Mathematics › Applied Mathematics

Mathematical functions and polynomials

Works Count: 32333

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Description

This cluster of papers focuses on the theory, computation, and applications of orthogonal polynomials. It covers topics such as quadrature methods, approximation, interpolation, asymptotics, numerical stability, matrix valued polynomials, Riemann-Hilbert approach, random matrix theory, and hypergeometric functions.

Keywords

Orthogonal Polynomials; Quadrature Methods; Approximation; Interpolation; Asymptotics; Numerical Stability; Matrix Valued Polynomials; Riemann-Hilbert Approach; Random Matrix Theory; Hypergeometric Functions

Hypergeometric Orthogonal Polynomials and Their q-Analogues

2010-01-01

Roelof Koekoek , Peter Lesky , René F. Swarttouw

Orthogonal Polynomials and Special Functions

1975-01-01

Richard Askey

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Classical Orthogonal Polynomials of a Discrete Variable

1991-01-01

Arnold F. Nikiforov , Vasilii B. Uvarov , Sergeĭ K. Suslov

Geometry of Polynomials

1949-12-31

Morris Marden

During the years since the first edition of this well-known monograph appeared, the subject (the geometry of the zeros of a complex polynomial) has continued to display the same outstanding … During the years since the first edition of this well-known monograph appeared, the subject (the geometry of the zeros of a complex polynomial) has continued to display the same outstanding vitality as it did in the first 150 years of its history, beginning with the contributions of Cauchy and Gauss. Thus, the number of entries in the bibliography of this edition had to be increased from about 300 to about 600 and the book enlarged by one third. It now includes a more extensive treatment of Hurwitz polynomials and other topics. The new material on infrapolynomials, abstract polynomials, and matrix methods is of particular interest.

Orthogonal Polynomials

2004-04-29

Walter Gautschi

This is the first book on constructive methods for, and applications of orthogonal polynomials, and the first available collection of relevant Matlab codes. The book begins with a concise introduction … This is the first book on constructive methods for, and applications of orthogonal polynomials, and the first available collection of relevant Matlab codes. The book begins with a concise introduction to the theory of polynomials orthogonal on the real line (or a portion thereof), relative to a positive measure of integration. Topics which are particularly relevant to computation are emphasized. The second chapter develops computational methods for generating the coefficients in the basic three-term recurrence relation. The methods are of two kinds: moment-based methods and discretization methods. The former are provided with a detailed sensitivity analysis. Other topics addressed concern Cauchy integrals of orthogonal polynomials and their computation, a new discussion of modification algorithms, and the generation of Sobolev orthogonal polynomials. The final chapter deals with selected applications: the numerical evaluation of integrals, especially by Gauss-type quadrature methods, polynomial least squares approximation, moment-preserving spline approximation, and the summation of slowly convergent series. Detailed historic and bibliographic notes are appended to each chapter. The book will be of interest not only to mathematicians and numerical analysts, but also to a wide clientele of scientists and engineers who perceive a need for applying orthogonal polynomials.

The Askey-scheme of hypergeometric orthogonal polynomials and its q-analogue

1996-01-01

Roelof Koekoek , René F. Swarttouw

We list the so-called Askey-scheme of hypergeometric orthogonal polynomials. In chapter 1 we give the definition, the orthogonality relation, the three term recurrence relation and generating functions of all classes … We list the so-called Askey-scheme of hypergeometric orthogonal polynomials. In chapter 1 we give the definition, the orthogonality relation, the three term recurrence relation and generating functions of all classes of orthogonal polynomials in this scheme. In chapeter 2 we give all limit relation between different classes of orthogonal polynomials listed in the Askey-scheme. In chapter 3 we list the q-analogues of the polynomials in the Askey-scheme. We give their definition, orthogonality relation, three term recurrence relation and generating functions. In chapter 4 we give the limit relations between those basic hypergeometric orthogonal polynomials. Finally in chapter 5 we point out how the `classical` hypergeometric orthogonal polynomials of the Askey-scheme can be obtained from their q-analogues.

Orthogonal Polynomials on the Unit Circle

2005-01-01

Barry Simon

Generalized interpolation in 𝐻^{∞}

1967-01-01

Donald Sarason

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Some basic hypergeometric orthogonal polynomials that generalize Jacobi polynomials

1985-01-01

Richard Askey , James Wilson

Distributions of Matrix Variates and Latent Roots Derived from Normal Samples

1964-06-01

A. T. James

The paper is largely expository, but some new results are included to round out the paper and bring it up to date. The following distributions are quoted in Section 7. … The paper is largely expository, but some new results are included to round out the paper and bring it up to date. The following distributions are quoted in Section 7. 1. Type $_0F_0$, exponential: (i) $\chi^2$, (ii) Wishart, (iii) latent roots of the covariance matrix. 2. Type $_1F_0$, binomial series: (i) variance ratio, $F$, (ii) latent roots with unequal population covariance matrices. 3. Type $_0F_1$, Bessel: (i) noncentral $\chi^2$, (ii) noncentral Wishart, (iii) noncentral means with known covariance. 4. Type $_1F_1$, confluent hypergeometric: (i) noncentral $F$, (ii) noncentral multivariate $F$, (iii) noncentral latent roots. 5. Type $_2F_1$, Gaussian hypergeometric: (i) multiple correlation coefficient, (ii) canonical correlation coefficients. The modifications required for the corresponding distributions derived from the complex normal distribution are outlined in Section 8, and the distributions are listed. The hypergeometric functions $_pF_q$ of matrix argument which occur in the multivariate distributions are defined in Section 4 by their expansions in zonal polynomials as defined in Section 5. Important properties of zonal polynomials and hypergeometric functions are quoted in Section 6. Formulae and methods for the calculation of zonal polynomials are given in Section 9 and the zonal polynomials up to degree 6 are given in the appendix. The distribution of quadratic forms is discussed in Section 10, orthogonal expansions of $_0F_0$ and $_1F_1$ in Laguerre polynomials in Section 11 and the asymptotic expansion of $_0F_0$ in Section 12. Section 13 has some formulae for moments.

