2023/07/23 by Hai-Liang Wu, Liyuan Wang, Wu, Hai-Liang +3 · 1 citation
Biochemistry, Genetics and Molecular Biology · Computer Science · Mathematics · #Coding theory and cryptography #DNA and Biological Computing #FOS: Mathematics #Finite Group Theory Research #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.2307.12261
openalex publication_date 2023/07/23 · openalex created_date 2023/07/26 · openalex updated_date 2026/07/31
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 \mathbbFq. Then we prove that det[Jq(χi,χj)]1≤ i,j≤ q-2=(q-1)q-3, where Jq(χi,χj) is the Jacobi sum over \mathbbFq. This is a finite analogue of det [B(i,j)]1≤ i,j≤ n=(-1)(n(n-1))/(2)∏r=0n-1((r!)3)/((n+r)!), where B is the Beta function. Also, if q=p≥5 is an odd prime, then we show that det [Jp(χ2i,χ2j)]1≤ i,j≤ (p-3)/2=\frac1+(-1)(p+1)/(2)p4((p-1)/(2))(p-5)/(2).