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We study bivariate Mittag-Leffler function which is an extension of several known Mittag-Leffler functions. We derive the Riemann-Liouville fractional derivatives and integrals of these function and solve a singular integral equation with the bivariate Mittag-Leffler function in the kernel. We also introduce and investigate a fractional integral operator involving the bivariate Mittag-Leffler function. [ABSTRACT FROM AUTHOR]

Recently, Casadei [4] provided an explicit formula for statistical marginal model in terms of Poisson-Gamma mixture. This model involving certain polynomials which play the key role in reference analysis of the signal and background model in counting experiments. The principal object of this paper is to present a natural further step toward the mathematical properties concerning this polynomials. We Ąrst obtain explicit representations for these polynomials in form of the Laguerre polynomials and the conĆuent hyper-geometric function and then based on these representations we derive a number of useful properties including generating functions, recurrence relations, differential equation, Rodrigues formula, Ąnite sums and integral transforms. [ABSTRACT FROM AUTHOR]

In this paper, we present several operator identities by using certain inverse pairs of symbolic operators. Using the obtained operator identities, we establish many decomposition formulas for Gaussian hypergeometric functions in three variables. Furthermore, some transformation formulas are introduced. [ABSTRACT FROM AUTHOR]

In this paper, the higher-order type 2 Daehee polynomials are introduced and some of their relations and properties are derived. Then, some p-adic integral representations of not only higher-order type 2 Daehee polynomials and numbers but also type 2 Daehee polynomials and numbers are acquired. Several identities and relations related to both central factorial numbers of the second kind and Stirling numbers of the first and second kinds are investigated. Moreover, the conjugate higher-order type 2 Daehee polynomials are considered and some correlations covering the type 2 Daehee polynomials of order β and the conjugate higher-order type 2 Daehee polynomials are attained. [ABSTRACT FROM AUTHOR]

In this paper, the higher-order type 2 Daehee polynomials are introduced and some of their relations and properties are derived. Then, some p-adic integral representations of not only higher-order type 2 Daehee polynomials and numbers but also type 2 Daehee polynomials and numbers are acquired. Several identities and relations related to both central factorial numbers of the second kind and Stirling numbers of the first and second kinds are investigated. Moreover, the conjugate higher-order type 2 Daehee polynomials are considered and some correlations covering the type 2 Daehee polynomials of order β and the conjugate higher-order type 2 Daehee polynomials are attained. [ABSTRACT FROM AUTHOR]

The main objective of this article is to establish generalized fractional Hermite–Hadamard and related type integral inequalities for h -Godunova–Levin convexity and h -Godunova–Levin preinvexity with extended Wright generalized Bessel function acting as kernel. Moreover, Hermite–Hadamard-type and trapezoid-type inequalities for several known convexities including Godunova–Levin function, classical convex, s -Godunova–Levin function, P -function, and s -convex function are deduced as corollaries. These obtained results are analyzed in the form of generalization of fractional inequalities. [ABSTRACT FROM AUTHOR]

By using the reciprocally inverse operators of the Burchnall and Chaundy type introduced by Choi and Hasanov in 2011, we aim at deriving certain decomposition formulas for some Gauss's triple hypergeometric functions. We also present some transformation formulas by using our decomposition formulas. [ABSTRACT FROM AUTHOR]

The authors establish a set of fifteen new integral representations of Euler-type for the Sharma and Parihar hypergeometric function in four variables F(4)20; whose kernels include the quadruple functions K1;K10; F(4)14 and X(4)8; the Exton hypergeometric functions of three variables X4; the Lauricella functions of three variable FE; FN and FR; the Appell's series of two variables F2; F3 and F4; and the Gauss hypergeometric function. [ABSTRACT FROM AUTHOR]

The authors establish a set of fifteen new integral representations of Euler-type for the Sharma and Parihar hypergeometric function in four variables F(4)20; whose kernels include the quadruple functions K1;K10; F(4)14 and X(4)8; the Exton hypergeometric functions of three variables X4; the Lauricella functions of three variable FE; FN and FR; the Appell's series of two variables F2; F3 and F4; and the Gauss hypergeometric function. [ABSTRACT FROM AUTHOR]

In the present work, we establish a new integral representations of Euler type for the quadruple hypergeometric functions Xi(4)(.), i = 38, 40, 45, 48, 50. We obtain some operational relations include these quadruple functions. [ABSTRACT FROM AUTHOR]

A new class of quadruple hypergeometric series is presented. We also give integral representations of Euler- type and Laplace-type for the new class of series. [ABSTRACT FROM AUTHOR]

While investigating the Exton's list of twenty one hyper-geometric functions of four variables and the Sharma's and Parihar's list of eighty three hyper-geometric functions of four variables, we noticed existence of new hyper-geometric series of four variables. The principal object of this paper is to introduce new hyper-geometric series of four variables and present a natural further step toward the mathematical integral presentation concerning these new series of four variables. Integral representations of Euler type and Laplace type involving Appell's hyper-geometric functions and the Horn's series of two variables, Exton's and Lauricella's triple functions and Sharma and Parihar hyper-geometric functions of four variables are established. [ABSTRACT FROM AUTHOR]

In this paper, we define some new quadruple hypergeometric functions, which we denoted by X(4)i (i = 31, 32, ..., 50). Then, we obtain its integral representations of Euler-type and Laplace-type. [ABSTRACT FROM AUTHOR]

While investigating the Exton's list of 21 hypergeometric functions of four variables namely K1,K2,. . .,K21 and the Sharma's and Parihar's list of eighty three hypergeometric functions of four variables namely F1(4), F2(4), . . .,F83(4)0, we noticed the existence of new hypergeometric series of four variables. In the present work, we first introduce five new quadruple series and then we give integral representations of Euler type and Laplace type for these new hypergeometric functions, which we denote by Xj(4)j (.), j = 1,2,3,4,5. [ABSTRACT FROM AUTHOR]