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Abstract
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We classify all the zeros and nonzero values of a family of hypergeometric series in the
-adic
setting. These values of hypergeometric series in the
-adic
setting lead to transformations of hypergeometric series in the
-adic setting which can
be described as
-adic
analogues of Kummer’s and Pfaff’s linear transformations on classical hypergeometric
series. We also evaluate certain summation identities for hypergeometric series in the
-adic
setting as well as Gaussian hypergeometric series.
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Keywords
character sum, Gauss sums, Jacobi sums, $p$-adic gamma
function
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Mathematical Subject Classification 2010
Primary: 11S80, 11T24, 33C20, 33C99, 33E50
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Milestones
Received: 19 March 2020
Revised: 1 May 2020
Accepted: 1 May 2020
Published: 4 September 2020
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