Let [Formula: see text] denote the number of [Formula: see text]-regular bipartitions of [Formula: see text]. In this work, we establish several infinite families of congruences modulo powers of 2 âŠ
Let [Formula: see text] denote the number of [Formula: see text]-regular bipartitions of [Formula: see text]. In this work, we establish several infinite families of congruences modulo powers of 2 and 5 for [Formula: see text]. For example, we find that for all nonnegative integers [Formula: see text], [Formula: see text] and [Formula: see text] and [Formula: see text], [Formula: see text]
Let $\overline { A}_3(n)$ and $\overline { A}_9(n)$ denote the number of 3- and 9-regular overpartitions of $n$. For each $\alpha \gt 0$, we obtain the generating functions for $\overline âŠ
Let $\overline { A}_3(n)$ and $\overline { A}_9(n)$ denote the number of 3- and 9-regular overpartitions of $n$. For each $\alpha \gt 0$, we obtain the generating functions for $\overline { A}_{3}(3^{2\alpha }n )$, $\overline { A}_{3}(3^{2\alpha -1}n )$ a
In this paper, we establish several new PâQ mixed modular equations involving thetaâfunctions which are similar to those recorded by Ramanujan in his notebooks. As an application, we establish several âŠ
In this paper, we establish several new PâQ mixed modular equations involving thetaâfunctions which are similar to those recorded by Ramanujan in his notebooks. As an application, we establish several new explicit evaluations of cubic class invariants and cubic singular moduli.
Let $b_{3,5}(n)$ denote the number of partitions of $n$ into parts that are not multiples of 3 or 5. We establish several infinite families of congruences modulo 2 for $b_{3,5}(n)$ âŠ
Let $b_{3,5}(n)$ denote the number of partitions of $n$ into parts that are not multiples of 3 or 5. We establish several infinite families of congruences modulo 2 for $b_{3,5}(n)$ . In the process, we also prove numerous parity results for broken 7-diamond partitions.
Andrews, Lewis and Lovejoy introduced a new class of partitions, partitions with designated summands. Let PD(n) denote the number of partitions of n with desig- nated summands and PDO(n) denote âŠ
Andrews, Lewis and Lovejoy introduced a new class of partitions, partitions with designated summands. Let PD(n) denote the number of partitions of n with desig- nated summands and PDO(n) denote the number of partitions of n with designated summands in which all parts are odd. Andrews et al. established many congruences modulo 3 for PDO(n) by using the theory of modular forms. Baruah and Ojah ob- tained numerous congruences modulo 3, 4, 8 and 16 for PDO(n) by using theta function identities. In this paper, we prove several infinite families of congruences modulo 9, 16 and 32 for PDO(n). AÂŻĂ?ĂÂœ 2017 University of Waterloo. All rights reserved.
In this paper, we establish some new modular equations of degree 9. We also establish several new $P$--$Q$ mixed modular equations involving thetaâfunctions which are similar to those recorded by âŠ
In this paper, we establish some new modular equations of degree 9. We also establish several new $P$--$Q$ mixed modular equations involving thetaâfunctions which are similar to those recorded by Ramanujan in his notebooks. As an application, we establish some new general formulas for explicit evaluations of a remarkable product of theta--functions.
In this article, we study the arithmetic properties of the partition function $p_8(n)$, the number of 8-colour partitions of $n$. We prove several Ramanujan type congruences modulo higher powers of âŠ
In this article, we study the arithmetic properties of the partition function $p_8(n)$, the number of 8-colour partitions of $n$. We prove several Ramanujan type congruences modulo higher powers of 2 for the function $p_8(n)$ by finding explicit formulas for the generating functions.
We study the divisibility properties of the partition function associated with the eighth order mock theta function $V_0(q)$, introduced by Gordon and McIntosh. We obtain congruences modulo powers of 2 âŠ
We study the divisibility properties of the partition function associated with the eighth order mock theta function $V_0(q)$, introduced by Gordon and McIntosh. We obtain congruences modulo powers of 2 for certain coefficients of the partition function, akin to Ramanujan's partition congruences. Further, we also present several infinite families of congruences molulo 13, 25 and 27.
