Abstract
The classical Gaussian $_2F_1$-series containing two free variables $\{x,y\}$ and two integer parameters $\{m,n\}$ are investigated by the linearization method. Several closed formulae are derived in terms of trigonometric functions. Some of them are lifted up, via a trigonometric integral approach, to identities of nonterminating $_3F_2$-series.
Highlights
Introduction and motivation Let Z andN be the sets of integers and natural numbers with N0 = {0} ∪ N
Let Z and N be the sets of integers and natural numbers with N0 = {0} ∪ N
The aim of this paper is to investigate, by the linearization method, the following Gaussian 2F1 -series with two free complex variables {x, y} :
Summary
∑λ (A + k)λ = (B + k)i(C + k)λ−iUλi i=0 where Uλi is independent of the variable k and given by the following expression: () Proof Substituting (3) into (2), we can express the resulting binomial sum in terms of hypergeometric series and evaluate it as follows:
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