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THE INDEX OF ELLIPTIC OPERATORS

1997-10-01

Michael Atiyah

A Test of Goodness of Fit

1954-12-01

T. W. Anderson , D. A. Darling

Some (large sample) significance points are tabulated for a distribution-free test of goodness of fit which was introduced earlier by the authors. The test, which uses the actual observations without … Some (large sample) significance points are tabulated for a distribution-free test of goodness of fit which was introduced earlier by the authors. The test, which uses the actual observations without grouping, is sensitive to discrepancies at the tails of the distribution rather than near the median. An illustration is given, using a numerical example used previously by Birnbaum in illustrating the Kolmogorov test.

Confluent Hypergeometric Functions

2008-05-02

L. J. Slater

The confluent hypergeometric differential equation has a regular singular point at and an essential singularity at Solutions analytic at are confluent hypergeometric functions of the first kind or Kummer functions … The confluent hypergeometric differential equation has a regular singular point at and an essential singularity at Solutions analytic at are confluent hypergeometric functions of the first kind or Kummer functions where are Pochhammer symbols defined by For the function becomes singular unless is an equal or smaller negative integer and it is convenient to define the regularized confluent h

Some results on Tchebycheffian spline functions

1971-01-01

George Kimeldorf , Grace Wahba

Computing the distribution of quadratic forms in normal variables

1961-01-01

J. P. Imhof

Journal Article Computing the distribution of quadratic forms in normal variables Get access J. P. IMHOF J. P. IMHOF University of Geneva Search for other works by this author on: … Journal Article Computing the distribution of quadratic forms in normal variables Get access J. P. IMHOF J. P. IMHOF University of Geneva Search for other works by this author on: Oxford Academic Google Scholar Biometrika, Volume 48, Issue 3-4, December 1961, Pages 419–426, https://doi.org/10.1093/biomet/48.3-4.419 Published: 01 December 1961

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Determination of quasiprobability distributions in terms of probability distributions for the rotated quadrature phase

1989-09-01

Karl Vogel , H. Risken

It is shown that the probability distribution for the rotated quadrature phase [${a}^{\mathrm{\ifmmode^\circ\else\textdegree\fi{}}}$exp(i\ensuremath{\theta})+a exp(-i\ensuremath{\theta})]/2 can be expressed in terms of quasiprobability distributions such as P, Q, and Wigner functions and … It is shown that the probability distribution for the rotated quadrature phase [${a}^{\mathrm{\ifmmode^\circ\else\textdegree\fi{}}}$exp(i\ensuremath{\theta})+a exp(-i\ensuremath{\theta})]/2 can be expressed in terms of quasiprobability distributions such as P, Q, and Wigner functions and that also the reverse is true, i.e., if the probability distribution for the rotated quadrature phase is known for every \ensuremath{\theta} in the interval 0\ensuremath{\le}\ensuremath{\theta}<\ensuremath{\pi}, then the quasiprobability distributions can be obtained.

Representation of Lie Groups and Special Functions

1992-01-01

N. Ja. Vilenkin , A. U. Klimyk

The Special Functions and Their Approximations

1972-01-01

C. W. Clenshaw , Yudell L. Luke

Toeplitz Forms and Their Applications

1958-10-01

Ulf Grenander , Gábor Szegő , Mark Kac

Part I: Toeplitz Forms: Preliminaries Orthogonal polynomials. Algebraic properties Orthogonal polynomials. Limit properties The trigonometric moment problem Eigenvalues of Toeplitz forms Generalizations and analogs of Toeplitz forms Further generalizations Certain … Part I: Toeplitz Forms: Preliminaries Orthogonal polynomials. Algebraic properties Orthogonal polynomials. Limit properties The trigonometric moment problem Eigenvalues of Toeplitz forms Generalizations and analogs of Toeplitz forms Further generalizations Certain matrices and integral equations of the Toeplitz type Part II: Applications of Toeplitz Forms: Applications to analytic functions Applications to probability theory Applications to statistics Appendix: Notes and references Bibliography Index.

Saddlepoint Approximations in Statistics

1954-12-01

H. E. Daniels

It is often required to approximate to the distribution of some statistic whose exact distribution cannot be conveniently obtained. When the first few moments are known, a common procedure is … It is often required to approximate to the distribution of some statistic whose exact distribution cannot be conveniently obtained. When the first few moments are known, a common procedure is to fit a law of the Pearson or Edgeworth type having the same moments as far as they are given. Both these methods are often satisfactory in practice, but have the drawback that errors in the "tail" regions of the distribution are sometimes comparable with the frequencies themselves. The Edgeworth approximation in particular notoriously can assume negative values in such regions. The characteristic function of the statistic may be known, and the difficulty is then the analytical one of inverting a Fourier transform explicitly. In this paper we show that for a statistic such as the mean of a sample of size $n$, or the ratio of two such means, a satisfactory approximation to its probability density, when it exists, can be obtained nearly always by the method of steepest descents. This gives an asymptotic expansion in powers of $n^{-1}$ whose dominant term, called the saddlepoint approximation, has a number of desirable features. The error incurred by its use is $O(n^{-1})$ as against the more usual $O(n^{-1/2})$ associated with the normal approximation. Moreover it is shown that in an important class of cases the relative error of the approximation is uniformly $O(n^{-1})$ over the whole admissible range of the variable. The method of steepest descents was first used systematically by Debye for Bessel functions of large order (Watson [17]) and was introduced by Darwin and Fowler (Fowler [9]) into statistical mechanics, where it has remained an indispensable tool. Apart from the work of Jeffreys [12] and occasional isolated applications by other writers (e.g. Cox [2]), the technique has been largely ignored by writers on statistical theory. In the present paper, distributions having probability densities are discussed first, the saddlepoint approximation and its associated asymptotic expansion being obtained for the probability density of the mean $\bar{x}$ of a sample of $n$. It is shown how the steepest descents technique is related to an alternative method used by Khinchin [14] and, in a slightly different context, by Cramer [5]. General conditions are established under which the relative error of the saddlepoint approximation is $O(n^{-1})$ uniformly for all admissible $\bar{x}$, with a corresponding result for the asymptotic expansion. The case of discrete variables is briefly discussed, and finally the method is used for approximating to the distribution of ratios.

