mml:math> <mml:mrow> <mml:mi>z</mml:mi> <mml:mi>f</mml:mi> <mml:mo>'</mml:mo> </mml:mrow> </mml:math>. Furthermore, we present the starlike and convex radius for the linear combination of the differential operator <mml:math> <mml:mrow> <mml:mi>z</mml:mi> <mml:mi>f</mml:mi> <mml:mo>'</mml:mo> </mml:mrow> </mml:math> and analytic functions <mml:math> <mml:mrow> <mml:mi>f</mml:mi> </mml:mrow> </mml:math> in <mml:math> <mml:mrow> <mml:mi mathvariant="bold">D</mml:mi> </mml:mrow> </mml:math>. Our results imply the related result obtained by Gavrilov [Mat. Zametki, 7, 295–298 (1970)].

Exchange Formulae for the Stieltjes–Poisson Transform over Weighted Lebesgue Spaces

2024-10-30

H. M. Srivastava , E. R. Negrín , Jeetendrasingh Maan

This paper aims to develop exchange formulae for the Stieltjes–Poisson transform by using Mellin-type convolutions in the context of weighted Lebesgue spaces. A key result is the introduction of bilinear … This paper aims to develop exchange formulae for the Stieltjes–Poisson transform by using Mellin-type convolutions in the context of weighted Lebesgue spaces. A key result is the introduction of bilinear and continuous Mellin-type convolutions, expanding the scope of the analysis to include the space of weighted L1 functions and the space of continuous functions vanishing at infinity.

Numerical simulations of Rosenau–Burgers equations via Crank–Nicolson spectral Pell matrix algorithm

2024-10-15

Mohammad Izadi , H. M. Srivastava , Kamal Mamehrashi

New Criteria for Functions in a Class of Meromorphically Strongly Starlike Functions

2024-10-01

Nak Eun Cho , Inhwa Kim , H. M. Srivastava

Abstract The authors propose to investigate some new criteria for a certain class of meromorphically strongly starlike functions in the punctured open unit disk. Some intriguing applications that arise as … Abstract The authors propose to investigate some new criteria for a certain class of meromorphically strongly starlike functions in the punctured open unit disk. Some intriguing applications that arise as special cases of the main results, which are presented in this study, are also considered.

A highly accurate and efficient Genocchi‐based spectral technique applied to singular fractional order boundary value problems

2024-08-06

Mohammad Izadi , Khursheed J‎. ‎Ansari , H. M. Srivastava

This article focuses on an efficient and highly accurate approximate solver for a class of generalized singular boundary value problems (SBVPs) having nonlinearity and with two‐term fractional derivatives. The involved … This article focuses on an efficient and highly accurate approximate solver for a class of generalized singular boundary value problems (SBVPs) having nonlinearity and with two‐term fractional derivatives. The involved fractional derivative operators are given in the form of Liouville–Caputo. The developed algorithm for solving the generalized SBVPs consists of two main stages. The first stage is devoted to an iterative quasilinearization method (QLM) to conquer the (strong) nonlinearity of the governing SBVPs. Secondly, we employ the generalized Genocchi polynomials (GGPs) to treat the resulting sequence of linearized SBVPs numerically. An upper error estimate for the Genocchi series solution in the norm is obtained via a rigorous error analysis. The main benefit of the presented QLM‐GGPs procedure is that the required number of iteration in the first stage is within a few steps, and an accurate polynomial solution is obtained through computer implementations in the second stage. Three widely applicable test cases are investigated to observe the efficacy as well as the high‐order accuracy of the QLM‐GGPs algorithm. The comparable accuracy and robustness of the presented algorithm are validated by doing comparisons with the results of some well‐established available computational methods. It is apparently shown that the QLM‐GGPs algorithm provides a promising tool to solve strongly nonlinear SBVPs with two‐term fractional derivatives accurately and efficiently.

Quantum (or q$$ q $$‐) operator equations and associated partial differential equations for bivariate Laguerre polynomials with applications to the q$$ q $$‐Hille‐Hardy type formulas

2024-08-04

Jian Cao , H. M. Srivastava , Yue Zhang

Based on the extensive application of the ‐series and ‐polynomials including (for example) the ‐Laguerre polynomials in several fields of the mathematical and physical sciences, we attach great importance to … Based on the extensive application of the ‐series and ‐polynomials including (for example) the ‐Laguerre polynomials in several fields of the mathematical and physical sciences, we attach great importance to the equations and related application issues involving the ‐Laguerre polynomials. The mission of this paper is to find the general ‐operational equation together with the expansion issue of the bivariate ‐Laguerre polynomials from the perspective of ‐partial differential equations. We also give some applications including some ‐Hille‐Hardy type formulas. In addition, we present the Rogers‐type formulas and the ‐type generating functions for the bivariate ‐Laguerre polynomials by the technique based upon ‐operational equations. Moreover, we derive a new generalized Andrews‐Askey integral and a new transformation identity involving the bivariate ‐Laguerre polynomials by applying ‐operational equations.

Approximation by a new Stancu variant of generalized <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si2.svg" display="inline" id="d1e1054"><mml:mrow><mml:mo>(</mml:mo><mml:mi>λ</mml:mi><mml:mo>,</mml:mo><mml:mi>μ</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math>-Bernstein operators

2024-07-17

Qing‐Bo Cai , Reşat Aslan , Faruk Özger +1 more

Analytical and approximate monotone solutions of the mixed order fractional nabla operators subject to bounded conditions

2024-07-05

Pshtiwan Othman Mohammed , H. M. Srivastava , Dumitru Băleanu +3 more

In this study, the sequential operator of mixed order is analysed on the domain (μ2,μ1)∈(0,1)×(0,1) with 1<μ2+μ1<2. Then, the positivity of the nabla operator is obtained analytically on a finite … In this study, the sequential operator of mixed order is analysed on the domain (μ2,μ1)∈(0,1)×(0,1) with 1<μ2+μ1<2. Then, the positivity of the nabla operator is obtained analytically on a finite time scale under some conditions. As a consequence, our analytical results are introduced on a set, named Em,ζ, on which the monotonicity analysis is obtained. Due to the complicatedness of the set Em,ζ several numerical simulations so are applied to estimate the structure of this set and they are provided by means of heat maps.

