2019Functiones et Approximatio Commentarii MathematiciOpen access

Summation identities and transformations for hypergeometric series -- II

Rupam Barman, Neelam Saikia

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Abstract

McCarthy defined a function in terms of quotients of the $p$-adic gamma functions which can best be described as an analogue of hypergeometric series in the $p$-adic setting. We find six transformations and three summation identities for the McCarthy's $p$-adic hypergeometric series by evaluating certain Gauss sums. We also find a transformation and two summation identities for the Greene's finite field hypergeometric series. Finally, we obtain some special values of the Greene's finite field hypergeometric series as an application of our main results.

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McCarthy defined a function in terms of quotients of the $p$-adic gamma functions which can best be described as an analogue of hypergeometric series in the $p$-adic setting. We find six transformations and three summation identities for the McCarthy's $p$-adic hypergeometric series by evaluating certain Gauss sums. We also find a transformation and two summation identities for the Greene's finite field hypergeometric series. Finally, we obtain some special values of the Greene's finite field hypergeometric series as an application of our main results.

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Available abstract

McCarthy defined a function in terms of quotients of the $p$-adic gamma functions which can best be described as an analogue of hypergeometric series in the $p$-adic setting. We find six transformations and three summation identities for the McCarthy's $p$-adic hypergeometric series by evaluating certain Gauss sums. We also find a transformation and two summation identities for the Greene's finite field hypergeometric series. Finally, we obtain some special values of the Greene's finite field hypergeometric series as an application of our main results.

Key concepts: Bilateral hypergeometric series, Hypergeometric identity, Basic hypergeometric series, Generalized hypergeometric function, Hypergeometric function of a matrix argument, Appell series, Confluent hypergeometric function, Mathematics

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