On convergence of basic hypergeometric series (original) (raw)

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.

Dn Basic Hypergeometric Series

Ramanujan Journal, 1999

We study multiple series extensions of basic hypergeometric series related to the root system Dn. We make a small change in the notation used for Cn and Dn series to bring them closer to An series. This allows us to combine the three types of series, and get Dn extensions of the following classical summation and transformation theorems: The q-Pfaff-Saalschütz summation, Rogers' 6 φ5 sum, the q-Gauss summation, q-Chu-Vandermonde summations, Watson's q-analogue of Whipple's transformation, and the q-Dougall summation theorem. We also define An and Cn extensions of the Rogers-Selberg function, and prove a reduction formula for both of them. This generalizes some work of Andrews. We use some techniques originally developed to study multiple basic hypergeometric series related to the root system An (U(n + 1) basic hypergeometric series).

A Note on the 2F1 Hypergeometric Function

2010

The special case of the hypergeometric function 2F1_{2}F_{1}2F1 represents the binomial series (1+x)alpha=sumn=0infty(:alphan:)xn(1+x)^{\alpha}=\sum_{n=0}^{\infty}(\:\alpha n\:)x^{n}(1+x)alpha=sumn=0infty(:alphan:)xn that always converges when ∣x∣<1|x|<1∣x∣<1. Convergence of the series at the endpoints, x=pm1x=\pm 1x=pm1, depends on the values of alpha\alphaalpha and needs to be checked in every concrete case. In this note, using new approach, we reprove the convergence of the hypergeometric series 2F1(alpha,beta;beta;x)_{2}F_{1}(\alpha,\beta;\beta;x)2F1(alpha,beta;beta;x) for ∣x∣<1|x|<1∣x∣<1 and obtain new result on its convergence at point x=−1x=-1x=−1 for every integer alphaneq0\alpha\neq 0alphaneq0. The proof is within a new theoretical setting based on the new method for reorganizing the integers and on the regular method for summation of divergent series.

An extension of Pochhammer’s symbol and its application to hypergeometric functions, II

Filomat, 2018

Recently we have introduced a productive form of gamma and beta functions and applied them for generalized hypergeometric series [Filomat, 31 (2017), 207-215]. In this paper, we define an additive form of gamma and beta functions and study some of their general properties in order to obtain a new extension of the Pochhammer symbol. We then apply the new symbol for introducing two different types of generalized hypergeometric functions. In other words, based on the defined additive beta function, we first introduce an extension of Gauss and confluent hypergeometric series and then, based on two additive types of the Pochhammer symbol, we introduce two extensions of generalized hypergeometric functions of any arbitrary order. The convergence of each series is studied separately and some illustrative examples are given in the sequel.

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.

An extension of Pochhammer’s symbol and its application to hypergeometric functions

Filomat, 2017

By using a special property of the gamma function, we first define a productive form of gamma and beta functions and study some of their general properties in order to define a new extension of the Pochhammer symbol. We then apply this extended symbol for generalized hypergeometric series and study the convergence problem with some illustrative examples in this sense. Finally, we introduce two new extensions of Gauss and confluent hypergeometric series and obtain some of their general properties.