In this paper, a computational technique is given to re-obtain the explicit forms of two cases of the Gauss hypergeometric function ) ; , , ( 1 2 x c b a F for 2 / 1 + = a b and 2 / 3 , 2 / 1 = c . Some special identities related to the two aforesaid cases are also introduced. Finally a special Maple package, called FormalPowerSeries (FPS), is used to automatically compute some results given in...

Abstract The main aim of this article is to study a new generalizations the Gauss hypergeometric matrix and confluent functions by using two-parameter Mittag–Leffler function. In particular, we investigate certain important properties these extended such as integral representations, differentiation formulas, beta transform, Laplace transform. Furthermore, introduce an extension Jacobi orthogona...

The aim of this research note is to prove the following new transformation formula begin{equation*} (1-x)^{-2a},_{3}F_{2}left[begin{array}{ccccc} a, & a+frac{1}{2}, & d+1 & & \ & & & ; & -frac{4x}{(1-x)^{2}} \ & c+1, & d & & end{array}right] \ =,_{4}F_{3}left[begin{array}{cccccc} 2a, & 2a-c, & a-A+1, & a+A+1 & & \ & & & & ; & -x \ & c+1, & a-A, & a+A & & end{array} right], end{equation*} wher...

The main objective of this paper is to use the newly proposed $(p,q;l)$-extended beta function introduce $τ$-Gauss hypergeometric and $τ$-confluent functions with some their properties, such as Laplace-type Euler-type integral formulas. Another apply them fractional kinetic equations that appear in astrophysics physics using Laplace transform method.

Theorems in the theory of partitions are closely related to basic hypergeometric series. Some identities arising in basic hypergeometric series can be interpreted in the theory of partitions using Fpartitions. In this paper, Ramanujan’s 1ψ1 summation and the q-Gauss summation are established combinatorially.

It is shown that Ramanujan’s cubic transformation of the Gauss hypergeometric function 2F1 arises from a relation between modular curves, namely the covering of X0(3) by X0(9). In general, when 2 N 7, the N-fold cover of X0(N) by X0(N) gives rise to an algebraic hypergeometric transformation. The N = 2, 3, 4 transformations are arithmetic–geometric mean iterations, but the N = 5, 6, 7 transform...

The present paper establishes several new integral representations of the Euler type and Laplace for some Gauss hypergeometric functions three variables. main results are obtained by using properties Gamma beta functions. novel carried out through ten Therefore, all derived integrals generalization representation classical function one two In addition, numerical examples were given to describe ...