Analytic solutions of the heat equation and some formulas for laguerre and hermite polynomials

Abstract

We consider analytic solutions of the heat equation u xx + u yy=u t defined in a cylinder and show that any such solution can be expanded in a series of polynomial solutions to the heat equation. If we define the independent complex variables z and [zbar] by =z=x + iy, [zbar] x−y where x and y are independent complex variables, it is shown that any real-valued analytic solution of the heat equation is uniquely determined by its values on [zbar]=0 or i=0. Using this result, and expressing the above mentioned polynomial solutions to the heat equation in terms of Laguerre polynomials, we obtain some generating functions for Laguerre polynomials, as well as connection formulas between products of Hermite polynomials and Laguerre polynomials of argument r 2=x 2 + y 2. These connection formulas generalize a well known result of Feldheim.