ON CERTAIN DEFINITE INTEGRALS WHICH ARISE IN AUTOMORPHIC LIE THEORY

We calculate in closed form a family of definite integrals I(n) (b, a; c), n = 0, 1, 2, 3, . . ., which arise in the calculation of regularized functional determinants associated with a compact space form X of a rank 1 Riemannian symmetric space. In the special case when n = 0, b = 1/2, a = pi, c = 1, and X is a Riemann surface, the integral I0(1/2, pi; 1) is known, for example in the context of Polyakov string theory.