Gaussian hypergeometric functions and cyclotomic matrices

2024/07/30 by Hai-Liang Wu, Li-Yuan Wang, Wu, Hai-Liang +1
Mathematics · Chemistry · #Advanced Combinatorial Mathematics #Molecular spectroscopy and chirality #Advanced Algebra and Geometry

paper · pdf · doi:10.1007/s11139-025-01093-8

Abstract

Let q=pⁿ be an odd prime power and let \mathbbF_q be the finite field with q elements. Let \widehat\mathbbF_q^× be the group of all multiplicative characters of \mathbbF_q and let χ be a generator of \widehat\mathbbF_q^×. In this paper, we investigate arithmetic properties of certain cyclotomic matrices involving nonzero squares over \mathbbF_q. For example, let s₁,s₂,⋯,s_(q-1)/2 be all nonzero squares over \mathbbF_q. For any integer 1≤ r≤ q-2, define the matrix B_q,2(χ^r):=[χ^r(s_i+s_j)+χ^r(s_i-s_j)]_{1≤ i,j≤ (q-1)/2}. We prove that if q≡ 3\pmod 4, then det (B_q,2(χ^r))=∏_{0≤ k≤ (q-3)/2}J_q(χ^r,χ^2k)= \begincases (-1)^(q-3)/(4)\bf iⁿG_q(χ^r)^(q-1)/(2)/√(q) if r≡ 1\pmod 2,
G_q(χ^r)^(q-1)/(2)/q if r≡ 0\pmod 2, \endcases where J_q(χ^r,χ^2k) and G_q(χ^r) are the Jacobi sum and the Gauss sum over \mathbbF_q respectively.

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