Special Issue "q-Series and Related Topics in Special Functions and Analytic Number Theory"

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A special issue of Axioms (ISSN 2075-1680).

Deadline for manuscript submissions: closed (31 October 2012)

Special Issue Editor

Guest Editor
Prof. Dr. Hari M. Srivastava

Department of Mathematics and Statistics, University of Victoria, Victoria, British Columbia V8W 3R4, Canada
Website | E-Mail
Phone: 1-250-472-5313 (Office); 1-250-477-6960 (Home)
Fax: 1-250-721-8962
Interests: real and complex analysis; fractional calculus and its applications; integral equations and transforms; higher transcendental functions and their applications; q-series and q-polynomials; analytic number theory; analytic and geometric Inequalities; probability and statistics; inventory modelling and optimization

Special Issue Information

Dear Colleagues,

Basic (or q-) series and basic (or q-) polynomials are known to occur frequently in many diverse fields of mathematical and physical sciences. The recent overwhelming contributions toward the Rogers-Ramanujan, Jacobi's Triple-Product and Macdonald Identities, Mock Theta Functions, Rogers-Ramanujan Continued Fractions, and so on, together with the recent developments involving such basic (or q-) functions of Analytic Number Theory as (for example) the q-Zeta and related functions, cannot be over-emphasized. Thus, naturally, publication of a Special Issue of Axioms featuring invited articles in these and other related areas, especially in the year 2012 in which the world is celebrating the 125th birth anniversary of Srinivasa Ramanujan (1887-1920), would be a rather welcome and important event.

Prof. Dr. Hari M. Srivastava
Guest Editor

Keywords

  • q-series and q-polynomials
  • Rogers-Ramanujan continued fractions
  • Rogers-Ramanujan identities
  • q-hypergeometric functions
  • q-gamma and q-beta functions
  • Jacobi's triple-product identities
  • q-Jacobi polynomials
  • q-Laguerre polynomials
  • q-Hermite polynomials
  • q-Wilson polynomials

Published Papers (8 papers)

