A Note on the Hardy-Hille and Mehler formulas

Let and H n (x) be the n th Laguerre and Hermite polynomials, respectively. Two well-known bilinear generating formulas are the Hardy-Hille formula [1, p. 101] and the mehler formula [1, p. 377] This suggests the following problem. Consider the equation where f a ( x ) is a polynomial in x of degree n with highest coefficient equal to 1, A 0 = B 0 = 1. We shall also assume that a k = 1 and y 0 y 1 y 2 … y k–1 ≠ 0. We seek all sets of polynomials { f n ( x )} which satisfy (1.3), (1.4) and (1.5).