2,513 research outputs found

    Certain transformations and summations for generalized hypergeometric series with integral parameter differences

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    Certain transformation and summation formulas for generalized hypergeometric series with integral parameter differences are derived

    Certain summation and transformation formulas for generalized hypergeometric series

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    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

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    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 &amp; Miller

    Transformation formulas for the generalized hypergeometric function with integral parameter differences

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    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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