Ramanujan in his second notebook recorded total of seven $P$--$Q$ modular equations involving theta--function $f(-q)$ with moduli of orders 1, 3, 5 and 15. In this paper, modular equations analogous âŠ
Ramanujan in his second notebook recorded total of seven $P$--$Q$ modular equations involving theta--function $f(-q)$ with moduli of orders 1, 3, 5 and 15. In this paper, modular equations analogous to those recorded by Ramanujan are obtained involving his theta--functions $\varphi(q)$ and $\psi(-q)$ with moduli of orders 1, 3, 5 and 15. As a consequence, several values of quotients of theta--function and a continued fraction of order 12 are explicitly evaluated.
Abstract In this work, we investigate the arithmetic properties of $b_{5^k}(n)$ , which counts the partitions of n where no part is divisible by $5^k$ . By constructing generating functions âŠ
Abstract In this work, we investigate the arithmetic properties of $b_{5^k}(n)$ , which counts the partitions of n where no part is divisible by $5^k$ . By constructing generating functions for $b_{5^k}(n)$ across specific arithmetic progressions, we establish a set of Ramanujan-type congruences.
Abstract In this work, we investigate the arithmetic properties of $b_{5^k}(n)$ , which counts the partitions of n where no part is divisible by $5^k$ . By constructing generating functions âŠ
Abstract In this work, we investigate the arithmetic properties of $b_{5^k}(n)$ , which counts the partitions of n where no part is divisible by $5^k$ . By constructing generating functions for $b_{5^k}(n)$ across specific arithmetic progressions, we establish a set of Ramanujan-type congruences.
Ramanujan in his second notebook recorded total of seven $P$--$Q$ modular equations involving theta--function $f(-q)$ with moduli of orders 1, 3, 5 and 15. In this paper, modular equations analogous âŠ
Ramanujan in his second notebook recorded total of seven $P$--$Q$ modular equations involving theta--function $f(-q)$ with moduli of orders 1, 3, 5 and 15. In this paper, modular equations analogous to those recorded by Ramanujan are obtained involving his theta--functions $\varphi(q)$ and $\psi(-q)$ with moduli of orders 1, 3, 5 and 15. As a consequence, several values of quotients of theta--function and a continued fraction of order 12 are explicitly evaluated.
In this article, we study the arithmetic properties of the partition function $p_8(n)$, the number of 8-colour partitions of $n$. We prove several Ramanujan type congruences modulo higher powers of âŠ
In this article, we study the arithmetic properties of the partition function $p_8(n)$, the number of 8-colour partitions of $n$. We prove several Ramanujan type congruences modulo higher powers of 2 for the function $p_8(n)$ by finding explicit formulas for the generating functions.
We study the divisibility properties of the partition function associated with the eighth order mock theta function $V_0(q)$, introduced by Gordon and McIntosh. We obtain congruences modulo powers of 2 âŠ
We study the divisibility properties of the partition function associated with the eighth order mock theta function $V_0(q)$, introduced by Gordon and McIntosh. We obtain congruences modulo powers of 2 for certain coefficients of the partition function, akin to Ramanujan's partition congruences. Further, we also present several infinite families of congruences molulo 13, 25 and 27.
Let $\overline { A}_3(n)$ and $\overline { A}_9(n)$ denote the number of 3- and 9-regular overpartitions of $n$. For each $\alpha \gt 0$, we obtain the generating functions for $\overline âŠ
Let $\overline { A}_3(n)$ and $\overline { A}_9(n)$ denote the number of 3- and 9-regular overpartitions of $n$. For each $\alpha \gt 0$, we obtain the generating functions for $\overline { A}_{3}(3^{2\alpha }n )$, $\overline { A}_{3}(3^{2\alpha -1}n )$ a
Andrews, Lewis and Lovejoy introduced a new class of partitions, partitions with designated summands. Let PD(n) denote the number of partitions of n with desig- nated summands and PDO(n) denote âŠ
Andrews, Lewis and Lovejoy introduced a new class of partitions, partitions with designated summands. Let PD(n) denote the number of partitions of n with desig- nated summands and PDO(n) denote the number of partitions of n with designated summands in which all parts are odd. Andrews et al. established many congruences modulo 3 for PDO(n) by using the theory of modular forms. Baruah and Ojah ob- tained numerous congruences modulo 3, 4, 8 and 16 for PDO(n) by using theta function identities. In this paper, we prove several infinite families of congruences modulo 9, 16 and 32 for PDO(n). AÂŻĂ?ĂÂœ 2017 University of Waterloo. All rights reserved.
