ON A FUNDAMENTAL SET OF SOLUTIONS TO THE GENERALIZED HYPERGEOMETRIC EQUATION <b><i>_n</i></b><b>₊</b>₁<b><i>E</i></b><b><i>_n</i></b>PRESENTED BY INTEGRALS OF EULER TYPE

Abstract

Abstract. We give a fundamental set of solutions to the generalized hypergeometric equation n+1 E n in terms of integrals of Euler type and explicitly determine the matrix elements of the circuit matrices with respect to this set of solutions. IntroductionThe generalized hypergeometric series n+1 F n is defined to be. It satisfies the generalized hypergeometric differential equation n+1 E n , which has regular singular points z = 0, 1, ∞ and whose rank is n + 1:where θ z = zd/dz and β n+1 = 1. The purpose of the present paper is to give a fundamental set of solutions of n+1 E n in terms of integrals of Euler type, and to express explicitly the matrix elements of the circuit matrices with respect to this set of solutions.We introduce some notation. When z is real and 0 < z < 1, let D j , j = 1, . . . , n + 1, be the domainof the real manifold (T z ) R , which is the real locus of