• Title, Summary, Keyword: basic hypergeometric series

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ON FOUR NEW MOCK THETA FUNCTIONS

  • Hu, QiuXia
    • Bulletin of the Korean Mathematical Society
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    • v.57 no.2
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    • pp.345-354
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    • 2020
  • In this paper, we first give some representations for four new mock theta functions defined by Andrews [1] and Bringmann, Hikami and Lovejoy [5] using divisor sums. Then, some transformation and summation formulae for these functions and corresponding bilateral series are derived as special cases of 2��2 series $${\sum\limits_{n=-{{\infty}}}^{{\infty}}}{\frac{(a,c;q)_n}{(b,d;q)_n}}z^n$$ and Ramanujan's sum $${\sum\limits_{n=-{{\infty}}}^{{\infty}}}{\frac{(a;q)_n}{(b;q)_n}}z^n$$.

q-EXTENSION OF A GENERALIZATION OF GOTTLIEB POLYNOMIALS IN THREE VARIABLES

  • Choi, June-Sang
    • Honam Mathematical Journal
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    • v.34 no.3
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    • pp.327-340
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    • 2012
  • Gottlieb polynomials were introduced and investigated in 1938, and then have been cited in several articles. Very recently Khan and Akhlaq introduced and investigated Gottlieb polynomials in two and three variables to give their generating functions. Subsequently, Khan and Asif investigated the generating functions for the $q$-analogue of Gottlieb polynomials. Very recently, Choi defined a $q$-extension of the generalized two variable Gottlieb polynomials ${\varphi}^2_n({\cdot})$ and presented their several generating functions. Also, by modifying Khan and Akhlaq's method, Choi presented a generalization of the Gottlieb polynomials in m variables to give two generating functions of the generalized Gottlieb polynomials ${\varphi}^m_n({\cdot})$. Here, in the sequel of the above results for their possible general $q$-extensions in several variables, again, we aim at trying to define a $q$-extension of the generalized three variable Gottlieb polynomials ${\varphi}^3_n({\cdot})$ and present their several generating functions.

The Fourth and Eighth Order Mock Theta Functions

  • Srivastava, Bhaskar
    • Kyungpook Mathematical Journal
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    • v.50 no.1
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    • pp.165-175
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    • 2010
  • In the paper we consider deemed three mock theta functions introduced by Hikami. We have given their alternative expressions in double summation analogous to Hecke type expansion. In proving we also give a new Bailey pair relative to $a^2$. I presume they will be helpful in getting fundamental transformations.

Modular Tranformations for Ramanujan's Tenth Order Mock Theta Functions

  • Srivastava, Bhaskar
    • Kyungpook Mathematical Journal
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    • v.45 no.2
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    • pp.211-220
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    • 2005
  • In this paper we obtain the transformations of the Ramanujan's tenth order mock theta functions under the modular group generators ${\tau}\;{\rightarrow}\;{\tau}\;+\;1\;and\;{\tau}\;{\rightarrow}\;-1/ {\tau}\;where\;q\;=\;e^{{\pi}it}$.

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QUANTUM MODULARITY OF MOCK THETA FUNCTIONS OF ORDER 2

  • Kang, Soon-Yi
    • Korean Journal of Mathematics
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    • v.25 no.1
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    • pp.87-97
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    • 2017
  • In [9], we computed shadows of the second order mock theta functions and showed that they are essentially same with the shadow of a mock theta function related to the Mathieu moonshine phenomenon. In this paper, we further survey the second order mock theta functions on their quantum modularity and their behavior in the lower half plane.

A Note on Certain Properties of Mock Theta Functions of Order Eight

  • Srivastava, Pankaj;Wahidi, Anwar Jahan
    • Kyungpook Mathematical Journal
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    • v.54 no.2
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    • pp.249-262
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    • 2014
  • In this paper, we have developed a non-homogeneous q-difference equation of first order for the generalized Mock theta function of order eight and besides these established limiting case of Mock theta functions of order eight. We have also established identities for Partial Mock theta function and Mock theta function of order eight and provided a number of cases of the identities.

q-SOBOLEV ORTHOGONALITY OF THE q-LAGUERRE POLYNOMIALS {Ln(-N)(·q)}n=0 FOR POSITIVE INTEGERS N

  • Moreno, Samuel G.;Garcia-Caballe, Esther M.
    • Journal of the Korean Mathematical Society
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    • v.48 no.5
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    • pp.913-926
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    • 2011
  • The family of q-Laguerre polynomials $\{L_n^{(\alpha)}({\cdot};q)\}_{n=0}^{\infty}$ is usually defined for 0 < q < 1 and ${\alpha}$ > -1. We extend this family to a new one in which arbitrary complex values of the parameter ${\alpha}$ are allowed. These so-called generalized q-Laguerre polynomials fulfil the same three term recurrence relation as the original ones, but when the parameter ${\alpha}$ is a negative integer, no orthogonality property can be deduced from Favard's theorem. In this work we introduce non-standard inner products involving q-derivatives with respect to which the generalized q-Laguerre polynomials $\{L_n^{(-N)}({\cdot};q)\}_{n=0}^{\infty}$, for positive integers N, become orthogonal.

A STUDY OF Q-CONTIGUOUS FUNCTION RELATIONS

  • Harsh, Harsh Vardhan;Kim, Yong Sup;Rakha, Medhat Ahmed;Rathie, Arjun Kumar
    • Communications of the Korean Mathematical Society
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    • v.31 no.1
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    • pp.65-94
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    • 2016
  • In 1812, Gauss obtained fifteen contiguous functions relations. Later on, 1847, Henie gave their q-analogue. Recently, good progress has been done in finding more contiguous functions relations by employing results due to Gauss. In 1999, Cho et al. have obtained 24 new and interesting contiguous functions relations with the help of Gauss's 15 contiguous relations. In fact, such type of 72 relations exists and therefore the rest 48 contiguous functions relations have very recently been obtained by Rakha et al.. Thus, the paper is in continuation of the paper [16] published in Computer & Mathematics with Applications 61 (2011), 620.629. In this paper, first we obtained 15 q-contiguous functions relations due to Henie by following a different method and then with the help of these 15 q-contiguous functions relations, we obtain 72 new and interesting q-contiguous functions relations. These q-contiguous functions relations have wide applications.