Quasi-Monomiality and Heat-Semigroup Extensions of Mittag–Leffler–Sheffer Polynomials
- 1Computer Science Department, College of Computer Sciences and Information Technology (CCSIT), King Faisal University, P.O. Box 400, Al-Ahsa 31982, Saudi Arabia
- 2Department of Mathematics, Jamia Millia Islamia, New Delhi 110025, India
Abstract
This draft presents the construction of a two-variable extension E2sn(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∂yE2sn=(P^Es)2E2sn 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.
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BibTeX
@article{sarfaraz2026quasimonomiality,
title = {{Quasi-Monomiality and Heat-Semigroup Extensions of Mittag–Leffler–Sheffer Polynomials}},
author = {Mohd. Sarfaraz and Umme Zainab},
journal = {Mathematics},
year = {2026},
month = oct,
doi = {10.3390/math14203650},
url = {https://www.mdpi.com/2227-7390/14/20/3650},
note = {CC BY 4.0},
}
RIS
TY - JOUR
TI - Quasi-Monomiality and Heat-Semigroup Extensions of Mittag–Leffler–Sheffer Polynomials
AU - Mohd. Sarfaraz
AU - Umme Zainab
PY - 2026
DA - 2026/10/09
JO - Mathematics
DO - 10.3390/math14203650
UR - https://www.mdpi.com/2227-7390/14/20/3650
AB - This draft presents the construction of a two-variable extension E2sn(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∂yE2sn=(P^Es)2E2sn 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.
SN - 2227-7390
ER -