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Upper bound for the second and third Hankel determinants of analytic functions associated with the error function and q-convolution combination

2024-07-04

H. M. Srivastava , Daniel Breaz , Alhanouf Alburaikan +1 more

Abstract Recently, El-Deeb and Cotîrlă (Mathematics 11:11234834, 2023) used the error function together with a q -convolution to introduce a new operator. By means of this operator the following class … Abstract Recently, El-Deeb and Cotîrlă (Mathematics 11:11234834, 2023) used the error function together with a q -convolution to introduce a new operator. By means of this operator the following class $\mathcal{R}_{\alpha ,\Upsilon}^{\lambda ,q}(\delta ,\eta )$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>R</mml:mi> <mml:mrow> <mml:mi>α</mml:mi> <mml:mo>,</mml:mo> <mml:mi>ϒ</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>λ</mml:mi> <mml:mo>,</mml:mo> <mml:mi>q</mml:mi> </mml:mrow> </mml:msubsup> <mml:mo>(</mml:mo> <mml:mi>δ</mml:mi> <mml:mo>,</mml:mo> <mml:mi>η</mml:mi> <mml:mo>)</mml:mo> </mml:math> of analytic functions was studied: $$\begin{aligned} &\mathcal{R}_{\alpha ,\Upsilon }^{\lambda ,q}(\delta ,\eta ) \\ &\quad := \biggl\{ \mathcal{ F}: {\Re} \biggl( (1-\delta +2\eta ) \frac{\mathcal{H}_{\Upsilon }^{\lambda ,q}\mathcal{F}(\zeta )}{\zeta}+(\delta -2\eta ) \bigl(\mathcal{H} _{\Upsilon}^{\lambda ,q}\mathcal{F}(\zeta ) \bigr) ^{{ \prime}}+\eta \zeta \bigl( \mathcal{H}_{\Upsilon}^{\lambda ,q} \mathcal{F}( \zeta ) \bigr) ^{{{\prime \prime}}} \biggr) \biggr\} \\ &\quad >\alpha \quad (0\leqq \alpha < 1). \end{aligned}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mtable> <mml:mtr> <mml:mtd/> <mml:mtd> <mml:msubsup> <mml:mi>R</mml:mi> <mml:mrow> <mml:mi>α</mml:mi> <mml:mo>,</mml:mo> <mml:mi>ϒ</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>λ</mml:mi> <mml:mo>,</mml:mo> <mml:mi>q</mml:mi> </mml:mrow> </mml:msubsup> <mml:mo>(</mml:mo> <mml:mi>δ</mml:mi> <mml:mo>,</mml:mo> <mml:mi>η</mml:mi> <mml:mo>)</mml:mo> </mml:mtd> </mml:mtr> <mml:mtr> <mml:mtd/> <mml:mtd> <mml:mspace/> <mml:mo>:</mml:mo> <mml:mo>=</mml:mo> <mml:mrow> <mml:mo>{</mml:mo> <mml:mi>F</mml:mi> <mml:mo>:</mml:mo> <mml:mi>ℜ</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mo>(</mml:mo> <mml:mn>1</mml:mn> <mml:mo>−</mml:mo> <mml:mi>δ</mml:mi> <mml:mo>+</mml:mo> <mml:mn>2</mml:mn> <mml:mi>η</mml:mi> <mml:mo>)</mml:mo> <mml:mfrac> <mml:mrow> <mml:msubsup> <mml:mi>H</mml:mi> <mml:mi>ϒ</mml:mi> <mml:mrow> <mml:mi>λ</mml:mi> <mml:mo>,</mml:mo> <mml:mi>q</mml:mi> </mml:mrow> </mml:msubsup> <mml:mi>F</mml:mi> <mml:mo>(</mml:mo> <mml:mi>ζ</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mi>ζ</mml:mi> </mml:mfrac> <mml:mo>+</mml:mo> <mml:mo>(</mml:mo> <mml:mi>δ</mml:mi> <mml:mo>−</mml:mo> <mml:mn>2</mml:mn> <mml:mi>η</mml:mi> <mml:mo>)</mml:mo> <mml:msup> <mml:mrow> <mml:mo>(</mml:mo> <mml:msubsup> <mml:mi>H</mml:mi> <mml:mi>ϒ</mml:mi> <mml:mrow> <mml:mi>λ</mml:mi> <mml:mo>,</mml:mo> <mml:mi>q</mml:mi> </mml:mrow> </mml:msubsup> <mml:mi>F</mml:mi> <mml:mo>(</mml:mo> <mml:mi>ζ</mml:mi> <mml:mo>)</mml:mo> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>′</mml:mo> </mml:msup> <mml:mo>+</mml:mo> <mml:mi>η</mml:mi> <mml:mi>ζ</mml:mi> <mml:msup> <mml:mrow> <mml:mo>(</mml:mo> <mml:msubsup> <mml:mi>H</mml:mi> <mml:mi>ϒ</mml:mi> <mml:mrow> <mml:mi>λ</mml:mi> <mml:mo>,</mml:mo> <mml:mi>q</mml:mi> </mml:mrow> </mml:msubsup> <mml:mi>F</mml:mi> <mml:mo>(</mml:mo> <mml:mi>ζ</mml:mi> <mml:mo>)</mml:mo> <mml:mo>)</mml:mo> </mml:mrow> <mml:mrow> <mml:mo>″</mml:mo> </mml:mrow> </mml:msup> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>}</mml:mo> </mml:mrow> </mml:mtd> </mml:mtr> <mml:mtr> <mml:mtd/> <mml:mtd> <mml:mspace/> <mml:mo>></mml:mo> <mml:mi>α</mml:mi> <mml:mspace/> <mml:mo>(</mml:mo> <mml:mn>0</mml:mn> <mml:mo>≦</mml:mo> <mml:mi>α</mml:mi> <mml:mo><</mml:mo> <mml:mn>1</mml:mn> <mml:mo>)</mml:mo> <mml:mo>.</mml:mo> </mml:mtd> </mml:mtr> </mml:mtable> </mml:math> For these general analytic functions $\mathcal{F}\in \mathcal{R}_{\beta ,\Upsilon}^{\lambda ,q}(\delta , \eta )$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>F</mml:mi> <mml:mo>∈</mml:mo> <mml:msubsup> <mml:mi>R</mml:mi> <mml:mrow> <mml:mi>β</mml:mi> <mml:mo>,</mml:mo> <mml:mi>ϒ</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>λ</mml:mi> <mml:mo>,</mml:mo> <mml:mi>q</mml:mi> </mml:mrow> </mml:msubsup> <mml:mo>(</mml:mo> <mml:mi>δ</mml:mi> <mml:mo>,</mml:mo> <mml:mi>η</mml:mi> <mml:mo>)</mml:mo> </mml:math> , we give upper bounds for the Fekete–Szegö functional and for the second and third Hankel determinants.

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Fourth order Hankel determinants for certain subclasses of modified sigmoid-activated analytic functions involving the trigonometric sine function

2024-06-14

H. M. Srivastava , Nazar Khan , Muhtarr A. Bah +3 more

Abstract The aim of this paper is to introduce two new subclasses $\mathcal{R}_{\sin }^{m}(\Im )$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>R</mml:mi> <mml:mo>sin</mml:mo> <mml:mi>m</mml:mi> </mml:msubsup> <mml:mo>(</mml:mo> <mml:mi>ℑ</mml:mi> <mml:mo>)</mml:mo> </mml:math> and $\mathcal{R}_{\sin }(\Im )$ … Abstract The aim of this paper is to introduce two new subclasses $\mathcal{R}_{\sin }^{m}(\Im )$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>R</mml:mi> <mml:mo>sin</mml:mo> <mml:mi>m</mml:mi> </mml:msubsup> <mml:mo>(</mml:mo> <mml:mi>ℑ</mml:mi> <mml:mo>)</mml:mo> </mml:math> and $\mathcal{R}_{\sin }(\Im )$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>R</mml:mi> <mml:mo>sin</mml:mo> </mml:msub> <mml:mo>(</mml:mo> <mml:mi>ℑ</mml:mi> <mml:mo>)</mml:mo> </mml:math> of analytic functions by making use of subordination involving the sine function and the modified sigmoid activation function $\Im (v)=\frac{2}{1+e^{-v}}$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ℑ</mml:mi> <mml:mo>(</mml:mo> <mml:mi>v</mml:mi> <mml:mo>)</mml:mo> <mml:mo>=</mml:mo> <mml:mfrac> <mml:mn>2</mml:mn> <mml:mrow> <mml:mn>1</mml:mn> <mml:mo>+</mml:mo> <mml:msup> <mml:mi>e</mml:mi> <mml:mrow> <mml:mo>−</mml:mo> <mml:mi>v</mml:mi> </mml:mrow> </mml:msup> </mml:mrow> </mml:mfrac> </mml:math> , $v\geq 0$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>v</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>0</mml:mn> </mml:math> in the open unit disc E . Our purpose is to obtain some initial coefficients, Fekete–Szego problems, and upper bounds for the third- and fourth-order Hankel determinants for the functions belonging to these two classes. All the bounds that we will find here are sharp. We also highlight some known consequences of our main results.

