On finite field analogues of determinants involving the Beta function

Motivated by the works of L. Carlitz, R. Chapman and Z.-W. Sun on cyclotomic matrices, in this paper, we investigate certain cyclotomic matrices concerning the Jacobi sums over finite fields, which can be viewed as finite field analogues of certain matrices involving the Beta function. For example, let q>1 be a prime power and let χ be a generator of the group of all multiplicative characters of 𝔽_q. Then we prove that [J_q(χ^i,χ^j)]_1≤ i,j≤ q-2=(q-1)^q-3, where J_q(χ^i,χ^j) is the Jacobi sum over 𝔽_q. This is a finite analogue of [B(i,j)]_1≤ i,j≤ n=(-1)^n(n-1)/2∏_r=0^n-1(r!)^3/(n+r)!, where B is the Beta function. Also, if q=p≥5 is an odd prime, then we show that [J_p(χ^2i,χ^2j)]_1≤ i,j≤ (p-3)/2=1+(-1)^p+1/2p/4(p-1/2)^p-5/2.