Certain product formulas and values of Gaussian hypergeometric series
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RESEARCH
Certain product formulas and values of Gaussian hypergeometric series Mohit Tripathi∗ and Rupam Barman * Correspondence:
[email protected] Department of Mathematics, Indian Institute of Technology Guwahati, North Guwahati, Guwahati, Assam 781039, India We thank the anonymous referee for his/her thorough review and highly appreciate the comments and suggestions, which significantly contributed to improving the quality of the paper. The second author is partially supported by a research grant under the MATRICS scheme of SERB, Department of Science and Technology , Government of India
Abstract In this article we find finite field analogues of certain product formulas satisfied by the classical hypergeometric series. We express product of two 2 F1 -Gaussian hypergeometric series as 4 F3 - and 3 F2 -Gaussian hypergeometric series. We use properties of Gauss and Jacobi sums and our earlier works on finite field Appell series to deduce these product formulas satisfied by the Gaussian hypergeometric series. We then use these transformations to evaluate explicitly some special values of 4 F3 - and 3 F2 -Gaussian hypergeometric series. By counting points on CM elliptic curves over finite fields, Ono found certain special values of 2 F1 - and 3 F2 -Gaussian hypergeometric series containing trivial and quadratic characters as parameters. Later, Evans and Greene found special values of certain 3 F2 -Gaussian hypergeometric series containing arbitrary characters as parameters from where some of the values obtained by Ono follow as special cases. We show that some of the results of Evans and Greene follow from our product formulas including a finite field analogue of the classical Clausen’s identity. Keywords: Hypergeometric series, Gauss and Jacobi sums, Hypergeometric series over finite fields Mathematics Subject Classification: 33C05, 33C20, 11T24
1 Introduction and statement of results For a complex number a, the rising factorial is defined as (a)0 = 1 and (a)k = a(a + 1) · · · (a + k − 1), k ≥ 1. For a non-negative integer n, and ai , bi ∈ C with bi ∈ / {. . . , −3, −2, −1, 0}, the (generalized) hypergeometric series n+1 Fn is defined by n+1 Fn
∞ (a1 )k · · · (an+1 )k xk a1 , a2 , . . . , an+1 | x := · , (b1 )k · · · (bn )k k! b1 , . . . , bn
(1.1)
k=0
which converges absolutely for |x| < 1. In 1980s, Greene [12,13] introduced a finite field, character sum analogue of classical hypergeometric series that satisfies summation and transformation properties similar to those satisfied by the classical hypergeometric series. Let p be an odd prime, and let Fq denote the finite field with q elements, where × × q = pr , r ≥ 1. Let F q be the group of all multiplicative characters on Fq . We extend the
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× domain of each χ ∈ F q to Fq by setting χ(0) = 0 including the trivial character ε. For multiplicative characters A and B on Fq , the binomial coefficient AB is defined by
A B(−1) B
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