Singularities of Jacobi series on \(C^ 2\) and the Poisson process equation (Q1096757)

In this paper the author studies Jacobi series \[ F(x,y)=\sum^{\infty}_{n=0}\omega_ na_ nP_ n^{(\alpha,\beta)}(x)P_ n^{(\alpha,\beta)}(y), \] (here \(\omega_ n\) is the orthonormalization constant of the Jacobi polynomials) associated with a certain partial differential equation. Any Poisson process analytic at the origin in \(E^ 2\) admits such a local expansion. The author first associated F locally with the Taylor series \[ f(t)=\sum^{\infty}_{n=0}a_ nt^{2n}, \] and then shows that f, F constitute a dual transform pair with appropriate kernels. He then states necessary and sufficient conditions for analytic continuations of F in \(C^ 2\) to encounter singularities. Interestingly, the conditions involve easily verified properties of the related function f.