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A New Representation Formula for the Bernoulli Numbers

2024-06-01

Aleksandar Petojević , H. M. Srivastava , Dragan Rastovac

Certain New Applications of Symmetric q-Calculus for New Subclasses of Multivalent Functions Associated with the Cardioid Domain

2024-05-29

H. M. Srivastava , Daniel Breaz , Shahid Khan +1 more

In this work, we study some new applications of symmetric quantum calculus in the field of Geometric Function Theory. We use the cardioid domain and the symmetric quantum difference operator … In this work, we study some new applications of symmetric quantum calculus in the field of Geometric Function Theory. We use the cardioid domain and the symmetric quantum difference operator to generate new classes of multivalent q-starlike and q-convex functions. We examine a wide range of interesting properties for functions that can be classified into these newly defined classes, such as estimates for the bounds for the first two coefficients, Fekete–Szego-type functional and coefficient inequalities. All the results found in this research are sharp. A number of well-known corollaries are additionally taken into consideration to show how the findings of this research relate to those of earlier studies.

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On existence of certain delta fractional difference models

2024-04-25

Pshtiwan Othman Mohammed , H. M. Srivastava , Rebwar Salih Muhammad +3 more

The discretization of initial and boundary value problems and their existence behaviors are of great significance in various fields. This paper explores the existence of a class of self-adjoint delta … The discretization of initial and boundary value problems and their existence behaviors are of great significance in various fields. This paper explores the existence of a class of self-adjoint delta fractional difference equations. The study begins by demonstrating the uniqueness of an initial value problem of delta Riemann–Liouville fractional operator type. Based on this result, the uniqueness of the self-adjoint equation will be examined and determined. Next, we define the Cauchy function based on the delta Riemann–Liouville fractional differences. Accordingly, the solution of the self-adjoint equation will be investigated according to the delta Cauchy function. Furthermore, the research investigates the uniqueness of the self-adjoint equation including the component of Green's functions of and examines how this equation has only a trivial solution. To validate the theoretical analysis, specific examples are conducted to support and verify our results

A class of the Newtonian HK-Sobolev spaces on metric measure spaces

2024-04-06

H. M. Srivastava , Parthapratim Saha , Bipan Hazarika

Some m-fold symmetric bi-univalent function classes and their associated Taylor-Maclaurin coefficient bounds

2024-04-02

H. M. Srivastava , Pishtiwan Sabir , Sevtap Sümer Eker +4 more

Abstract The Ruscheweyh derivative operator is used in this paper to introduce and investigate interesting general subclasses of the function class $\Sigma_{m}$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>Σ</mml:mi> <mml:mi>m</mml:mi> </mml:msub> </mml:math> of … Abstract The Ruscheweyh derivative operator is used in this paper to introduce and investigate interesting general subclasses of the function class $\Sigma_{m}$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>Σ</mml:mi> <mml:mi>m</mml:mi> </mml:msub> </mml:math> of m -fold symmetric bi-univalent analytic functions. Estimates of the initial Taylor-Maclaurin coefficients $\vert a_{m+1} \vert $ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mo>|</mml:mo> <mml:msub> <mml:mi>a</mml:mi> <mml:mrow> <mml:mi>m</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msub> <mml:mo>|</mml:mo> </mml:math> and $\vert a_{2 m+1} \vert $ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mo>|</mml:mo> <mml:msub> <mml:mi>a</mml:mi> <mml:mrow> <mml:mn>2</mml:mn> <mml:mi>m</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msub> <mml:mo>|</mml:mo> </mml:math> are obtained for functions of the subclasses introduced in this study, and the consequences of the results are discussed. Additionally, the Fekete-Szegö inequalities for these classes are investigated. The results presented could generalize and improve some recent and earlier works. In some cases, our estimates are better than the existing coefficient bounds. Furthermore, within the engineering domain, the utilization of the Ruscheweyh derivative operator can encompass a broad spectrum of engineering applications, including the robotic manipulation control, optimizing optical systems, antenna array signal processing, image compression, and control system filter design. It emphasizes the potential for innovative solutions that can significantly enhance the reliability and effectiveness of engineering applications.

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Coefficient bounds and second Hankel determinant for a subclass of symmetric bi-starlike functions involving Euler polynomials

2024-03-01

H. M. Srivastava , Timilehin Gideon Shaba , Musthafa Ibrahim +2 more

Well-posedness of a nonlinear Hilfer fractional derivative model for the Antarctic circumpolar current

2024-02-26

H. M. Srivastava , Kanika Dhawan , Ramesh Kumar Vats +1 more

Bi-Concave Functions Connected with the Combination of the Binomial Series and the Confluent Hypergeometric Function

2024-02-13

H. M. Srivastava , Sheza M. El-Deeb , Daniel Breaz +2 more

In this article, we first define and then propose to systematically study some new subclasses of the class of analytic and bi-concave functions in the open unit disk. For this … In this article, we first define and then propose to systematically study some new subclasses of the class of analytic and bi-concave functions in the open unit disk. For this purpose, we make use of a combination of the binomial series and the confluent hypergeometric function. Among some other properties and results, we derive the estimates on the initial Taylor-Maclaurin coefficients |a2| and |a3| for functions in these analytic and bi-concave function classes, which are introduced in this paper. We also derive a number of corollaries and consequences of our main results in this paper.

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Faber polynomial coefficient inequalities for bi-Bazilevič functions associated with the Fibonacci-number series and the square-root functions

2024-01-31

H. M. Srivastava , Shahid Khan , Sarfraz Nawaz Malik +3 more

Abstract Two new subclasses of the class of bi-Bazilevič functions, which are related to the Fibonacci-number series and the square-root functions, are introduced and studied in this article. Under a … Abstract Two new subclasses of the class of bi-Bazilevič functions, which are related to the Fibonacci-number series and the square-root functions, are introduced and studied in this article. Under a special choice of the parameter involved, these two classes of Bazilevič functions reduce to two new subclasses of star-like biunivalent functions related with the Fibonacci-number series and the square-root functions. Using the Faber polynomial expansion (FPE) technique, we find the general coefficient bounds for the functions belonging to each of these classes. We also find bounds for the initial coefficients for bi-Bazilevič functions and demonstrate how unexpectedly these initial coefficients behave in relation to the square-root functions and the Fibonacci-number series.

