Quasi-Monomiality and Heat-Semigroup Extensions of Mittag–Leffler–Sheffer Polynomials

This draft presents the construction of a two-variable extension 2Esn(Θ,Ω)(ρ,y) of the Mittag–Leffler–Sheffer polynomials Esn(Θ,Ω)(ρ) by applying the heat (Gauss–Weierstrass) semigroup generated by the family’s own quasi-monomial lowering operator P^Es, rather than the ordinary derivative ∂ρ. This produces a one-parameter family of hybrid polynomials, an explicit finite-sum representation, and as the central result, a generalized evolution equation∂y2Esn=(P^Es)22Esn in which the diffusion operator is determined by the Sheffer datum p and the Mittag–Leffler parameters (Θ,Ω); the datum q enters the initial data and the raising operator but not the evolution operator. We establish this identity as an exact identity on polynomials, and we show that it is a second-order equation in ρ precisely in the classical case p(t)=t, Θ=Ω=1: for p(t)=t and Θ∈N, the operator P^Es has a differential order of exactly Θ. The manuscript delivers a step-by-step proof of quasi-monomiality for the two-variable family via an explicit commutator computation (with an independent differential-operator sanity check), derives the corresponding raising operator and recursive formula, and verifies all results on three fully worked examples—two classical and one with Θ=2—supported by closed-form tables, three-dimensional surface plots, a finite-difference study of the classical reduction with a common final time and an observed-order column, a numerical test of the generalized operator itself, and MATLAB implementations.