On finite field analogues of determinants involving the Beta function
Abstract
Motivated by the works of L. Carlitz, R. Chapman and Z.-W. Sun on cyclotomic matrices, we investigate certain cyclotomic matrices concerning Jacobi sums over finite fields, which can be viewed as finite field analogues of certain matrices involving the Beta function. For example, let $q \gt 1$ be a prime power and let $\chi $ be a generator of the group of all multiplicative characters of $\mathbb {F}_q$. Then we prove that $$\det \,[J_q(\chi ^i,\chi ^j)]_{1\le i,j\le q-2}=(q-1)^{q-3},$$ where $J_q(\chi ^i,\chi ^j)$ is the Jacobi sum over $\mathbb {F}_q$. This is a finite analogue of $$\det \, [B(i,j)]_{1\le i,j\le n}=(-1)^{{n(n-1)}/{2}}\prod _{r=0}^{n-1}\frac {(r!)^3}{(n+r)!},$$ where $B$ is the Beta function. Also, if $q=p\ge 5$ is an odd prime, then we show that $$\det \, [J_p(\chi ^{2i},\chi ^{2j})]_{1\le i,j\le (p-3)/2}=\frac {1+(-1)^{{(p+1)}/{2}}p}{4}\bigg (\frac {p-1}{2}\bigg )^{{(p-5)}/{2}}.$$