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Research

Open AccessArticle On Solutions of Holonomic Divided-Difference Equations on Nonuniform Lattices
Axioms 2013, 2(3), 404-434; doi:10.3390/axioms2030404
Received: 28 May 2013 / Revised: 4 June 2013 / Accepted: 3 July 2013 / Published: 23 July 2013
Cited by 4 | PDF Full-text (368 KB) | HTML Full-text | XML Full-text
Abstract
The main aim of this paper is the development of suitable bases that enable the direct series representation of orthogonal polynomial systems on nonuniform lattices (quadratic lattices of a discrete or a q-discrete variable). We present two bases of this type, the first
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The main aim of this paper is the development of suitable bases that enable the direct series representation of orthogonal polynomial systems on nonuniform lattices (quadratic lattices of a discrete or a q-discrete variable). We present two bases of this type, the first of which allows one to write solutions of arbitrary divided-difference equations in terms of series representations, extending results given by Sprenger for the q-case. Furthermore, it enables the representation of the Stieltjes function, which has already been used to prove the equivalence between the Pearson equation for a given linear functional and the Riccati equation for the formal Stieltjes function. If the Askey-Wilson polynomials are written in terms of this basis, however, the coefficients turn out to be not q-hypergeometric. Therefore, we present a second basis, which shares several relevant properties with the first one. This basis enables one to generate the defining representation of the Askey-Wilson polynomials directly from their divided-difference equation. For this purpose, the divided-difference equation must be rewritten in terms of suitable divided-difference operators developed in previous work by the first author. Full article
Open AccessArticle On the q-Analogues of Srivastava’s Triple Hypergeometric Functions
Axioms 2013, 2(2), 85-99; doi:10.3390/axioms2020085
Received: 28 February 2013 / Revised: 22 March 2013 / Accepted: 3 April 2013 / Published: 11 April 2013
Cited by 1 | PDF Full-text (207 KB) | HTML Full-text | XML Full-text
Abstract
We find Euler integral formulas, summation and reduction formulas for q-analogues of Srivastava’s three triple hypergeometric functions. The proofs use q-analogues of Picard’s integral formula for the first Appell function, a summation formula for the first Appell function based on the Bayley–Daum formula,
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We find Euler integral formulas, summation and reduction formulas for q-analogues of Srivastava’s three triple hypergeometric functions. The proofs use q-analogues of Picard’s integral formula for the first Appell function, a summation formula for the first Appell function based on the Bayley–Daum formula, and a general triple series reduction formula of Karlsson. Many of the formulas are purely formal, since it is difficult to find convergence regions for these functions of several complex variables. We use the Ward q-addition to describe the known convergence regions of q-Appell and q-Lauricella functions. Full article
Open AccessArticle Golden Ratio and a Ramanujan-Type Integral
Axioms 2013, 2(1), 58-66; doi:10.3390/axioms2010058
Received: 1 November 2012 / Revised: 2 March 2013 / Accepted: 5 March 2013 / Published: 20 March 2013
Cited by 1 | PDF Full-text (170 KB) | HTML Full-text | XML Full-text
Abstract In this paper, we give a pedagogical introduction to several beautiful formulas discovered by Ramanujan. Using these results, we evaluate a Ramanujan-type integral formula. The result can be expressed in terms of the Golden Ratio. Full article
Open AccessArticle Some Modular Relations Analogues to the Ramanujan’s Forty Identities with Its Applications to Partitions
Axioms 2013, 2(1), 20-43; doi:10.3390/axioms2010020
Received: 1 November 2012 / Revised: 22 January 2013 / Accepted: 28 January 2013 / Published: 18 February 2013
Cited by 1 | PDF Full-text (291 KB) | HTML Full-text | XML Full-text
Abstract
Recently, the authors have established a large class of modular relations involving the Rogers-Ramanujan type functions J(q) and K(q) of order ten. In this paper, we establish further modular relations connecting these two functions with Rogers-Ramanujan functions, Göllnitz-Gordon functions and cubic functions, which
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Recently, the authors have established a large class of modular relations involving the Rogers-Ramanujan type functions J(q) and K(q) of order ten. In this paper, we establish further modular relations connecting these two functions with Rogers-Ramanujan functions, Göllnitz-Gordon functions and cubic functions, which are analogues to the Ramanujan’s forty identities for Rogers-Ramanujan functions. Furthermore, we give partition theoretic interpretations of some of our modular relations. Full article
Open AccessArticle Generalized q-Stirling Numbers and Their Interpolation Functions
Axioms 2013, 2(1), 10-19; doi:10.3390/axioms2010010
Received: 21 November 2012 / Revised: 28 January 2013 / Accepted: 31 January 2013 / Published: 8 February 2013
Cited by 1 | PDF Full-text (164 KB) | HTML Full-text | XML Full-text
Abstract
In this paper, we define the generating functions for the generalized q-Stirling numbers of the second kind. By applying Mellin transform to these functions, we construct interpolation functions of these numbers at negative integers. We also derive some identities and relations related to
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In this paper, we define the generating functions for the generalized q-Stirling numbers of the second kind. By applying Mellin transform to these functions, we construct interpolation functions of these numbers at negative integers. We also derive some identities and relations related to q-Bernoulli numbers and polynomials and q-Stirling numbers of the second kind. Full article
Open AccessArticle Generating Functions for q-Apostol Type Frobenius–Euler Numbers and Polynomials
Axioms 2012, 1(3), 395-403; doi:10.3390/axioms1030395
Received: 30 October 2012 / Revised: 26 November 2012 / Accepted: 28 November 2012 / Published: 7 December 2012
Cited by 7 | PDF Full-text (158 KB) | HTML Full-text | XML Full-text
Abstract
The aim of this paper is to construct generating functions, related to nonnegative real parameters, for q-Eulerian type polynomials and numbers (or q-Apostol type Frobenius–Euler polynomials and numbers). We derive some identities for these polynomials and numbers based on the generating functions and
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The aim of this paper is to construct generating functions, related to nonnegative real parameters, for q-Eulerian type polynomials and numbers (or q-Apostol type Frobenius–Euler polynomials and numbers). We derive some identities for these polynomials and numbers based on the generating functions and functional equations. We also give multiplication formula for the generalized Apostol type Frobenius–Euler polynomials. Full article
Open AccessArticle The Cranks for 5-Core Partitions
Axioms 2012, 1(3), 372-383; doi:10.3390/axioms1030372
Received: 7 August 2012 / Revised: 21 September 2012 / Accepted: 26 November 2012 / Published: 3 December 2012
Cited by 1 | PDF Full-text (288 KB) | HTML Full-text | XML Full-text
Abstract
It is well known that the number of 5-core partitions of 5kn + 5k − 1 is a multiple of 5k. In [1] a statistic called a crank was developed to sort the 5-core partitions of 5n
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It is well known that the number of 5-core partitions of 5kn + 5k − 1 is a multiple of 5k. In [1] a statistic called a crank was developed to sort the 5-core partitions of 5n + 4 and 25n + 24 into 5 and 25 classes of equal size, respectively. In this paper we will develop the cranks that can be used to sort the 5-core partitions of 5kn + 5k − 1 into 5k classes of equal size. Full article
Open AccessArticle New Curious Bilateral q-Series Identities
Axioms 2012, 1(3), 365-371; doi:10.3390/axioms1030365
Received: 17 September 2012 / Revised: 9 October 2012 / Accepted: 17 October 2012 / Published: 31 October 2012
Cited by 1 | PDF Full-text (144 KB) | HTML Full-text | XML Full-text
Abstract
By applying a classical method, already employed by Cauchy, to a terminating curious summation by one of the authors, a new curious bilateral q-series identity is derived. We also apply the same method to a quadratic summation by Gessel and Stanton, and to
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By applying a classical method, already employed by Cauchy, to a terminating curious summation by one of the authors, a new curious bilateral q-series identity is derived. We also apply the same method to a quadratic summation by Gessel and Stanton, and to a cubic summation by Gasper, respectively, to derive a bilateral quadratic and a bilateral cubic summation formula. Full article

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