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Dynamics for a diffusive prey–predator model with advection and free boundaries

2024-01-23

Y. B. Zhao , H. M. Srivastava

This article deals with a free boundary problem of the Lotka–Volterra type prey–predator model with advection in one space dimension. The model considered here may be applied to describe the … This article deals with a free boundary problem of the Lotka–Volterra type prey–predator model with advection in one space dimension. The model considered here may be applied to describe the expanding of an invasive or new predator species adopting a combination of random movement and advection upward or downward along the resource gradient, with the free boundaries representing expanding fronts of predator species. The main purpose of this article is to understand the influence of the advection environment on the dynamics of the model. We provide sufficient conditions for spreading and vanishing of the predator species, and we find a sharp threshold between the spreading and vanishing concerning this model. Moreover, for the case of successful spreading for the predator, we give estimates of asymptotic spreading speeds and nonlocal long‐time behavior of the prey and the predator.

Some summation theorems and transformations for hypergeometric functions of Kampé de Fériet and Srivastava

2024-01-01

H. M. Srivastava , Bhawna Gupta , Mohammad Idris Qureshi +1 more

Sharp bounds on Hankel determinants of bounded turning functions involving the hyperbolic tangent function

2024-01-01

Zhi-Gang Wang , H. M. Srivastava , Muhammad Arif +2 more

In this paper, we determine the sharp upper bounds of Hankel determinants for logarithmic and inverse functions of bounded turning functions associated with the hyperbolic tangent function. In this paper, we determine the sharp upper bounds of Hankel determinants for logarithmic and inverse functions of bounded turning functions associated with the hyperbolic tangent function.

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Second Hankel determinant of the logarithmic coefficients for a subclass of univalent functions

2024-01-01

H. M. Srivastava , Sevtap Sümer Eker , Bilal Şeker +1 more

In the present paper, we give the bounds for the second Hankel determinant of the logarithmic coefficients of a certain subclass of normalized univalent functions, which we have introduced here. … In the present paper, we give the bounds for the second Hankel determinant of the logarithmic coefficients of a certain subclass of normalized univalent functions, which we have introduced here. Relevant connections of the results, which we have presented here, with those available in the existing literature are also described briefly.

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Numerical Computations of Time-Dependent Auto-Catalytic Glycolysis Chemical Reaction-Diffusion System

2024-01-01

Mohammad Izadi , Hijaz Ahmad , H. M. Srivastava

A combined semi-discretized spectral matrix collocation algorithm based on (new) family of Krawtchouk polynomials is proposed to solve the time-dependent nonlinear auto-catalytic glycolysis reaction-diffusion system arising in mathematical chemistry. The … A combined semi-discretized spectral matrix collocation algorithm based on (new) family of Krawtchouk polynomials is proposed to solve the time-dependent nonlinear auto-catalytic glycolysis reaction-diffusion system arising in mathematical chemistry. The first stage of the numerical algorithm is devoted to the Taylor series time advancement procedure yielding to a (linear and steady) system of ODEs. In the second stage and in each time frame, a matrix collocation technique based on the Krawtchouk polynomials is utilized to the resulting system of ODEs in an iterative manner. The results of the performed numerical experiments with Neumann boundary conditions are given to show the utility and applicability of the combined Taylor-Krawtchouk spectral collocation algorithm. The positive property of the glycolysis chemical model is sustained by the proposed algorithm and verified through comparisons with existing numerical methods in the literature. The combined technique is simple and flexible enough to easily produce the approximate solutions of diverse physical and applied models in engineering and science.

A study of the q-analogue of the paranormed Cesàro sequence spaces

2024-01-01

H. M. Srivastava , Taja Yaying , Bipan Hazarika

In this article, we introduce and investigate the q-Ces?ro matrix C(q) = (cq uv) with q ? (0, 1) for which we have cq uv ={qv/[u + 1]q (0 ? … In this article, we introduce and investigate the q-Ces?ro matrix C(q) = (cq uv) with q ? (0, 1) for which we have cq uv ={qv/[u + 1]q (0 ? v ? u) 0 (v > u), where the q-number [?]q is given, as usual in the q-theory, by [?]q := {1 ? q?/1 ? q (? ? C) ?n?1 k=0 qk = 1 + q + q2 + ....+ qn?1 (? = n ? N), C and N being the sets of complex numbers and positive integers, respectively. The q-Ces?ro matrix C(q) is a q-analogue of the Ces?ro matrix C1. We study the sequence spaces Xq(p), Xq 0(p), Xq c (p) and Xq ?(p), which are obtained by the domain of the matrix C(q) in the Maddox spaces ?(p), c0(p), c(p) and ??(p), respectively. We derive the Schauder basis and the alpha-, beta- and gamma-duals of these newly-defined spaces. Moreover, we state and prove several theorems characterizing matrix transformation from the spaces Xq(p),Xq 0(p),Xq c(p) and Xq ?(p) to anyone of the spaces c0, c or ??.

Some definite integrals involving elliptic integrals in association with hypergeometric functions

2024-01-01

H. M. Srivastava , Salah Uddin , M.P. Chaudhary

In this paper, the authors make use of the Gamma function as well as the hypergeometric and the generalized hypergeometric functions in order to investigate and develop several definite integrals … In this paper, the authors make use of the Gamma function as well as the hypergeometric and the generalized hypergeometric functions in order to investigate and develop several definite integrals involving the elliptic integrals of the first and the second kind. The numerical approximation of these definite integrals and the corresponding hypergeometric functions are also presented. The results derived in this article are believed to be new and extend and unify those that are available in the scientific literature.

A class of generalized Mittag-Leffler-type functions associated with the Lauricella functions of three variable

2024-01-01

H. M. Srivastava , Hilola Yuldashova

In this article, we aim to study the Mittag-Leffler-type functions ~F(3)A , ~F(3) B , ~F(3) C and ~F(3) D , which correspond, respectively, to the familiar Lauricella hypergeometric functions … In this article, we aim to study the Mittag-Leffler-type functions ~F(3)A , ~F(3) B , ~F(3) C and ~F(3) D , which correspond, respectively, to the familiar Lauricella hypergeometric functions F(3) A , F(3) B , F(3) C and F(3) D of three variables. Amongthe various properties and characteristics of these three-variable Mittag-Leffler-type functions, which we investigate in this article, include their relationships with other extensions and generalizations of the classical Mittag-Leffler functions, their three-dimensional convergence regions, the systems of partial differential equations which are are satisfied by them, their Euler-type integral representations, their one- as well as three-dimensional Laplace transforms, and their connections with the Riemann-Liouville operators of fractional calculus.

Some sharp bounds of the third-order Hankel determinant for the inverses of the Ozaki type close-to-convex functions

2023-12-22

H. M. Srivastava , Biswajit Rath , K. Sanjay Kumar +1 more

Some characterizations of the symmetric q -Dunkl-semiclassical orthogonal q -polynomials

2023-12-12

H. M. Srivastava , Jihad Souissi , Mohamed Khalfallah

In this paper, we introduce a new class of symmetric orthogonal polynomials, which generalizes the q-Dunkl-classical polynomials and also provides new characterizations of symmetric q-semiclassical orthogonal polynomials. Both the notion … In this paper, we introduce a new class of symmetric orthogonal polynomials, which generalizes the q-Dunkl-classical polynomials and also provides new characterizations of symmetric q-semiclassical orthogonal polynomials. Both the notion of a class and a criterion for determining the class will be given. As an application, several examples of q-Dunkl-semiclassical forms of the class s≦2 are highlighted.