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The Monte Carlo Method

1949-09-01

N. Metropolis , Stanislaw M. Ulam

Abstract We shall present here the motivation and a general description of a method dealing with a class of problems in mathematical physics. The method is, essentially, a statistical approach … Abstract We shall present here the motivation and a general description of a method dealing with a class of problems in mathematical physics. The method is, essentially, a statistical approach to the study of differential equations, or more generally, of integro-differential equations that occur in various branches of the natural sciences.

A CENTRAL LIMIT THEOREM AND A STRONG MIXING CONDITION

1956-01-01

M. Rosenblatt

* research for this paper was supported by the United States Air Force under Contract No. AF18(600-685) monitored by the Office of Scientific Research. 1 A discussion of the problem … * research for this paper was supported by the United States Air Force under Contract No. AF18(600-685) monitored by the Office of Scientific Research. 1 A discussion of the problem along with the necessary references will be found in Harry Pollard, The Harmonic Analysis of Bounded Functions, Duke Math. / . , 20, 499-512, 1953. 2 See ibid. 3 See S. Bochner, Fouriersche Integrate (Leipzig, 1932), p. 33. 4 Levitan polynomial. See N.I. Achieser, Approximationstheorie (Berlin, 1953), p. 146.

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On certain integrals of Lipschitz-Hankel type involving products of bessel functions

1955-04-19

G. Eason , B. Noble , I. N. Sneddon

This paper is concerned with the evaluation and tabulation of certain integrals of the type (* 00 I(p, v; A) = J J fa t) ) e~cttxdt. In part I … This paper is concerned with the evaluation and tabulation of certain integrals of the type (* 00 I(p, v; A) = J J fa t) ) e~cttxdt. In part I of this paper, a formula is derived for the integrals in terms of an integral of a hypergeometric function. This new integral is evaluated in the particular cases which are of most frequent use in mathematical physics. By means of these results, approximate expansions are obtained for cases in which the ratio b/a is small or in which b~a and is small. In part II, recurrence relations are developed between integrals with integral values of the parameters pt, v and A. Tables are given by means of which 7(0, 0; 1), 7(0, 1; 1), 7(1, 0; 1), 7(1,1; 1), 7(0, 0 ;0), 7(1, 0;'0), 7(0, 1; 0), 7(1, 1; 0), 7(0,1; - 1 ), 7(1,0; - 1 ) and 7(1,1; - 1 ) may be evaluated for 0 < b/a ^ 2, 0 ^ c/a ^ 2.

Handbook of Elliptic Integrals for Engineers and Scientists

1972-07-01

Y. L. L. , Paul F. Byrd , Morris D. Friedman

Download HANDBOOK OF ELLIPTIC INTEGRALS FOR ENGINEERS AND SCIENTISTS PDF. Consequently, it is of interest to approximate complete elliptic integrals by simple and accurate algebraic formulas over the As such, … Download HANDBOOK OF ELLIPTIC INTEGRALS FOR ENGINEERS AND SCIENTISTS PDF. Consequently, it is of interest to approximate complete elliptic integrals by simple and accurate algebraic formulas over the As such, it should be of interest to practitioners in the water engineering community. Handbook of elliptic integrals for ResearchGate is the professional network for scientists and researchers. We also introduce an integral operator on the set of means and investigate its properties. Handbook of Elliptic Integrals for Engineers and Scientists. Springer.

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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.

Orthogonal Polynomials of Several Variables

2014-08-05

Charles F. Dunkl , Yuan Xu

Serving both as an introduction to the subject and as a reference, this book presents the theory in elegant form and with modern concepts and notation. It covers the general … Serving both as an introduction to the subject and as a reference, this book presents the theory in elegant form and with modern concepts and notation. It covers the general theory and emphasizes the classical types of orthogonal polynomials whose weight functions are supported on standard domains. The approach is a blend of classical analysis and symmetry group theoretic methods. Finite reflection groups are used to motivate and classify symmetries of weight functions and the associated polynomials. This revised edition has been updated throughout to reflect recent developments in the field. It contains 25% new material, including two brand new chapters on orthogonal polynomials in two variables, which will be especially useful for applications, and orthogonal polynomials on the unit sphere. The most modern and complete treatment of the subject available, it will be useful to a wide audience of mathematicians and applied scientists, including physicists, chemists and engineers.

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Splines minimizing rotation-invariant semi-norms in Sobolev spaces

