On some determinants involving the tangent function.

Let p be an odd prime and let a , b ∈ Z with p ∤ a b . In this paper,we mainly evaluate T p (δ) (a , b , x) : = det x + tan π a j 2 + b k 2 p δ ⩽ j , k ⩽ (p - 1) / 2 (δ = 0 , 1). For example, in the case p ≡ 3 (mod 4) , we show that T p (1) (a , b , 0) = 0 and T p (0) (a , b , x) = 2 (p - 1) / 2 p (p + 1) / 4 if (ab p) = 1 , p (p + 1) / 4 if (ab p) = - 1 , where (· p) is the Legendre symbol. When (- a b p) = - 1 , we also evaluate the determinant det [ x + cot π a j 2 + b k 2 p ] 1 ⩽ j , k ⩽ (p - 1) / 2. In addition, we pose several conjectures one of which states that for any prime p ≡ 3 (mod 4) , there is an integer x p ≡ 1 (mod p) such that det sec 2 π (j - k) 2 p 0 ⩽ j , k ⩽ p - 1 = - p (p + 3) / 2 x p 2. [ABSTRACT FROM AUTHOR]