Common Coauthors

Coauthor	Papers Together
Junesang Choi	47
Pshtiwan Othman Mohammed	45
Shigeyoshi Owa	44
Dumitru Băleanu	38
Shy‐Der Lin	33
Xiao‐Jun Yang	30
Nazar Khan	30
Bilal Khan	25
Qazi Zahoor Ahmad	23
Nak Eun Cho	22
Mohammad Izadi	21
R. K. Raina	20
Muhammad Arif	19
Y. S. Hamed	16
Yong Chan Kim	16
Maslina Darus	16
Serkan Aracı	16
Firdous A. Shah	16
Khaled M. Saad	15
M. K. Aouf	14
Bidu Bhusan Jena	14
Chao-Ping Chen	13
Muhammad Tahir	13
Rekha Panda	13
Shih-Tong Tu	13
Mohammad Masjed‐Jamei	12
Yılmaz Şimşek	12
Sébastien Gaboury	12
Kung-Yu Chen	11
Megumi Saigo	11
Eman Al-Sarairah	11
Susanta Kumar Paikray	11
Harendra Singh	11
Qinghua Xu	11
Shuoh-Jung Liu	11
Nejmeddine Chorfi	10
Pin-Yu Wang	10
Tibor K. Pogány	10
G. Dattoli	10
Shahid Mahmood	10
J. P. Singhal	9
Sama Arjika	9
Bipan Hazarika	9
Djurdje Cvijović	9
S. H. Ong	9
G. Murugusundaramoorthy	9
Sarfraz Nawaz Malik	9
R. G. Buschman	9
Ming‐Po Chen	9
Bai‐Ni Guo	9

Commonly Cited References

Series Associated with the Zeta and Related Functions

2001-01-01

H. M. Srivastava , Junesang Choi

Current Topics in Analytic Function Theory

1992-12-01

H. M. Srivastava , Shigeyoshi Owa

Univalent logharmonic extensions onto the unit disk or onto an annulus, Z. Abdulhadi and W. Hengartner hypergeometric functions and elliptic integrals, G.D. Anderson et al a certain class of caratheodory … Univalent logharmonic extensions onto the unit disk or onto an annulus, Z. Abdulhadi and W. Hengartner hypergeometric functions and elliptic integrals, G.D. Anderson et al a certain class of caratheodory functions defined by conditions on the circle, J. Fuka and Z.J. Jakubowski recent advances in the theory of zero sets of the Bergman spaces, E.A. LeBlanc a coefficient functional for meromorphic univalent functions, L. Liu spherical linear invariance and uniform local spherical convexity, W. Ma and D. Minda a special differential subordination and its application to univalency conditions, S.S. Miller and P.T. Mocanu on the Bernardi integral functions, S. Owa analytic solutions of a class of Briot-Bouquet differential equations, S. Owa and H.M. Srivastava a certain class of generalized hypergeometric functions associated with the Hardy space of analytic functions, H.M. Srivastava on the coefficients of the univalent functions of the Nevanlinna classes N1 and N2, P.G. Todorov.

Theory and Applications of Fractional Differential Equations

2006-01-01

Anatoly A. Kilbas , H. M. Srivastava , Juan J. Trujillo

Operators of Basic (or q-) Calculus and Fractional q-Calculus and Their Applications in Geometric Function Theory of Complex Analysis

2020-01-27

H. M. Srivastava

Zeta and q-Zeta Functions and Associated Series and Integrals

2012-01-01

H. M. Srivastava , June-Sang Choi

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Generalized Hypergeometric Functions

1966-10-01

Y. L. L. , L. J. Slater

Generalized Hypergeometric Functions

1967-03-01

L. J. Slater , Werner C. Rheinboldt

1. The Gauss Function 2. The Generalized Gauss Function 3. Basic Hypergeometric Functions 4. Hypergeometric Integrals 5. Basic Hypergeometric Integrals 6. Bilateral Series 7. Basic Bilateral Series 8. Appell Series … 1. The Gauss Function 2. The Generalized Gauss Function 3. Basic Hypergeometric Functions 4. Hypergeometric Integrals 5. Basic Hypergeometric Integrals 6. Bilateral Series 7. Basic Bilateral Series 8. Appell Series 9. Basic Appell Series.

Some formulas for the Bernoulli and Euler polynomials at rational arguments

2000-07-01

H. M. Srivastava

In a recent paper [5], the classical Bernoulli and Euler polynomials were expressed as finite sums involving the Hurwitz zeta function. The object of this sequel is first to give … In a recent paper [5], the classical Bernoulli and Euler polynomials were expressed as finite sums involving the Hurwitz zeta function. The object of this sequel is first to give several remarkably shorter proofs of each of these summation formulas. Various generalizations and analogues, which are relevant to the present investigation, are also considered.

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Tables of Integral Transforms

2005-01-01

Bateman Manuscript , H. Bateman , Arthur Erdélyi

AbstractIn this chapter we provide a set of short tables of integral transforms of the functions that are either cited in the text or in most common use in mathematical, … AbstractIn this chapter we provide a set of short tables of integral transforms of the functions that are either cited in the text or in most common use in mathematical, physical, and engineering applications. For exhaustive lists of integral transforms, the reader is referred to Erdélyi et al. (1954), Campbell and Foster (1948), Ditkin and Prudnikov (1965), Doetsch (1970), Marichev (1983), Debnath (1995), and Oberhettinger (1972).KeywordsDifferential EquationFourier TransformPartial Differential EquationMathematical MethodEngineering ApplicationThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Orthogonal Polynomials

1971-01-01

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An Introductory Overview of Fractional-Calculus Operators Based Upon the Fox-Wright and Related Higher Transcendental Functions

2021-09-30

H. M. Srivastava

This survey-cum-expository review article is motivated essentially by the widespread usages of the operators of fractional calculus (that is, fractional-order integrals and fractional-order derivatives) in the modeling and analysis of … This survey-cum-expository review article is motivated essentially by the widespread usages of the operators of fractional calculus (that is, fractional-order integrals and fractional-order derivatives) in the modeling and analysis of a remarkably large variety of applied scientific and real-world problems in mathematical, physical, biological, engineering and statistical sciences, and in other scientific disciplines. Here, in this article, we present a brief introductory overview of the theory and applications of the fractional-calculus operators which are based upon the general Fox-Wright function and its such specialized forms as (for example) the widely- and extensively investigated and potentially useful Mittag-Leffter type functions.This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium provided the original work is properly cited.

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The H-functions of one and two variables, with applications

1982-01-01

H. M. Srivastava , K. C. Gupta , Sukriti Goyal

Univalent and Starlike Generalized Hypergeometric Functions

1987-10-01

Shigeyoshi Owa , H. M. Srivastava

A single-valued function f(z) is said to be univalent in a domain if it never takes on the same value twice, that is, if f ( z 1 ) = … A single-valued function f(z) is said to be univalent in a domain if it never takes on the same value twice, that is, if f ( z 1 ) = f ( z 2 ) for implies that z 1 = z 2 . A set is said to be starlike with respect to the line segment joining w 0 to every other point lies entirely in . If a function f(z) maps onto a domain that is starlike with respect to w 0 , then f(z) is said to be starlike with respect to w 0 . In particular, if w 0 is the origin, then we say that f(z) is a starlike function. Further, a set is said to be convex if the line segment joining any two points of lies entirely in . If a function f(z) maps onto a convex domain , then we say that f(z) is a convex function in .

