L 2 expansions in terms of generalized heat polynomials and of their Appell transforms

The object of this paper is to characterize functions which have L 2 expansions in terms of polynomial solutions Pn,*(x, t) of the generalized heat equation ( * ϊ ΓJ!_ + 2ύL J-Ί u (r t) -JL u(x f) L dx 2 x dx J at and in terms of the Appell transforms W n ,»(x, t) of the P n ,»(x, t).H* denotes the C 2 class of functions u(x, t) which, for a t υbγiJΓn, v v) U/μϋ) - , 0 w = 0 and C oo oo y> -f\ I Ίl(/γ f\ I 2 rϊ ftf/y\ X 1 I sy (2 ?Λ 1 / 2W Jo where 6 W = p^w!]" 1and Γ oo n,v(y, -t)dμ{y) .