A generalized jacobi theta function

The delta function initial condition solution v^*(x,t;y) at x = y ≥ 0 of the generalized Feller equation is used to define a generalized Jacobi Theta function documentclass{article}pagestyle{empty}begin{document}$ Theta (x,t) = upsilon *(x,t;0) + 2sumlimits_{n = 1}^infty {v*(x,t;y_n)} $end{document} for a sufficiently rapidly increasing and unbounded positive sequence {y_y}. It is shown that Θ(x,t) is analytic in each variable in certain regions of the complex x and t planes and that it is a solution of the generalized Feller equation. For those parameters for which this equation reduces to the heat equation, Θ(x,t) reduces to the third Jacobi Theta function.