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Fractional Differential Equations - An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of their Solution and some of their Applications

1999-01-01

Fractional-Order Derivatives and Integrals: Introductory Overview and Recent Developments

2020-03-31

H. M. Srivastava

Convex and starlike univalent functions

1969-01-01

S. D. Bernardi

Introduction. Let (S) denote the class of functions/(z) = z + 2? anzn which are regular and univalent in \z\ < 1 and which map \z\ < 1 onto domains … Introduction. Let (S) denote the class of functions/(z) = z + 2? anzn which are regular and univalent in \z\ < 1 and which map \z\ < 1 onto domains D(f).Let (C), (S*), and (K) represent the subclasses of (S) where D(f) are respectively, close-to-convex, starlike with respect to the origin, and convex.It follows that (K)<=(S*)<=(C)<=(S).We will simply say "starlike" when we mean starlike with respect to the origin, and the statement "f(z) is convex" will mean that the domain D(f) is convex.The abbreviations "i.o.i." and "n.s.c." have the usual meanings.Let (F) denote the class of functions p(z) which are regular and satisfy p(0)=l, Rep(z)>0 for \z\ < 1.The following results, which we will use repeatedly, are well known [1]:Let/(z)e(S).ThenIf h(z) = z+ -is regular in |z| < 1 and h'(z) e (P) then h(z) e (S).In a recent paper [4] R. J. Libera established that (1.1) iff(z) is a member of(C), (S*) or (K) then F(z) = (2/z)Jlf(t) dt is also a member of the same class, respectively.The purpose of this paper is to construct a many-parameter class of functions F(c¡;/) which includes the result (1.1) for special choice of parameters c¡ and, in addition, establish a subordinate relation between the image domains of certain members of F(ct;f) and the image domain of the parent function/(z).2. Definitions and lemmas.The notation of g(z)<f(z), ("g(z) is subordinate to/(z)"), will mean that every value taken by g(z) for \z\ < 1 is also taken by/(z).The convolution of two power series/=2fAnzn and g=2í° Bnzn is defined as the power series/* g=2i° AnBnzn.Definition 1.An infinite sequence {bn}x of complex numbers is called a subordinating factor sequence (s.f.s.) if whenever f(z) e (K) we have E(z)=f(z) * 2f ^nZn<f(z).This definition is due to H. S. Wilf [11] who gave n.s.c. for a sequence to be a s.f.s.It follows that the product sequence {bncn}x of two s.f.s.{¿>"}f and {cn}x is also a s.f.s.

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A generalization of starlike functions

1990-04-01

Mourad E. H. Ismail , E. P. Merkes , D. Styer

Abstract For every q. 0<q<1 we define a class of complex functions as the class of functions f, analytic on the open unit disc ℬ, f(0)=0, f′(0)=1 and |f(qz)| ⩽ … Abstract For every q. 0<q<1 we define a class of complex functions as the class of functions f, analytic on the open unit disc ℬ, f(0)=0, f′(0)=1 and |f(qz)| ⩽ |f(z)| on ℬ. This class is denoted by PSq . We study this class and explore the relationships between this class and other classes of analytic functions. We find the function F q that maximizes the coefficients of members of PS q. We also prove that a basic hypergeomelric function Ф(a,b;c;q,rz)ε PSq for certain values of r. AMS No.: 30C4533A70 †Research partially supported by a grant from the National Science Foundation. †Research partially supported by a grant from the National Science Foundation. Notes †Research partially supported by a grant from the National Science Foundation.

New criteria for univalent functions

1975-05-01

Stephan Ruscheweyh

The classes <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K Subscript n"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>K</mml:mi> <mml:mi>n</mml:mi> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">{K_n}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of functions <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="f … The classes <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K Subscript n"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>K</mml:mi> <mml:mi>n</mml:mi> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">{K_n}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of functions <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="f left-parenthesis z right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>f</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>z</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">f(z)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> regular in the unit disc <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="German upper U"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="fraktur">U</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathfrak {U}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> with <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="f left-parenthesis 0 right-parenthesis equals 0 comma f prime left-parenthesis 0 right-parenthesis equals 1"> <mml:semantics> <mml:mrow> <mml:mi>f</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mn>0</mml:mn> <mml:mo stretchy="false">)</mml:mo> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:msup> <mml:mi>f</mml:mi> <mml:mo>′</mml:mo> </mml:msup> <mml:mo stretchy="false">(</mml:mo> <mml:mn>0</mml:mn> <mml:mo stretchy="false">)</mml:mo> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">f(0) = 0,f’(0) = 1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> satisfying <disp-formula content-type="math/mathml"> \[ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper R e left-bracket left-parenthesis z Superscript n Baseline f right-parenthesis Superscript left-parenthesis n plus 1 right-parenthesis Baseline slash left-parenthesis z Superscript n minus 1 Baseline f right-parenthesis Superscript left-parenthesis n right-parenthesis Baseline right-bracket greater-than left-parenthesis n plus 1 right-parenthesis slash 2"> <mml:semantics> <mml:mrow> <mml:mi>Re</mml:mi> <mml:mo>⁡</mml:mo> <mml:mo stretchy="false">[</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">(</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>z</mml:mi> <mml:mi>n</mml:mi> </mml:msup> </mml:mrow> <mml:mi>f</mml:mi> <mml:msup> <mml:mo stretchy="false">)</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">(</mml:mo> <mml:mi>n</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:msup> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>/</mml:mo> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">(</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>z</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>n</mml:mi> <mml:mo>−</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> </mml:mrow> <mml:mi>f</mml:mi> <mml:msup> <mml:mo stretchy="false">)</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">(</mml:mo> <mml:mi>n</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:msup> </mml:mrow> <mml:mo stretchy="false">]</mml:mo> <mml:mo>></mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mi>n</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> <mml:mo stretchy="false">)</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>/</mml:mo> </mml:mrow> <mml:mn>2</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">\operatorname {Re} [{({z^n}f)^{(n + 1)}}/{({z^{n - 1}}f)^{(n)}}] > (n + 1)/2</mml:annotation> </mml:semantics> </mml:math> \] </disp-formula> in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="German upper U"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="fraktur">U</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathfrak {U}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> are considered and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K Subscript n plus 1 Baseline subset-of upper K Subscript n Baseline comma n equals 0 comma 1 comma midline-horizontal-ellipsis"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>K</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>n</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msub> </mml:mrow> <mml:mo>⊂</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>K</mml:mi> <mml:mi>n</mml:mi> </mml:msub> </mml:mrow> <mml:mo>,</mml:mo> <mml:mi>n</mml:mi> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mo>⋯</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">{K_{n + 1}} \subset {K_n},n = 0,1, \cdots</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, is proved. Since <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K 0"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>K</mml:mi> <mml:mn>0</mml:mn> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">{K_0}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is the class of functions starlike of order 1/2 all functions in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K Subscript n"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>K</mml:mi> <mml:mi>n</mml:mi> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">{K_n}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> are univalent. Some coefficient estimates are given and special elements of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K Subscript n"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>K</mml:mi> <mml:mi>n</mml:mi> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">{K_n}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> are determined.