Let [Formula: see text] denote the number of [Formula: see text]-regular bipartitions of [Formula: see text]. In this work, we establish several infinite families of congruences modulo powers of 2 âŠ
Let [Formula: see text] denote the number of [Formula: see text]-regular bipartitions of [Formula: see text]. In this work, we establish several infinite families of congruences modulo powers of 2 and 5 for [Formula: see text]. For example, we find that for all nonnegative integers [Formula: see text], [Formula: see text] and [Formula: see text] and [Formula: see text], [Formula: see text]
Let $b_{3,5}(n)$ denote the number of partitions of $n$ into parts that are not multiples of 3 or 5. We establish several infinite families of congruences modulo 2 for $b_{3,5}(n)$ âŠ
Let $b_{3,5}(n)$ denote the number of partitions of $n$ into parts that are not multiples of 3 or 5. We establish several infinite families of congruences modulo 2 for $b_{3,5}(n)$ . In the process, we also prove numerous parity results for broken 7-diamond partitions.
In this paper, we establish several new PâQ mixed modular equations involving thetaâfunctions which are similar to those recorded by Ramanujan in his notebooks. As an application, we establish several âŠ
In this paper, we establish several new PâQ mixed modular equations involving thetaâfunctions which are similar to those recorded by Ramanujan in his notebooks. As an application, we establish several new explicit evaluations of cubic class invariants and cubic singular moduli.
In this paper, we establish some new modular equations of degree 9. We also establish several new $P$--$Q$ mixed modular equations involving thetaâfunctions which are similar to those recorded by âŠ
In this paper, we establish some new modular equations of degree 9. We also establish several new $P$--$Q$ mixed modular equations involving thetaâfunctions which are similar to those recorded by Ramanujan in his notebooks. As an application, we establish some new general formulas for explicit evaluations of a remarkable product of theta--functions.
In this paper, we use the explicit Shimura Reciprocity Law to compute the cubic singular moduli α * n , which are used in the constructions of new rapidly convergent âŠ
In this paper, we use the explicit Shimura Reciprocity Law to compute the cubic singular moduli α * n , which are used in the constructions of new rapidly convergent series for 1/Ï.We also complete a table of values for the class invariant λ n initiated by S. Ramanujan on page 212 of his Lost Notebook.
Abstract In a recent paper, Calkin et al . [N. Calkin, N. Drake, K. James, S. Law, P. Lee, D. Penniston and J. Radder, âDivisibility properties of the 5-regular and âŠ
Abstract In a recent paper, Calkin et al . [N. Calkin, N. Drake, K. James, S. Law, P. Lee, D. Penniston and J. Radder, âDivisibility properties of the 5-regular and 13-regular partition functionsâ, Integers 8 (2008), #A60] used the theory of modular forms to examine 5-regular partitions modulo 2 and 13-regular partitions modulo 2 and 3; they obtained and conjectured various results. In this note, we use nothing more than Jacobiâs triple product identity to obtain results for 5-regular partitions that are stronger than those obtained by Calkin and his collaborators. We find infinitely many Ramanujan-type congruences for b 5 ( n ), and we prove the striking result that the number of 5-regular partitions of the number n is even for at least 75% of the positive integers n .
In this paper, we establish several explicit evaluations, reciprocity theorems and integral representations for a continued fraction of order twelve which are analogues to Rogers-Ramanujan's continued fraction and Ramanujan's cubic âŠ
In this paper, we establish several explicit evaluations, reciprocity theorems and integral representations for a continued fraction of order twelve which are analogues to Rogers-Ramanujan's continued fraction and Ramanujan's cubic continued fraction.
The generating function of partitions with repeated (resp. distinct) parts such that each odd part is less than twice the smallest part is shown to be the third order mock âŠ
The generating function of partitions with repeated (resp. distinct) parts such that each odd part is less than twice the smallest part is shown to be the third order mock theta function Ï(q) (resp. Îœ(âq)). Similar results for partitions with the corresponding restriction on each even part are also obtained, one of which involves the third order mock theta function Ï(q). Congruences for the smallest parts partition function(s) associated to such partitions are obtained. Two analogues of the partition-theoretic interpretation of Euler's pentagonal number theorem are also obtained.
