A Note on the 2F1{2}F{1}2​F1​ Hypergeometric Function (original) (raw)

On convergence of basic hypergeometric series

arXiv: Classical Analysis and ODEs, 2015

is the q-Pochhammer symbol. Here a1, . . . , ar, b1, . . . , bs and q are complex parameters. In this paper we always assume (1) aiq n 6= 1 and bjq 6= 1 (i = 1, . . . , r, j = 1, . . . , s, n = 0, 1, 2, . . .) so that the factors (ai; q)n and (bj; q)n in the terms of the series are never zero. Let vn be the terms of the series rφs which contain z . Then we have vn+1 vn = (1− a1q)(1− a2q) · · · (1 − arq) (1 − qn+1)(1 − b1q) · · · (1− bsq) (−q)z

Some properties of the hypergeometric functions of one variable

2011

In this paper we introduced the novel concept of basic hypergeometric series and the hypergeometric function. We express many of common mathematical function in terms of the hypergeometric function. Gauss’ contigous relations, some integral formulas, Recurrence relations, transformation formulas, values at the special points.

Some expansions of hypergeometric functions in series of hypergeometric functions†

Glasgow Mathematical Journal, 1976

Throughout the present note we abbreviate the set of p parameters a1,…,ap by (ap), with similar interpretations for (bq), etc. Also, by [(ap)]m we mean the product , where [λ]m = Г(λ + m)/ Г(λ), and so on. One of the main results we give here is the expansion formula(1)which is valid, by analytic continuation, when, p,q,r,s,t and u are nonnegative integers such that p+r < q+s+l (or p+r = q+s+l and |zω| <1), p+t < q+u (or p + t = q + u and |z| < 1), and the various parameters including μ are so restricted that each side of equation (1) has a meaning.

An analytic method for convergence acceleration of certain hypergeometric series

Mathematics of Computation, 1995

A method is presented for convergence acceleration of the generalized hypergeometric series 3F2 with the argument ±1 , using analytic properties of their terms. Iterated transformation of the series is performed analytically, which results in obtaining new fast converging expansions for some special functions and mathematical constants.

Properties of a Newly Defined Hypergeometric Power Series Function

2009

Ahmad (2007b) has recently defined a generalized hypergeometric series function and referred to it as a hypergeometric power series function or rs H -function which is an alternative notation for the rs F -function, the rs H notation has advantages when the arguments are large and parameters are repeated and discussed the some basic properties. In this paper further properties of the hypergeometric power series functions have been developed.

Summation of some infinite series by the methods of Hypergeometric functions and partial fractions

BULLETIN OF THE KARAGANDA UNIVERSITY-MATHEMATICS

In this article we obtain the summations of some infinite series by partial fraction method and by using certain hypergeometric summation theorems of positive and negative unit arguments, Riemann Zeta functions, polygamma functions, lower case beta functions of one-variable and other associated functions. We also obtain some hypergeometric summation theorems for: 8F7[9/2, 3/2, 3/2, 3/2, 3/2, 3, 3, 1; 7/2, 7/2, 7/2, 7/2, 1/2, 2, 2; 1], 5F4[5/3, 4/3, 4/3, 1/3, 1/3; 2/3, 1, 2, 2; 1], 5F4[9/4, 5/2, 3/2, 1/2, 1/2; 5/4, 2, 3, 3; 1], 5F4[13/8, 5/4, 5/4, 1/4, 1/4; 5/8, 2, 2, 1; 1], 5F4[1/2, 1/2, 5/2, 5/2, 1; 3/2, 3/2, 7/2, 7/2; −1], 4F3[3/2, 3/2, 1, 1; 5/2, 5/2, 2; 1], 4F3[2/3, 1/3, 1, 1; 7/3, 5/3, 2; 1], 4F3[7/6, 5/6, 1, 1; 13/6, 11/6, 2; 1] and 4F3[1, 1, 1, 1; 3, 3, 3; −1].