Certain subclasses of analytic and bi-univalent functions

2010-05-28

H. M. Srivastava , Akshaya Mishra , P. Gochhayat

A Treatise on the Theory of Bessel Functions.

1923-09-01

Matthew Porter , G. N. Watson

1. Bessel functions before 1826 2. The Bessel coefficients 3. Bessel functions 4. Differential equations 5. Miscellaneous properties of Bessel functions 6. Integral representations of Bessel functions 7. Asymptotic expansions … 1. Bessel functions before 1826 2. The Bessel coefficients 3. Bessel functions 4. Differential equations 5. Miscellaneous properties of Bessel functions 6. Integral representations of Bessel functions 7. Asymptotic expansions of Bessel functions 8. Bessel functions of large order 9. Polynomials associated with Bessel functions 10. Functions associated with Bessel functions 11. Addition theorems 12. Definite integrals 13. Infinitive integrals 14. Multiple integrals 15. The zeros of Bessel functions 16. Neumann series and Lommel's functions of two variables 17. Kapteyn series 18. Series of Fourier-Bessel and Dini 19. Schlomlich series 20. The tabulation of Bessel functions Tables of Bessel functions Bibliography Indices.

Formulas and Theorems for the Special Functions of Mathematical Physics

1966-01-01

Wilhelm Magnus , Fritz Oberhettinger , Raj Pal Soni

Applications of Fractional Calculus in Physics

2000-01-01

R. Hilfer

An introduction to fractional calculus, P.L. Butzer & U. Westphal fractional time evolution, R. Hilfer fractional powers of infinitesimal generators of semigroups, U. Westphal fractional differences, derivatives and fractal time … An introduction to fractional calculus, P.L. Butzer & U. Westphal fractional time evolution, R. Hilfer fractional powers of infinitesimal generators of semigroups, U. Westphal fractional differences, derivatives and fractal time series, B.J. West and P. Grigolini fractional kinetics of Hamiltonian chaotic systems, G.M. Zaslavsky polymer science applications of path integration, integral equations, and fractional calculus, J.F. Douglas applications to problems in polymer physics and rheology, H. Schiessel et al applications of fractional calculus and regular variation in thermodynamics, R. Hilfer.

Starlike and Prestarlike Hypergeometric Functions

1984-07-01

B. C. Carlson , Dorothy Browne Shaffer

A linear operator is defined which acts on an analytic function in the open unit disk by forming its Hadamard product with an incomplete beta function. The operator is shown … A linear operator is defined which acts on an analytic function in the open unit disk by forming its Hadamard product with an incomplete beta function. The operator is shown to be convenient in discussing starlike, convex, and prestarlike functions. It is applied to the study of certain classes of hypergeometric functions which constitute dense subsets in the classes of starlike functions of order $\alpha $, convex functions of order $\alpha $, and prestarlike functions of order $\alpha $. Integral representations of the functions in these classes are obtained from the integral representation of the starlike functions of order $\alpha $.

Some expansion formulas for a class of generalized Hurwitz–Lerch Zeta functions

2006-09-25

Shy‐Der Lin , H. M. Srivastava , Pin-Yu Wang

Abstract By making use of fractional calculus, the authors present a systematic investigation of expansion and transformation formulas for several general families of the Hurwitz–Lerch Zeta functions. Relevant connections of … Abstract By making use of fractional calculus, the authors present a systematic investigation of expansion and transformation formulas for several general families of the Hurwitz–Lerch Zeta functions. Relevant connections of the results discussed here with those obtained in earlier works are also indicated precisely. Keywords: Fractional calculusExpansion formulasLerch's functional equation (or Lerch's transformation formula)Hurwitz–Lerch Zeta functionsLipschitz–Lerch Zeta functionsRiemann–Liouville fractional derivativeLeibniz ruleHurwitz (or generalized) Zeta functionRiemann Zeta functionsum-integral representationsBernoulli polynomials and Bernoulli numbers of higher orderFox–Wright generalized hypergeometric functionEulerian integral of the first kind Acknowledgements The present investigation was supported, in part, by the National Science Council of the Republic of China under Grant NSC 95-2115-M-006, the Faculty Research Program of Chung Yuan Christian University under Grant CYCU-94RD-RA001-10108, and the Natural Sciences and Engineering Research Council of Canada under Grant OGP0007353.

Some relationships between the generalized Apostol–Bernoulli polynomials and Hurwitz–Lerch Zeta functions

2006-09-25

Mridula Garg , Kumkum Jain , H. M. Srivastava

The main object of this paper is to further investigate the generalized Apostol–Bernoulli polynomials of higher order, which were introduced and studied recently by Luo and Srivastava [2005, Journal of … The main object of this paper is to further investigate the generalized Apostol–Bernoulli polynomials of higher order, which were introduced and studied recently by Luo and Srivastava [2005, Journal of Mathematical Analysis and Applications, 308, 290–302; 2006, Computers and Mathematics with Applications, 51, 631–642]. Here, we first derive an explicit representation of these generalized Apostol–Bernoulli polynomials of higher order in terms of a generalization of the Hurwitz–Lerch Zeta function and then proceed to establish a functional relationship between the generalized Apostol–Bernoulli polynomials of rational arguments and the Hurwitz (or generalized) Zeta function. Our results would provide extensions of those given earlier by (for example) Apostol [1951, Pacific Journal of Mathematics, 1, 161–167] and Srivastava [2000, Mathematical Proceedings of the Cambridge Philosophical Society, 129, 77–84].

Univalent functions with negative coefficients

1975-08-01

Herb Silverman

Coefficient, distortion, covering, and coefficient inequalities are determined for univalent functions with negative coefficients that are starlike of order <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="alpha"> <mml:semantics> <mml:mi>α</mml:mi> <mml:annotation encoding="application/x-tex">\alpha</mml:annotation> </mml:semantics> </mml:math> … Coefficient, distortion, covering, and coefficient inequalities are determined for univalent functions with negative coefficients that are starlike of order <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="alpha"> <mml:semantics> <mml:mi>α</mml:mi> <mml:annotation encoding="application/x-tex">\alpha</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and convex of order <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="alpha"> <mml:semantics> <mml:mi>α</mml:mi> <mml:annotation encoding="application/x-tex">\alpha</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. Extreme points for these classes are also determined.

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A Treatise on the Theory of Bessel Functions

1944-12-01

H. T. Davis , G. N. Watson

Fonctions hypergéométriques et hypersphériques : polynomes d'Hermite

1926-01-01

Paul Appell , J. Kampé de Fériet

fonction de plusieurs variables # fonction hypergeometrique # fonction hyperpherique # hyperfonction # polynome d'Hermite fonction de plusieurs variables # fonction hypergeometrique # fonction hyperpherique # hyperfonction # polynome d'Hermite

Classes of analytic functions associated with the generalized hypergeometric function

1999-08-01

Jacek Dziok , H. M. Srivastava

Some families of the Hurwitz–Lerch Zeta functions and associated fractional derivative and other integral representations

2003-12-12

Shy‐Der Lin , H. M. Srivastava

A. Erdelyi, W. Magnus, F. Oberhettinger and F. G. Tricomi ; Higher Transcendental Functions, Vols. I, II, III. McGraw-Hill, New York-Toronto-London, 1953, 1953, 1955. xxvi+302, xvii+396, xvii+292頁. 16×23.5cm. $6.50, $7.50, $6.50.

