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 .