• Honam Mathematical Journal
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• v.37 no.1
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• pp.113-126
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• 2015

Recently, Liu and Wang generalized Appell's and Lauricella's hypergeometric functions. Motivated by the work of Liu and Wang, the main object of this paper is to present new generalizations of Appell's and Lauricella's hypergeometric functions. Some integral representations, transformation formulae, differential formulae and recurrence relations are obtained for these new generalized Appell's and Lauricella's functions.

• Communications of the Korean Mathematical Society
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• v.32 no.2
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• pp.321-332
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• 2017

In this paper, we generalize the extended Appell's and Lauricella's hypergeometric functions which have recently been introduced by Liu [9] and Khan [7]. Also, we aim at establishing some (presumbly) new integral representations and transforms for the extended generalized Appell's and Lauricella's hypergeometric functions.

• Communications of the Korean Mathematical Society
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• v.32 no.4
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• pp.835-850
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• 2017

A remarkably large number of special functions (such as the Gamma and Beta functions, the Gauss hypergeometric function, and so on) have been investigated by many authors. Motivated the works of both works of both Burchnall and Chaundy and Chaundy and very recently, Brychkov and Saad gave interesting generalizations of Appell type functions. In the present sequel to the aforementioned investigations and some of the earlier works listed in the reference, we present some new differential formulas for the generalized Appell's type functions ${\kappa}_i$, $i=1,2,{\ldots},18$ by considering the product of two $_4F_3$ functions.

• Communications of the Korean Mathematical Society
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• v.33 no.4
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• pp.1239-1248
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• 2018

Recently, many authors have obtained several hypergeometric identities involving hypergeometric functions of one and multi-variables such as the Appell's functions and Horn's functions. In this paper, we obtain several new transformations suitably by applying the fractional calculus operator to these hypergeometric identities, which was introduced recently by Tremblay.

• Communications of the Korean Mathematical Society
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• v.30 no.4
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• pp.403-414
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• 2015

Extensions of some classical special functions, for example, Beta function B(x, y) and generalized hypergeometric functions $_pF_q$ have been actively investigated and found diverse applications. In recent years, several extensions for B(x, y) and $_pF_q$ have been established by many authors in various ways. Here, we aim to generalize Appell's hypergeometric functions of two variables and Lauricella's hypergeometric function of three variables by using the extended generalized beta type function $B_p^{({\alpha},{\beta};m)}$ (x, y). Then some properties of the extended generalized Appell's hypergeometric functions and Lauricella's hypergeometric functions are investigated.

• Communications of the Korean Mathematical Society
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• v.30 no.3
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• pp.239-252
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• 2015

Motivated mainly by certain interesting extensions of the ${\tau}$-hypergeometric function defined by Virchenko et al. [11] and some ${\tau}$-Appell's function introduced by Al-Shammery and Kalla [1], we introduce here the ${\tau}$-Lauricella functions $F_A^{(n),{\tau}_1,{\cdots},{\tau}_n}$, $F_B^{(n),{\tau}_1,{\cdots},{\tau}_n}$ and $F_D^{(n),{\tau}_1,{\cdots},{\tau}_n}$ and the confluent forms ${\Phi}_2^{(n),{\tau}_1,{\cdots},{\tau}_n}$ and ${\Phi}_D^{(n),{\tau}_1,{\cdots},{\tau}_n}$ of n variables. We then systematically investigate their various integral representations of each of these ${\tau}$-Lauricella functions including their generating functions. Various (known or new) special cases and consequences of the results presented here are also considered.

• Communications of the Korean Mathematical Society
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• v.19 no.1
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• pp.75-92
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• 2004

A generalised integration by parts formula for sequences of absolutely continuous functions that satisfy the ${\omega}-Appell$ condition and different estimates for the remainder are provided. Applications for particular instances of such sequences are pointed out as well.

• Bulletin of the Korean Mathematical Society
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• v.60 no.3
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• pp.575-591
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• 2023

Motivated by several generalizations of the Pochhammer symbol and their associated families of hypergeometric functions and hypergeometric polynomials, by choosing to use a very generalized Pochhammer symbol, we aim to introduce certain extensions of the generalized Lauricella function F⁽ⁿ⁾_A and the Humbert's confluent hypergeometric function Ψ⁽ⁿ⁾ of n variables with, as their respective particular cases, the second Appell hypergeometric function F₂ and the generalized Humbert's confluent hypergeometric functions Ψ₂ and investigate their several properties including, for example, various integral representations, finite summation formulas with an s-fold sum and integral representations involving the Laguerre polynomials, the incomplete gamma functions, and the Bessel and modified Bessel functions. Also, pertinent links between the major identities discussed in this article and different (existing or novel) findings are revealed.

• The Pure and Applied Mathematics
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• v.20 no.1
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• pp.37-50
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• 2013

In the theory of hypergeometric functions of one or several variables, a remarkable amount of mathematicians's concern has been given to develop their transformation formulas and summation identities. Here we aim at presenting explicit expressions (in a single form) of the following weighted Appell's function $F_1$: $$(1+2x)^{-a}(1+2z)^{-b}F_1\;\(c,\;a,\;b;\;2c+j;\;\frac{4x}{1+2x},\;\frac{4z}{1+2z}\)\;(j=0,\;{\pm}1,\;{\ldots},\;{\pm}5)$$ in terms of Exton's triple hypergeometric $X_9$. The results are derived with the help of generalizations of Kummer's second theorem very recently provided by Kim et al. A large number of very interesting special cases including Exton's result are also given.

• Communications of the Korean Mathematical Society
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• v.38 no.3
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• pp.807-819
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• 2023

In 2019, Mathur and Solanki [7, 8] obtained a few transformation formulas for Appell, Horn and the Kampé de Fériet functions. Unfortunately, some of the results are well-known and very old results in literature while others are erroneous. Thus the aim of this note is to provide the results in corrected forms and some of the results have been written in more compact form.