2,513 research outputs found
Certain transformations and summations for generalized hypergeometric series with integral parameter differences
Certain transformation and summation formulas for generalized hypergeometric series with integral parameter differences are derived
Certain summation and transformation formulas for generalized hypergeometric series
AbstractWe derive summation formulas for generalized hypergeometric series of unit argument, one of which upon specialization reduces to Minton’s summation theorem. As an application we deduce a reduction formula for a certain Kampé de Fériet function that in turn provides a Kummer-type transformation formula for the generalized hypergeometric function pFp(x)
An alternative proof of the extended SaalschĂĽtz summation theorem for the <sub>r + 3</sub>F<sub>r + 2</sub>(1) series with applications
A simple proof is given of a new summation formula recently added in the literature for a terminating r + 3Fr + 2(1) hypergeometric series for the case when r pairs of numeratorial and denominatorial parameters differ by positive integers. This formula represents an extension of the well-known Saalschütz summation formula for a 3F2(1) series. Two applications of this extended summation formula are discussed. The first application extends two identities given by Ramanujan and the second, which also employs a similar extension of the Vandermonde–Chu summation theorem for the 2F1 series, extends certain reduction formulas for the Kampé de Fériet function of two variables given by Exton and Cvijović & Miller
Transformation formulas for the generalized hypergeometric function with integral parameter differences
Transformation formulas of Euler and Kummer-type are derived respectively for the generalized hypergeometric functions r+2Fr+1(x) and r+1Fr+1(x), where r pairs of numeratorial and denominatorial parameters differ by positive integers. Certain quadratic transformations for the former function, as well as a summation theorem when x = 1, are also considered.<br/
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