• Title/Summary/Keyword: Hypergeometric function

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CERTAIN NEW INTEGRAL FORMULAS INVOLVING THE GENERALIZED BESSEL FUNCTIONS

  • Choi, Junesang;Agarwal, Praveen;Mathur, Sudha;Purohit, Sunil Dutt
    • Bulletin of the Korean Mathematical Society
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    • v.51 no.4
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    • pp.995-1003
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    • 2014
  • A remarkably large number of integral formulas involving a variety of special functions have been developed by many authors. Also many integral formulas involving various Bessel functions have been presented. Very recently, Choi and Agarwal derived two generalized integral formulas associated with the Bessel function $J_{\nu}(z)$ of the first kind, which are expressed in terms of the generalized (Wright) hypergeometric functions. In the present sequel to Choi and Agarwal's work, here, in this paper, we establish two new integral formulas involving the generalized Bessel functions, which are also expressed in terms of the generalized (Wright) hypergeometric functions. Some interesting special cases of our two main results are presented. We also point out that the results presented here, being of general character, are easily reducible to yield many diverse new and known integral formulas involving simpler functions.

A GRÜSS TYPE INTEGRAL INEQUALITY ASSOCIATED WITH GAUSS HYPERGEOMETRIC FUNCTION FRACTIONAL INTEGRAL OPERATOR

  • Choi, Junesang;Purohit, Sunil Dutt
    • Communications of the Korean Mathematical Society
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    • v.30 no.2
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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.

CERTAIN HYPERGEOMETRIC IDENTITIES DEDUCIBLE BY USING THE BETA INTEGRAL METHOD

  • Choi, Junesang;Rathie, Arjun K.;Srivastava, Hari M.
    • Bulletin of the Korean Mathematical Society
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    • v.50 no.5
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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.

ON EULERIAN q-INTEGRALS FOR SINGLE AND MULTIPLE q-HYPERGEOMETRIC SERIES

  • Ernst, Thomas
    • Communications of the Korean Mathematical Society
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    • v.33 no.1
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    • pp.179-196
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    • 2018
  • In this paper we extend the two q-additions with powers in the umbrae, define a q-multinomial-coefficient, which implies a vector version of the q-binomial theorem, and an arbitrary complex power of a JHC power series is shown to be equivalent to a special case of the first q-Lauricella function. We then present several q-analogues of hypergeometric integral formulas from the two books by Exton and the paper by Choi and Rathie. We also find multiple q-analogues of hypergeometric integral formulas from the recent paper by Kim. Finally, we prove several multiple q-hypergeometric integral formulas emanating from a paper by Koschmieder, which are special cases of more general formulas by Exton.

SOME COMPOSITION FORMULAS OF JACOBI TYPE ORTHOGONAL POLYNOMIALS

  • Malik, Pradeep;Mondal, Saiful R.
    • Communications of the Korean Mathematical Society
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    • v.32 no.3
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    • pp.677-688
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    • 2017
  • The composition of Jacobi type finite classes of the classical orthogonal polynomials with two generalized Riemann-Liouville fractional derivatives are considered. The outcomes are expressed in terms of generalized Wright function or generalized hypergeometric function. Similar composition formulas are also obtained by considering the generalized Riemann-Liouville and $Erd{\acute{e}}yi-Kober$ fractional integral operators.

Inference on the reliability P(Y < X) in the gamma case

  • Moon, Yeung-Gil;Lee, Chang-Soo
    • Journal of the Korean Data and Information Science Society
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    • v.20 no.1
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    • pp.219-223
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    • 2009
  • We shall derive a quotient distribution of two independent gamma variables and its moment and reliability are represented by hypergeometric function and Wittaker's function. And we shall consider an inference on the reliability in two independent gamma random variables.

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NEW RESULTS FOR THE SERIES 2F2(x) WITH AN APPLICATION

  • Choi, Junesang;Rathie, Arjun Kumar
    • Communications of the Korean Mathematical Society
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    • v.29 no.1
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    • pp.65-74
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    • 2014
  • The well known quadratic transformation formula due to Gauss: $$(1-x)^{-2a}{_2F_1}\[{{a,b;}\\\hfill{21}{2b;}}\;-\frac{4x}{(1-x)^2}\]={_2F_1}\[{{a,a-b+\frac{1}{2};}\\\hfill{65}{b+\frac{1}{2};}}\;x^2\]$$ plays an important role in the theory of (generalized) hypergeometric series. In 2001, Rathie and Kim have obtained two results closely related to the above quadratic transformation for $_2F_1$. Our main objective of this paper is to deduce some interesting known or new results for the series $_2F_1(x)$ by using the above Gauss's quadratic transformation and its contiguous relations and then apply our results to provide a list of a large number of integrals involving confluent hypergeometric functions, some of which are (presumably) new. The results established here are (potentially) useful in mathematics, physics, statistics, engineering, and so on.

SOME INTEGRALS ASSOCIATED WITH MULTIINDEX MITTAG-LEFFLER FUNCTIONS

  • KHAN, N.U.;USMAN, T.;GHAYASUDDIN, M.
    • Journal of applied mathematics & informatics
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    • v.34 no.3_4
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    • pp.249-255
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    • 2016
  • The object of the present paper is to establish two interesting unified integral formulas involving Multiple (multiindex) Mittag-Leffler functions, which is expressed in terms of Wright hypergeometric function. Some deduction from these results are also considered.

SOME SUBORDINATION PROPERTIES OF THE LINEAR OPERATOR

  • PANIGRAHI, TRAILOKYA
    • Journal of the Korean Mathematical Society
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    • v.53 no.1
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    • pp.147-159
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    • 2016
  • In this paper, subordination results of analytic function $f{\in}{\mathcal{A}}_p$ involving linear operator ${\mathcal{K}}^{{\delta},{\lambda}}_{c,p}$ are obtained. By applying the differential subordination method, results are derived under some sufficient subordination conditions. On using some hypergeometric identities, corollaries of the main results are derived. Furthermore, convolution preserving properties for a class of multivalent analytic function associated with the operator ${\mathcal{K}}^{{\delta},{\lambda}}_{c,p}$ are investigated.