1977-01-01

Jean Duchon

On the Weierstrass-Mandelbrot fractal function

1980-04-24

Marsha Berry , Z. V. Lewis

The function <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi>W</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>t</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>≡</mml:mo> <mml:munderover> <mml:mo movablelimits="false">∑</mml:mo> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>=</mml:mo> <mml:mo>−</mml:mo> <mml:mi mathvariant="normal">∞</mml:mi> </mml:mrow> <mml:mi mathvariant="normal">∞</mml:mi> </mml:munderover> <mml:mrow> <mml:mstyle> <mml:mrow> <mml:mfrac> <mml:mrow> … The function <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi>W</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>t</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>≡</mml:mo> <mml:munderover> <mml:mo movablelimits="false">∑</mml:mo> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>=</mml:mo> <mml:mo>−</mml:mo> <mml:mi mathvariant="normal">∞</mml:mi> </mml:mrow> <mml:mi mathvariant="normal">∞</mml:mi> </mml:munderover> <mml:mrow> <mml:mstyle> <mml:mrow> <mml:mfrac> <mml:mrow> <mml:mo stretchy="false">[</mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mn>1</mml:mn> <mml:mo>−</mml:mo> <mml:msup> <mml:mi>e</mml:mi> <mml:mrow> <mml:mi>i</mml:mi> <mml:msup> <mml:mi>γ</mml:mi> <mml:mi>n</mml:mi> </mml:msup> <mml:mi>t</mml:mi> </mml:mrow> </mml:msup> <mml:mo stretchy="false">)</mml:mo> <mml:msup> <mml:mi>e</mml:mi> <mml:mrow> <mml:mi>i</mml:mi> <mml:mi>ϕ</mml:mi> <mml:mi>n</mml:mi> </mml:mrow> </mml:msup> <mml:mo stretchy="false">]</mml:mo> </mml:mrow> <mml:mrow> <mml:msup> <mml:mi>γ</mml:mi> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mn>2</mml:mn> <mml:mo>−</mml:mo> <mml:mi>D</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mi>n</mml:mi> </mml:mrow> </mml:msup> </mml:mrow> </mml:mfrac> </mml:mrow> </mml:mstyle> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mn>1</mml:mn> <mml:mo><</mml:mo> <mml:mi>D</mml:mi> <mml:mo><</mml:mo> <mml:mn>2</mml:mn> <mml:mo>,</mml:mo> <mml:mspace width="thinmathspace" /> <mml:mi>γ</mml:mi> <mml:mo>></mml:mo> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mspace width="thinmathspace" /> <mml:msub> <mml:mi>ϕ</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:mo>=</mml:mo> <mml:mrow> <mml:mi mathvariant="normal">arbitrary</mml:mi> </mml:mrow> <mml:mspace width="thinmathspace" /> <mml:mrow> <mml:mi mathvariant="normal">phases</mml:mi> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> </mml:math> is continuous but non-differentiable and possesses no scale. The graph of ReW or Im W has Hausdorff-Besicovitch (fractal) dimension D. Choosing Ø n = un gives a deterministic W the scaling properties of which can be studied analytically in terms of a representation obtained by using the Poisson summation formula. Choosing Ø n random gives a stochastic IF whose increments W( t +r) — W (t) are statistically stationary, with a mean square which, as a function of r, is smooth if 1.0 < D < 1.5 and fractal if 1.5 < D < 2.0. The properties of IF are illustrated by computed graphs for several values of D (including some ‘marginal’ cases = 1 where the series for W converges) and several values of y, with deterministic and random Ø n , for 0 ≤ t ≤ 1 and the magnified range 0.30 ≤ t ≤ 0.31. The Weierstrass spectrum y n can be generated by the energy levels of the quantum-mechanical potential — A / x 2 ,where A = 4π 2 /In 2 y.

Introduction to Approximation Theory

1969-04-01

T. J. Rivlin , E. W. Cheney

Introduction: 1 Examples and prospectus 2 Metric spaces 3 Normed linear spaces 4 Inner-product spaces 5 Convexity 6 Existence and unicity of best approximations 7 Convex functions The Tchebycheff Solution … Introduction: 1 Examples and prospectus 2 Metric spaces 3 Normed linear spaces 4 Inner-product spaces 5 Convexity 6 Existence and unicity of best approximations 7 Convex functions The Tchebycheff Solution of Inconsistent Linear Equations: 1 Introduction 2 Systems of equations with one unknown 3 Characterization of the solution 4 The special case 5 Polya's algorithm 6 The ascent algorithm 7 The descent algorithm 8 Convex programming Tchebycheff Approximation by Polynomials and Other Linear Families: 1 Introduction 2 Interpolation 3 The Weierstrass theorem 4 General linear families 5 the unicity problem 6 Discretization errors: General theory 7 Discretization: Algebraic polynomials. The inequalities of Markoff and Bernstein 8 Algorithms Least-squares Approximation and Related Topics: 1 Introduction 2 Orthogonal systems of polynomials 3 Convergence of orthogonal expansions 4 Approximation by series of Tchebycheff polynomials 5 Discrete least-squares approximation 6 The Jackson theorems Rational Approximation: 1 Introduction 2 Existence of best rational approximations 3 The characterization of best approximations 4 Unicity Continuity of best-approximation operators 5 Algorithms 6 Pade Approximation and its generalizations 7 Continued fractions Some Additional Topics: 1 The Stone approximation theorem 2 The Muntz theorem 3 The converses of the Jackson theorems 4 Polygonal approximation and bases in $C[a, b]$ 5 The Kharshiladze-Lozinski theorems 6 Approximation in the mean Notes References Index.

Table of Integrals, Series, and Products

1980-01-01

The Problem of Moments

1943-12-31

J. Shohat , J. Tamarkin

This book was first published in 1943 and then was reprinted several times with corrections. It presents the development of the classical problem of moments for the first 50 years, … This book was first published in 1943 and then was reprinted several times with corrections. It presents the development of the classical problem of moments for the first 50 years, after its introduction by Stieltjes in the 1890s. In addition to initial developments by Stieltjes, Markov, and Chebyshev, later contributions by Hamburger, Nevanlinna, Hausdorff, Stone, and others are discussed. The book also contains some results on the trigonometric moment problem and a chapter devoted to approximate quadrature formulas.

On orthogonal polynomials

1975-12-01

P. Túrán

Generalized Hypergeometric Functions

1966-10-01

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

Theory of Non-Commutative Polynomials

1933-07-01

Öystein Ore

The Orthogonal Development of Non-Linear Functionals in Series of Fourier-Hermite Functionals

1947-04-01

R. H. Cameron , W. T. Martin

(see, for example, Kaczmarz and Steinhaus [I, pp. 143-144]). Let (2.4) tap(t)} p = 1, 2, 3, be any orthonormal set of real functions, each belonging to L2(0, 1). Paley … (see, for example, Kaczmarz and Steinhaus [I, pp. 143-144]). Let (2.4) tap(t)} p = 1, 2, 3, be any orthonormal set of real functions, each belonging to L2(0, 1). Paley and 1 Wiener [II] have shown for each index p = 1, 2, that f ap(t) dx(t) exist as a generalized Stieltjes integral for almost all functions x(&) of C and that the equality W 1~~~~~~~~~~~~ |G a, ot(t) dx(t), * ,Iap(t) dx(t)] dwx (2.5) c 00 -p/2 L G(ui, * , up)euhu du, ... du.

