Uniform asymptotic estimates of hypergeometric functions appearing in Potential Theory

Abstract

The solution of a Dirichlet problem for the Laplace-Beltrami operator with Bergman metric in the unit ball in the complex $n$-dimensional space can be expressed in terms of integrals of which the kernel can be expanded in spherical harmonics. The coefficients in this expansion contain ratios of Gauss hypergeometric functions of the form $F(p,q;p+q+n;r )/ F(p,q;p+q+n;1)$. The paper studies the uniform asymptotic behaviour of $F(q,mq;q+mq+n;t)$ for large values of $q$. Several results are formulated as inequalities for certain integrals containing ratios of hypergeometric functions [Zentralblatt MATH].