• 호남수학학술지
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• 제43권2호
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• pp.183-197
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• 2021

Recently several authors have extended the Beta function by using its integral representation. However, in many cases no expression of these extended functions in terms of classic special functions is known. In the present paper, we introduce a further extension by defining a family of functions G_r,s : ℝ^*₊ → ℂ, with r, s ∈ ℂ and ℜ(r) > 0. For given r, s, we prove that this function satisfies a second-order linear differential equation with rational coefficients. Solving this ODE, we express G_r,s as a combination of confluent hypergeometric functions. From this we deduce a new integral relation satisfied by Tricomi's function. We then investigate additional specific properties of G_r,1 which take the form of new non trivial integral relations involving exponential and error functions. We discuss the connection between G_r,1 and Stokes' first problem (or Rayleigh problem) in fluid mechanics which consists in determining the flow created by the movement of an infinitely long plate. For $r{\in}{\frac{1}{2}}{\mathbb{N}}^*$, we find additional relations between G_r,1 and Hermite polynomials. In view of these results, we believe the family of extended beta functions G_r,s will find further applications in two directions: (i) for improving our knowledge of confluent hypergeometric functions and Tricomi's function, (ii) and for engineering and physics problems.

• 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.

• Journal of applied mathematics & informatics
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• 제37권1_2호
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• pp.13-21
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• 2019

In the present research paper, we introduce a further extension of Hurwitz-Lerch zeta function by using the generalized extended Beta function defined by Parmar et al.. We investigate its integral representations, Mellin transform, generating functions and differential formula. In view of diverse applications of the Hurwitz-Lerch Zeta functions, the results presented here may be potentially useful in some related research areas.

• East Asian mathematical journal
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• 제35권1호
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• pp.23-30
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• 2019

Since Mittag-Leffler introduced the so-called Mittag-Leffler function, a number of its extensions have been investigated due mainly to their applications in a variety of research subjects. Shukla and Prajapati presented a lot of formulas involving a generalized Mittag-Leffler function in a systematic manner. Motivated mainly by Shukla and Prajapati's work, we aim to investigate a generalized multi-index Mittag-Leffler function and, among possible numerous formulas, choose to present several formulas involving this generalized multi-index Mittag-Leffler function such as a recurrence formula, derivative formula, three integral transformation formulas. The results presented here, being general, are pointed out to reduce to yield relatively simple formulas including known ones.

• 대한수학회보
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• 제50권5호
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• pp.1673-1681
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• 2013

The main objective of this paper is to show how one can obtain eleven new and interesting hypergeometric identities in the form of a single result from the old ones by mainly employing the known beta integral method which was recently introduced and used in a systematic manner by Krattenthaler and Rao [6]. The results are derived with the help of a generalization of a well-known hypergeometric transformation formula due to Kummer. Several identities including one obtained earlier by Krattenthaler and Rao [6] follow as special cases of our main results.

• 대한수학회논문집
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• 제30권2호
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• pp.81-92
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• 2015

In this paper, we aim at establishing a generalized fractional integral version of Gr$\ddot{u}$ss type integral inequality by making use of the Gauss hypergeometric function fractional integral operator. Our main result, being of a very general character, is illustrated to specialize to yield numerous interesting fractional integral inequalities including some known results.

• 충청수학회지
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• 제25권2호
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• pp.253-265
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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. Subse- quently, Khan and Asif investigated the generating functions for the $q$-analogue of Gottlieb polynomials. 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 ${\varphi}_{n}^{m}(\cdot)$. Here, we aim at defining a $q$-extension of the generalized two variable Gottlieb polynomials ${\varphi}_{n}^{2}(\cdot)$ and presenting their several generating functions.

• 호남수학학술지
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• 제34권3호
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• pp.467-472
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• 2012

The aim of this paper is to provide two interesting summation formulae with the argument unity for the generalized hypergeometric function of higher order. The results are obtained with the help of two new summation formulae very recently obtained by Kim et al.. Summation formulae obtained earlier by Carlson and re-derived by Exton turn out to be special cases of our main findings.

• 대한수학회논문집
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• 제36권4호
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• pp.671-684
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• 2021

The tenacity of the current paper is to find connections between various subclasses of analytic univalent functions by applying certain convolution operator involving generalized hypergeometric distribution series. To be more specific, we examine such connections with the classes of analytic univalent functions k - 𝓤𝓒𝓥^* (𝛽), k - 𝓢^*_p (𝛽), 𝓡 (𝛽), 𝓡^𝜏 (A, B), k - 𝓟𝓤𝓒𝓥^* (𝛽) and k - 𝓟𝓢^*_p (𝛽) in the open unit disc 𝕌.

• 대한수학회논문집
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• 제36권2호
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• pp.239-246
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• 2021

In this article, a study of generalized Bateman's matrix polynomials is presented. We obtained partial differential equations by using differential operators in the generalized Bateman's matrix polynomials for two variables. Then we introduced some different recurrence relationships of the generalized Bateman's matrix polynomials. Finally present the relationship between the generalized Bateman's matrix polynomials of one and two variables.