Hypergeometric functions and Tricomi operators: pole in the elliptic region

Abstract.

In this paper we obtain fundamental solutions for a class of operators of the form

$$\frac{1}{2}\Delta _{{x,s}} + \frac{\alpha }{s}\frac{\partial } {{\partial s}},\quad \alpha \in \mathbb{C}\backslash \{ 0\} .$$

These include generalizations of the classical Tricomi operator as well as the Laplace–Beltrami operator on a manifold with a suitable metric. Our results are then used to obtain explicit formulas for fundamental solutions of generalized Tricomi operators.