Ramanujan conjecture for powers of 5 Abstract.Ramanujan conjectured, and G. N. Watson proved, that if n is of a specific form then p(n), the number of partitions of n, is âŠ
Ramanujan conjecture for powers of 5 Abstract.Ramanujan conjectured, and G. N. Watson proved, that if n is of a specific form then p(n), the number of partitions of n, is divisible by a high power of 5.In the present note, we establish appropriate generating function formulae, from which the truth of Ramanujan's conjecture, as well as some results of a similar type due to Watson, are shown to follow easily.Furthermore, we derive two new congruences for the partition function.Our proofs are more straightforward than those of Watson and more recent writers and use only classical identities of Euler and Jacobi.
We study Ramanujan's cubic continued fraction and explicit evaluations of theta-functions
We study Ramanujan's cubic continued fraction and explicit evaluations of theta-functions
In 2004, the first author gave the combinatorial interpretations of four mock theta functions of Srinivasa Ramanujan using $n$-color partitions which were introduced by himself and G.E. Andrews in 1987. âŠ
In 2004, the first author gave the combinatorial interpretations of four mock theta functions of Srinivasa Ramanujan using $n$-color partitions which were introduced by himself and G.E. Andrews in 1987. In this paper we introduce a new class of partitions and call them "split $(n+t)$-color partitions". These new partitions generalize Agarwal-Andrews $(n+t)$-color partitions. We use these new combinatorial objects and give combinatorial meaning to two basic functions of Gordon-McIntosh found in 2000. They used these functions to establish the modular transformation formulas for certain eight order mock theta functions. The work done here has a great potential for future research.
Recently, Andrews introduced the partition function [Formula: see text] as the number of overpartitions of [Formula: see text] in which no part is divisible by [Formula: see text] and only âŠ
Recently, Andrews introduced the partition function [Formula: see text] as the number of overpartitions of [Formula: see text] in which no part is divisible by [Formula: see text] and only parts [Formula: see text] may be overlined. He proved that [Formula: see text] and [Formula: see text] are divisible by [Formula: see text]. Let [Formula: see text] be the number of overpartitions of [Formula: see text] into parts not divisible by [Formula: see text]. In this paper, we call the overpartitions enumerated by the function [Formula: see text] [Formula: see text]-regular overpartitions. For [Formula: see text] and [Formula: see text], we obtain some explicit results on the generating function dissections. We also derive some congruences for [Formula: see text] modulo [Formula: see text], [Formula: see text] and [Formula: see text] which imply the congruences for [Formula: see text] proved by Andrews. By introducing a rank of vector partitions, we give a combinatorial interpretation of the congruences of Andrews for [Formula: see text] and [Formula: see text].
In this paper, we derive new Ramanujan-type series for 1/Ï which belong to "Ramanujan's theory of elliptic functions to alternative base 3" developed recently by B.C.
In this paper, we derive new Ramanujan-type series for 1/Ï which belong to "Ramanujan's theory of elliptic functions to alternative base 3" developed recently by B.C.
Abstract In this paper, we revisit Russell-type modular equations, a collection of modular equations first studied systematically by R. Russell in 1887. We give a proof of Russellâs main theorem âŠ
Abstract In this paper, we revisit Russell-type modular equations, a collection of modular equations first studied systematically by R. Russell in 1887. We give a proof of Russellâs main theorem and indicate the relations between such equations and the constructions of Hilbert class fields of imaginary quadratic fields. Motivated by Russellâs theorem, we state and prove its cubic analogue which allows us to construct Russell-type modular equations in the theory of signature 3.
Let [Formula: see text] be the number of overpartitions of [Formula: see text] into parts not divisible by [Formula: see text]. In this paper, we find infinite families of congruences âŠ
Let [Formula: see text] be the number of overpartitions of [Formula: see text] into parts not divisible by [Formula: see text]. In this paper, we find infinite families of congruences modulo 4, 8 and 16 for [Formula: see text] and [Formula: see text] for any [Formula: see text]. Along the way, we obtain several Ramanujan type congruences for [Formula: see text] and [Formula: see text]. We also find infinite families of congruences modulo [Formula: see text] for [Formula: see text].