A Closed Set of Normal Orthogonal Functions

1923-01-01

J. L. Walsh

A set of normal orthogonal functions {χ} for the interval 0 5 x 5 1 has been constructed by Haar†, each function taking merely one constant value in each of … A set of normal orthogonal functions {χ} for the interval 0 5 x 5 1 has been constructed by Haar†, each function taking merely one constant value in each of a finite number of sub-intervals into which the entire interval (0, 1) is divided. Haar’s set is, however, merely one of an infinity of sets which can be constructed of functions of this same character. It is the object of the present paper to study a certain new closed set of functions {φ} normal and orthogonal on the interval (0, 1); each function φ has this same property of being constant over each of a finite number of sub-intervals into which the interval (0, 1) is divided. In fact each function φ takes only the values +1 and −1, except at a finite number of points of discontinuity, where it takes the value zero. The chief interest of the set φ lies in its similarity to the usual (e.g., sine, cosine, Sturm-Liouville, Legendre) set of orthogonal functions, while the chief interest of the set χ lies in its dissimilarity to these ordinary sets. The set φ shares with the familiar sets the following properties, none of which is possessed by the set χ: the nth function has n−1 zeroes (or better, sign-changes) interior to the interval considered, each function is either odd or even with respect to the mid-point of the interval, no function vanishes identically on any sub-interval of the original interval, and the entire set is uniformly bounded. Each function χ can be expressed as a linear combination of a finite number of functions φ, so the paper illustrates the changes in properties which may arise from a simple orthogonal transformation of a set of functions. In § 1 we define the set χ and give some of its principal properties. In § 2 we define the set φ and compare it with the set χ. In § 3 and § 4 we develop some of the properties of the set φ, and prove in particular that every continuous function of bounded variation can be expanded in terms of the φ’s and that every continuous function can be so developed in the sense not of convergence of the series but of summability by the first Cesaro mean. In § 5 it is proved that there exists a continuous function which cannot be

Fonctions de répartition à N dimensions et leurs marges

1959-01-01

Michael Sklar

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Sinc approximation of algebraically decaying functions

2018-01-01

Dmytro Sytnyk

An extension of sinc interpolation on $\mathbb{R}$ to the class of algebraically decaying functions is developed in the paper. Similarly to the classical sinc interpolation we establish two types of … An extension of sinc interpolation on $\mathbb{R}$ to the class of algebraically decaying functions is developed in the paper. Similarly to the classical sinc interpolation we establish two types of error estimates. First covers a wider class of functions with the algebraic order of decay on $\mathbb{R}$. The second type of error estimates governs the case when the order of function's decay can be estimated everywhere in the horizontal strip of complex plane around $\mathbb{R}$. The numerical examples are provided.

Orthogonal Polynomials

1971-01-01

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The Monte Carlo Method

1949-09-01

N. Metropolis , Stanislaw M. Ulam

Abstract Abstract We shall present here the motivation and a general description of a method dealing with a class of problems in mathematical physics. The method is, essentially, a statistical … Abstract Abstract We shall present here the motivation and a general description of a method dealing with a class of problems in mathematical physics. The method is, essentially, a statistical approach to the study of differential equations, or more generally, of integro-differential equations that occur in various branches of the natural sciences.

Tables of integrals, series and products

1967-02-01

Leković Danijela R.

Chebyshev Polynomials

2002-09-17

J. C. Mason , D. C. Handscomb

Orthogonal Polynomials

1939-12-31

Gábor Szegő

Derivatives of Exponential and Logarithmic Functions

2025-06-25

Nicolas A. Pereyra | CRC Press eBooks

On approximation of some Lauricella-Saran's hypergeometric functions $F_M$ and their ratios by branched continued fractions

2025-06-24

Roman Dmytryshyn , I. Nyzhnyk | Matematychni Studii

The paper considers the problem of approximating Lauricella-Saran's hypergeometric functions $F_M(a_1,a_2,b_1,b_2;a_1,c_2;z_1,z_2,z_3)$ by rational functions, which are approximants of branched continued fraction expansions - a special family functions. Under the conditions … The paper considers the problem of approximating Lauricella-Saran's hypergeometric functions $F_M(a_1,a_2,b_1,b_2;a_1,c_2;z_1,z_2,z_3)$ by rational functions, which are approximants of branched continued fraction expansions - a special family functions. Under the conditions of positive definite values of the elements of the expansions, the domain of analytic continuation of these functions and their ratios is established. Here, the domain is an open connected set. It is also proven that under the above conditions, every branched continued fraction expansion converges to the function that is holomorphic in a given domain of analytic continuation at least as fast as a geometric series with a ratio less then unity.