1955-07-05

英典橋本

Operational formulas connected with two generalizations of Hermite polynomials

1962-03-01

H. W. Gould , A. T. Hopper

Tables of integral transforms

1954-06-01

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Sulle funzioni ipergeometriche a piu variabili

1893-12-01

Giuseppe Lauricella

Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables

1965-02-01

D. B. Owen , Milton Abramowitz , Irene A. Stegun

Remarks on some relationships between the Bernoulli and Euler polynomials

2004-04-01

H. M. Srivastava , Ákos Pintér

Some General Classes of q-Starlike Functions Associated with the Janowski Functions

2019-02-23

H. M. Srivastava , Muhammad Tahir , Bilal Khan +2 more

By making use of the concept of basic (or q-) calculus, various families of q-extensions of starlike functions of order α in the open unit disk U were introduced and … By making use of the concept of basic (or q-) calculus, various families of q-extensions of starlike functions of order α in the open unit disk U were introduced and studied from many different viewpoints and perspectives. In this paper, we first investigate the relationship between various known classes of q-starlike functions that are associated with the Janowski functions. We then introduce and study a new subclass of q-starlike functions that involves the Janowski functions. We also derive several properties of such families of q-starlike functions with negative coefficients including (for example) distortion theorems.

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A new class of orthogonal polynomials: The Bessel polynomials

1949-01-01

H. L. Krall , Orrin Frink

The classical sets of orthogonal polynomials of Jacobi, Laguerre, and Hermite satisfy second order differential equations, and also have the property that their derivatives form orthogonal systems. There is a … The classical sets of orthogonal polynomials of Jacobi, Laguerre, and Hermite satisfy second order differential equations, and also have the property that their derivatives form orthogonal systems. There is a fourth class of polynomials with these two properties, and similar in other ways to the other three classes, which has hitherto been little studied. We call these the Bessel polynomials because of their close relationship with the Bessel functions of half-integral order. They are orthogonal, but not in quite the same sense as the other three systems. The Bessel polynomials satisfy:

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Certain Subclasses of Analytic Functions Associated with the Generalized Hypergeometric Function

2003-02-01

Jacek Dziok , H. M. Srivastava

Recently, the authors introduced and studied certain subclasses of analytic functions (associated with the generalized hypergeometric function) using the classical normalization. The main object of this sequel to the earlier … Recently, the authors introduced and studied certain subclasses of analytic functions (associated with the generalized hypergeometric function) using the classical normalization. The main object of this sequel to the earlier work is to present a systematic investigation of various subclasses of analytic functions using Montel's normalization. Coefficients estimates, distortion theorems, and the radii of convexity and starlikeness for each of these classes are given.

UNIVALENT FUNCTIONS (Grundlehren der mathematischen Wissenschaften, 259)

1984-09-01

W. K. Hayman

By Peter L. Duren: pp.382. DM. 138.-; US$53.60. (Springer-Verlag, Berlin, 1983.) By Peter L. Duren: pp.382. DM. 138.-; US$53.60. (Springer-Verlag, Berlin, 1983.)

Hankel and Toeplitz Determinants for a Subclass of q-Starlike Functions Associated with a General Conic Domain

2019-02-15

H. M. Srivastava , Qazi Zahoor Ahmad , Nasir Khan +2 more

By using a certain general conic domain as well as the quantum (or q-) calculus, here we define and investigate a new subclass of normalized analytic and starlike functions in … By using a certain general conic domain as well as the quantum (or q-) calculus, here we define and investigate a new subclass of normalized analytic and starlike functions in the open unit disk U . In particular, we find the Hankel determinant and the Toeplitz matrices for this newly-defined class of analytic q-starlike functions. We also highlight some known consequences of our main results.

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Some generalizations of the Apostol–Bernoulli and Apostol–Euler polynomials

2005-02-10

Qiu-Ming Luo , H. M. Srivastava

Some fractional differintegral equations

1985-03-01

H. M. Srivastava , Shigeyoshi Owa , Katsuyuki Nishimoto

Coefficient inequalities for $q$-starlike functions associated with the Janowski functions

2019-06-01

H. M. Srivastava , Bilal Khan , Nazar Khan +1 more

The main purpose of this investigation is to find several coefficient inequalities and a sufficient condition for $q$-starlike functions which are associated with the Janowski functions. Relevant connections of the … The main purpose of this investigation is to find several coefficient inequalities and a sufficient condition for $q$-starlike functions which are associated with the Janowski functions. Relevant connections of the results presented in this paper with those in a number of other related works on this subject are also pointed out.

Integral and computational representations of the extended Hurwitz–Lerch zeta function

2011-01-24

H. M. Srivastava , Ram K. Saxena , Tibor K. Pogány +1 more

This article presents a systematic investigation of various integrals and computational representations for some families of generalized Hurwitz–Lerch Zeta functions which are introduced here. We first establish their relationship with … This article presents a systematic investigation of various integrals and computational representations for some families of generalized Hurwitz–Lerch Zeta functions which are introduced here. We first establish their relationship with the -function, which enables us to derive the Mellin–Barnes type integral representations for nearly all of the generalized and specialized Hurwitz–Lerch Zeta functions. The integral expressions studied in this paper provide extensions of the corresponding results given by many authors, including (for example) Garg et al. [A further study of general Hurwitz-Lerch zeta function, Algebras Groups Geom. 25 (2008), pp. 311–319] and Lin and Srivastava [Some families of the Hurwitz-Lerch Zeta functions and associated fractional derivative and other integral representations, Appl. Math. Comput. 154 (2004), pp. 725–733]. We also derive a further analytic continuation formula which provides an elegant extension of the well-known analytic continuation formula for the Gauss hypergeometric function. Fractional derivatives associated with the generalized Hurwitz–Lerch Zeta functions are obtained. The relationship between the generalized Hurwitz–Lerch Zeta function and the -function, which was given by Garg et al., is seen to be erroneous and we give its corrected version here. Finally, a unification and extension of the Hurwitz–Lerch Zeta function, introduced in this article, is presented and two of its interesting special cases associated with the Mittag–Leffler type functions due to Barnes [The asymptotic expansion of integral functions defined by Taylor series, Philos. Trans. Roy. Soc. London Ser. A 206 (1906), pp. 249–297] and the generalized M-series considered recently by Sharma and Jain [A note on a generalzed M-series as a special function of fractional calculus, Fract. Calc. Appl. Anal. 12 (2009), pp. 449–452.] are deduced.

Upper Bound of the Third Hankel Determinant for a Subclass of q-Starlike Functions

2019-03-07

Shahid Mahmood , H. M. Srivastava , Nazar Khan +3 more

The main purpose of this article is to find the upper bound of the third Hankel determinant for a family of q-starlike functions which are associated with the Ruscheweyh-type q-derivative … The main purpose of this article is to find the upper bound of the third Hankel determinant for a family of q-starlike functions which are associated with the Ruscheweyh-type q-derivative operator. The work is motivated by several special cases and consequences of our main results, which are pointed out herein.

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