• Kyungpook Mathematical Journal
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• 제57권3호
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• pp.393-400
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• 2017

Very recently, Srivastava et al. [8] introduced an extension of the Pochhammer symbol and used it to define a generalization of the generalized hypergeometric functions. In this paper, by using the generalized Pochhammer symbol, we extend the generalized Hurwitz-Lerch Zeta function by Goyal and Laddha [6] and investigate some interesting properties which include various integral representations, Mellin transforms, differential formula and generating function. Some interesting special cases of our main results are also considered.

• 호남수학학술지
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• 제45권4호
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• pp.682-688
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• 2023

The main object of this paper is to establish twenty definite integrals involving Struve functions and modified Srtuve functions associated with hypergeometric functions, which are presumably new and not present in the scientific literature.

• Kyungpook Mathematical Journal
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• 제60권4호
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• pp.731-752
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• 2020

We introduce the p-deformation of generalized Humbert polynomials. For these polynomials, we derive the differential equation, generating function relations, Fibonacci-type representations, and recurrence relations and state the companion matrix. These properties are illustrated for certain polynomials belonging to p-deformed generalized Humbert polynomials.

• East Asian mathematical journal
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• 제33권5호
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• pp.527-531
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• 2017

We aim to prove a known summation formula for the series $_4F_3(1)$ by mainly using a similar method as in [2], which is different from that in [3]. The method of proof here as well as that in [2] is potentially useful in getting some other summation formulas for $_pF_q$.

• 호남수학학술지
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• 제40권1호
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• pp.163-186
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• 2018

The objective of this paper is to present 87 recurrence relations for the Jacobi polynomials $P_n^{({\alpha},{\beta})}(x)$. The results presented here most of which are presumably new are obtained with the help of Gauss's fifteen contiguous function relations and some other identities recently recorded in the literature.

• 대한수학회논문집
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• 제34권4호
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• pp.1175-1199
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• 2019

We construct a general bi-basic inverse series relation which provides extension to several q-polynomials including the Askey-Wilson polynomials and the q-Racah polynomials. We introduce a general class of polynomials suggested by this general inverse pair which would unify certain polynomials such as the q-extended Jacobi polynomials and q-Konhauser polynomials. We then emphasize on applications of the general inverse pair and obtain the generating function relations, summation formulas involving the associated polynomials and derive the p-deformation of some of the q-analogues of Riordan's classes of inverse series relations. We also illustrate the companion matrix corresponding to the general class of polynomials; this is followed by a chart showing the reducibility of the extended p-deformed Askey-Wilson polynomials as well as the extended p-deformed q-Racah polynomials.

• 대한수학회보
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• 제33권1호
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• pp.107-110
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• 1996

Usng the Pochhammer symbol $(\lambda)_n$ given by $$ (1.1) (\lambda)_n = {1, if n = 0 {\lambda(\lambda + 1) \cdots (\lambda + n - 1), if n \in N = {1, 2, 3, \ldots}, $$ we define a general double hypergeometric series by [3, pp.27] $$ (1.2) F_{q:s;\upsilon}^{p:r;u} [\alpha_1, \ldots, \alpha_p : \gamma_1, \ldots, \gamma_r; \lambda_1, \ldots, \lambda_u;_{x,y}][\beta_1, \ldots, \beta_q : \delta_1, \ldots, \delta_s; \mu_1, \ldots, \mu_v; ] = \sum_{l,m = 0}^{\infty} \frac {\prod_{j=1}^{q} (\beta_j)_{l+m} \prod_{j=1}^{s} (\delta_j)_l \prod_{j=1}^{v} (\mu_j)_m)}{\prod_{j=1}^{p} (\alpha_j)_{l+m} \prod_{j=1}^{r} (\gamma_j)_l \prod_{j=1}^{u} (\lambda_j)_m} \frac{l!}{x^l} \frac{m!}{y^m} $$ provided that the double series converges.

• 호남수학학술지
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• 제34권4호
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• pp.603-614
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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. In this sequel, by modifying Khan and Akhlaq's method, Choi presented a generalization of the Gottlieb polynomials in $m$ variables to present two generating functions of the generalized Gottlieb polynomials ${\varphi}^m_n({\cdot})$. Here, we show that many formulas regarding the Gottlieb polynomials in m variables and their reducible cases can easily be obtained by using one of two generating functions for Choi's generalization of the Gottlieb polynomials in m variables expressed in terms of well-developed Lauricella series $F^{(m)}_D[{\cdot}]$.

• 호남수학학술지
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• 제34권3호
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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.

• 대한수학회논문집
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• 제28권2호
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• pp.297-301
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• 2013

When certain general single or multiple hypergeometric functions were introduced, their reduction formulas have naturally been investigated. Here, in this paper, we aim at presenting a very interesting reduction formula for the Srivastava's triple hypergeometric function $F^{(3)}[x,y,z]$ by applying the so-called Beta integral method to the Henrici's triple product formula for hypergeometric series.