Some Identities for the Chebyshev Polynomials Using Graph Matchings

2025-06-23

Ihab-Eddine Djellas , Abdelhak Taane , Hacène Belbachir | Mediterranean Journal of Mathematics

ON THE ITERATED POLYNOMIAL REGULARISATION OF THE DIFFERENTIAL SYSTEM RELATED TO THE GENERALISED MEIXNER ORTHOGONAL POLYNOMIALS

2025-06-21

Galina Filipuk , Hongmei Chen | Journal of Mathematical Sciences

The explicit hypergeometric-modularity method I

2025-06-20

Michael W. Allen , Brian Grove , Ling Long +1 more | Advances in Mathematics

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Weighted Optimal Quadrature Formulas in Sobolev Space and Their Applications

2025-06-20

Kholmat Shadimetov , Khojiakbar Usmanov | Algorithms

The optimization of computational algorithms is one of the main problems of computational mathematics. This optimization is well demonstrated by the example of the theory of quadrature and cubature formulas. … The optimization of computational algorithms is one of the main problems of computational mathematics. This optimization is well demonstrated by the example of the theory of quadrature and cubature formulas. It is known that the numerical integration of definite integrals is of great importance in basic and applied sciences. In this paper we consider the optimization problem of weighted quadrature formulas with derivatives in Sobolev space. Using the extremal function, the square of the norm of the error functional of the considered quadrature formula is calculated. Then, minimizing this norm by coefficients, we obtain a system to find the optimal coefficients of this quadrature formula. The uniqueness of solutions of this system is proved, and an algorithm for solving this system is given. The proposed algorithm is used to obtain the optimal coefficients of the derivative weight quadrature formulas. It should be noted that the optimal weighted quadrature formulas constructed in this work are optimal for the approximate calculation of regular, singular, fractional and strongly oscillating integrals. The constructed optimal quadrature formulas are applied to the approximate solution of linear Fredholm integral equations of the second kind. Finally, the numerical results are compared with the known results of other authors.

Abelian theorems for a variant of the index Whittaker transform over $$ \mathscr{E'}(\mathbb{R}_+)$$

2025-06-19

Jeetendrasingh Maan , B. J. González , E. R. Negrín | Analysis Mathematica

Using a General Hurwitz-Lerch Zeta for BI-Univalent Analytic Functions to Estimate a Second Hankel Determinan

2025-06-17

Alaa Ali Aljamie , Nagat Muftah Alabbar | International journal of research and scientific innovation

In this paper, we introduce and investigate a new class of bi- univalent functions defined in the open unit disk involving a general integral operator associated with the general Hurwitz- … In this paper, we introduce and investigate a new class of bi- univalent functions defined in the open unit disk involving a general integral operator associated with the general Hurwitz- Lerch Zeta function denoted by . The main result of the investigation is to estimate the upper bounds for the initial Taylor–Maclaurin coefficients of functions and for this class. Following, we find the second Hankel determinant. Several new results are shown after specializing the parameters employed in our main results.

The Approximation of Analytic Functions Using Shifts of the Lerch Zeta-Function in Short Intervals

2025-06-17

Antanas Laurinčikas | Axioms

In this paper, we obtain approximation theorems of classes of analytic functions by shifts L(λ,α,s+iτ) of the Lerch zeta-function for τ∈[T,T+H] where H∈[T27/82,T1/2]. The cases of all parameters, λ,α∈(0,1], are … In this paper, we obtain approximation theorems of classes of analytic functions by shifts L(λ,α,s+iτ) of the Lerch zeta-function for τ∈[T,T+H] where H∈[T27/82,T1/2]. The cases of all parameters, λ,α∈(0,1], are considered. If the set {log(m+α):m∈N0} is linearly independent over Q, then every analytic function in the strip {s=σ+it∈C:σ∈(1/2,1)} is approximated by the above shifts.

Unified finite integrals involving multivariable 𝐴-functions and hypergeometric functions

2025-06-16

Dinesh Kumar , Frédéric Ayant | Analysis

Abstract In this paper, we established new results of the product of a hypergeometric function with the multivariable 𝐴-function by applying definite integrals. Several other new and known results can … Abstract In this paper, we established new results of the product of a hypergeometric function with the multivariable 𝐴-function by applying definite integrals. Several other new and known results can be obtained from our main theorems.

On an orthogonal polynomial sequence and its recurrence coefficients: II

2025-06-16

D. Mbouna | Letters in Mathematical Physics

Explicit representations of the norms of the Laguerre-Sobolev and Jacobi-Sobolev polynomials

2025-06-14

Clemens Markett | Numerical Algorithms

Abstract This paper deals with discrete Sobolev orthogonal polynomials with respect to inner products built upon the classical Laguerre and Jacobi measures on the intervals $$ [0,\infty ) $$ <mml:math … Abstract This paper deals with discrete Sobolev orthogonal polynomials with respect to inner products built upon the classical Laguerre and Jacobi measures on the intervals $$ [0,\infty ) $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>[</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mi>∞</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> and $$ [-1,1] $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>[</mml:mo> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> <mml:mo>]</mml:mo> </mml:mrow> </mml:math> , respectively. In addition, they are equipped with point masses at a finite endpoint of the interval involving the underlying functions and their derivatives of first or higher order. One of the intrinsic features of these polynomials are their $$ L^2 $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>L</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:math> -norms in the corresponding inner product spaces. Their knowledge is essential to orthonormalize the polynomials and thus indispensable to treat the corresponding Fourier-Sobolev series and other topics, notably in approximation theory, spectral theory or mathematical physics. Proceeding from an appropriate representation of the Sobolev polynomials which reflect the influence of the point masses, we explicitly establish their squared norm in an efficient form. In each case, the value differs from the familiar squared norm of the Laguerre or Jacobi polynomials by a factor which itself is a product of two essentially identical terms. Surprisingly, each of these factors turns out to be the quotient of the leading coefficients of the Sobolev polynomial and its classical counterpart. Obviously, our results enable to determine the asymptotic behavior of the norms of the orthogonal polynomials considered for large n .

Around the support problem for Hilbert class polynomials

2025-06-03

Francesco Campagna , Gabriel A. Dill

Let H_{D}(T) denote the Hilbert class polynomial of the imaginary quadratic order of discriminant D . We study the rate of growth of the greatest common divisor of H_{D}(a) and … Let H_{D}(T) denote the Hilbert class polynomial of the imaginary quadratic order of discriminant D . We study the rate of growth of the greatest common divisor of H_{D}(a) and H_{D}(b) as |D| \to \infty for a and b belonging to various Dedekind domains. We also study the modular support problem: if for all but finitely many D every prime ideal dividing H_{D}(a) also divides H_{D}(b) , what can we say about a and b ? If we replace H_{D}(T) by T^{n}-1 and the Dedekind domain is a ring of S -integers in some number field, then these are classical questions that have been investigated by Bugeaud–Corvaja–Zannier, Corvaja–Zannier, and Corrales-Rodrigáñez–Schoof.

Skew odd orthogonal characters and interpolating Schur polynomials

2025-06-02

Naihuan Jing , Zhijun Li , Danxia Wang +2 more

Abstract We introduce two vertex operators to realize skew odd orthogonal characters and derive the Cauchy identity for the skew characters via the Toeplitz–Hankel‐type determinant as an analog of Schur … Abstract We introduce two vertex operators to realize skew odd orthogonal characters and derive the Cauchy identity for the skew characters via the Toeplitz–Hankel‐type determinant as an analog of Schur functions. The method also gives new proofs of the Jacobi–Trudi identity and Gelfand–Tsetlin patterns for . Moreover, combining the vertex operators related to characters of types (Baker, J. Phys. A . 29 (12) (1996), 3099–3117; Jing and Nie, Ann. Combin . 19 (2015), 427–442) and the new vertex operators related to ‐type characters, we obtain three families of symmetric polynomials that interpolate among characters of , , and . Their transition formulae are also explicitly given among symplectic, orthogonal, and odd orthogonal characters.

Signs of the second coefficients of Hecke polynomials

2025-06-01

Erick Ross , Hui Xue | Illinois Journal of Mathematics

Multiple Orthogonal Polynomials of two real variables

2025-06-01

Lidia Fernández , Juan Antonio Villegas | Journal of Mathematical Analysis and Applications

On solutions to q-difference equations for q-appell functions in the spirit of Olsson and Exton

2025-06-01

Thomas Ernst | Mathematica Slovaca

Partial Derivatives of Functions

2025-06-01

A Numerical Algorithm with Linear Complexity for Multi-Marginal Optimal Transport with $L^1$ Cost

2025-06-01

Chunhui Chen , Jing Chen , Baojia Luo +1 more | CSIAM Transactions on Applied Mathematics

Comparative Results among Different Types of Generalized Integrals

2025-06-01

Hemanta Kalita , Anca Croitoru | WORLD SCIENTIFIC eBooks

Fourier transforms of orthogonal polynomials on the cone

2025-05-31

Rabia Aktaş Karaman , Iván Area | Numerical Algorithms

On computing the zeros of a class of Sobolev orthogonal polynomials

2025-05-31

Nicola Mastronardi , Marc Van Barel , Raf Vandebril +1 more | Numerical Algorithms

A property of Cauchy–Stieltjes kernel families based on dilation of measures

2025-05-29

Ibrahim-Elkhalil Ahmed , Ayed R. A. Alanzi , Raouf Fakhfakh

A criterion for the simple normality of fractional powers of two via the Riemann zeta function

2025-05-29

Yuya Kanado , Kota Saito

Certain unified integral formulas involving five-parameter Mittag-Leffler function

2025-05-28

Ankit Pal , Vinod Kumar Jatav , Udai Kumar

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

2025-05-26

Jian Cao , H. M. Srivastava , Yue Zhang

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.

Re-examining symmetric q -Dunkl-classical orthogonal polynomials

2025-05-26

Jihad Souissi

Algebras behind the bispectrality of the Wilson rational functions and their $$ _4\phi _3$$ limits

2025-05-26

Nicolas Crampé , Satoshi Tsujimoto , Luc Vinet +1 more

A Note on Generalizations of q-Beta Integral and Related Transformational Identities

2025-05-24

Jian Cao , Yue Yang

Analytic continuation component of the GreenX library: robust Padé approximants with symmetry constraints

2025-05-23

Moritz Leucke , Ramón L. Panadés‐Barrueta , Ekin Esme Bas +1 more

On the region containing the zeros of a polynomial with restricted coefficients

2025-05-23

Humayun Mohd Sofi , Narendra Kumar

For the polynomial P(z) =∑_(j=0)^n▒〖a_j z^j 〗 , aj ≥ aj−1, a0 > 0, j = 1, 2, ..., n, an > 0, a classical result of Enestro¨m and Kakeya … For the polynomial P(z) =∑_(j=0)^n▒〖a_j z^j 〗 , aj ≥ aj−1, a0 > 0, j = 1, 2, ..., n, an > 0, a classical result of Enestro¨m and Kakeya says that all the zeros of P (z) lie in |z| ≤ 1. In the literature [1-12], there exist several extensions and generalizations of this result. Recently N.A.Rather and M.A.Shah [12] generalized it by further relaxing the condition on the coefficients. In this paper, we prove some more extensions and generalizations of the above results and hence of Enestro¨m and Kakeya theorem

A radial variable for de Sitter two-point functions

2025-05-21

Manuel Loparco , Jiaxin Qiao , Zimo Sun

We introduce a “radial” two-point invariant for quantum field theory in de Sitter (dS) analogous to the radial coordinate used in conformal field theory. We show that the two-point function … We introduce a “radial” two-point invariant for quantum field theory in de Sitter (dS) analogous to the radial coordinate used in conformal field theory. We show that the two-point function of a free massive scalar in the Bunch-Davies vacuum has an exponentially convergent series expansion in this variable with positive coefficients only. Assuming a convergent Källén-Lehmann decomposition, this result is then generalized to the two-point function of any scalar operator non-perturbatively. A corollary of this result is that, starting from two-point functions on the sphere, an analytic continuation to an extended complex domain is admissible. dS two-point configurations live inside or on the boundary of this domain, and all the paths traced by analytic continuation between dS and the sphere or between dS and Euclidean Anti-de Sitter are also contained within this domain.

Unimodality Preservation by Ratios of Functional Series and Integral Transforms

2025-05-20

Dmitrii Karp , Anna Vishnyakova , Yi Zhang

Lee–Yang zeroes of the Curie–Weiss ferromagnet, unitary Hermite polynomials, and the backward heat flow

2025-05-20

Zakhar Kabluchko

The backward heat flow on the real line started from the initial condition <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mi>z</mml:mi> <mml:mi>n</mml:mi> </mml:msup></mml:math> results in the classical <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mi>n</mml:mi> <mml:mi> th </mml:mi> </mml:msup></mml:math> Hermite polynomial whose … The backward heat flow on the real line started from the initial condition <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mi>z</mml:mi> <mml:mi>n</mml:mi> </mml:msup></mml:math> results in the classical <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mi>n</mml:mi> <mml:mi> th </mml:mi> </mml:msup></mml:math> Hermite polynomial whose zeroes are distributed according to the Wigner semicircle law in the large <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>n</mml:mi></mml:math> limit. Similarly, the backward heat flow with the periodic initial condition <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mrow><mml:mo>(</mml:mo><mml:mo form="prefix">sin</mml:mo><mml:mfrac><mml:mi>θ</mml:mi> <mml:mn>2</mml:mn></mml:mfrac><mml:mo>)</mml:mo></mml:mrow> <mml:mi>n</mml:mi> </mml:msup></mml:math> leads to trigonometric or unitary analogues of the Hermite polynomials. These polynomials are closely related to the partition function of the Curie–Weiss model and appeared in the work of Mirabelli on finite free probability. We relate the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mi>n</mml:mi> <mml:mi> th </mml:mi> </mml:msup></mml:math> unitary Hermite polynomial to the expected characteristic polynomial of a unitary random matrix obtained by running a Brownian motion on the unitary group <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>U</mml:mi><mml:mo>(</mml:mo><mml:mi>n</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math>. We identify the global distribution of zeroes of the unitary Hermite polynomials as the free unitary normal distribution. We also compute the asymptotics of these polynomials or, equivalently, the free energy of the Curie–Weiss model in a complex external field. We identify the global distribution of the Lee–Yang zeroes of this model. Finally, we show that the backward heat flow applied to a high-degree real-rooted polynomial (respectively, trigonometric polynomial) induces, on the level of the asymptotic distribution of its roots, a free Brownian motion (respectively, free unitary Brownian motion).

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Integral Transforms of the Generalized Mittag Leffler Type Function

2025-05-18

Seema Kabra

Abstract: This paper refers to some integral transforms of the generalised Mittag-Leffler type function. Using generalized ࢗࡾ function ࢽ;ࢼ,ࢻ ࢖ (ࢠ) ࢑ defined by Kapoor [7], various integral transform, including … Abstract: This paper refers to some integral transforms of the generalised Mittag-Leffler type function. Using generalized ࢗࡾ function ࢽ;ࢼ,ࢻ ࢖ (ࢠ) ࢑ defined by Kapoor [7], various integral transform, including Euler transform, Laplace transform, Whittaker transform, Mellin Transform and Hankel transform are derived. The results are expressed in the form of generalized Wright function. Based on the new result some integral formula with different special functions established as special cases of main results for different values of parameters.

Legendre polynomials in terms of integrals involving Hermite polynomials

2025-05-16

Baghdadi Aloui

Third Order Hankel Determinant for Inverse Functions of a Classes of Univalent Functions with Bounded Turning

2025-05-16

Elena Gelova , Nikola Tuneski

The main goal of this paper is to determine an upper bound for the third Hankel determinant for the inverse functions of f, belonging to the two classes of univalent … The main goal of this paper is to determine an upper bound for the third Hankel determinant for the inverse functions of f, belonging to the two classes of univalent functions with bounded turning.

Calculation of Generalized Dirichlet Integral: From Special Cases to General Formula

2025-05-15

Yue Yu

The Dirichlet integral is widely used in the fields of mathematical analysis, probability theory and physics. This paper explores the calculation of the generalized Dirichlet integral from zero to infinity. … The Dirichlet integral is widely used in the fields of mathematical analysis, probability theory and physics. This paper explores the calculation of the generalized Dirichlet integral from zero to infinity. The author focuses on transitioning from special cases to deriving a general formula. The methodology used mainly include substitution and integration by parts. In the simplification of formulas, trigonometric identities and Frullani integral are also used. Moreover, the author obtains the general formula by discussing the odd and even power cases respectively. This paper deduces the general formula of Dirichlet integral by using Eulers formula and binomial expansion. The result demonstrates that it uses special cases to find a general formula with the different order power and even the particular case of it, which is the same order power. The formula simplifies calculations. The significance of this paper lies in the calculation of Dirichlet integral general formula and various variations and give the answer. It provides an accurate formula for other studies using the this integral, and enhancing the overall body of knowledge in integral calculus.

On two Gauss–Radau quadrature formulae and their weighted averaged-rule: Is Gauss–Legendre always better than Radau?

2025-05-15

Hiroshi Sugiura , Takemitsu Hasegawa

On the Lebesgue Constant of the Morrow–Patterson Points

2025-05-14

Tomasz Beberok , Leokadia Białas-Cież , Stefano De Marchı

Interlacing of zeros from different sequences of Meixner-Pollaczek, Pseudo-Jacobi and Continuous Hahn polynomials

2025-05-14

A. Jooste , Kerstin Jordaan

Generalizations and improvements of Bernstein and Turán-type inequalities for the polar derivative of a polynomial

2025-05-13

Maisnam Triveni Devi , Thangjam Birkramjit Singh , Barchand Chanam

On a class of Rainville type generating functions for classical orthogonal polynomials

2024-12-04

Mohammed Brahim Zahaf , Mohammed Mesk

The aim of this work is to characterize all generating functions of the form $A(t)F(xtA(t)-R(t))$ for the classical orthogonal polynomials. Further generating functions are also provided by derivation. The aim of this work is to characterize all generating functions of the form $A(t)F(xtA(t)-R(t))$ for the classical orthogonal polynomials. Further generating functions are also provided by derivation.

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