A Two-Way Convolution of the Mittag-Leffler Function with Sheffer Polynomials

This article presents a two-way convolution of the Mittag-Leffler function with Sheffer polynomials via the umbral algebraic method, thereby introducing the Mittag-Leffler–Sheffer polynomials (MLSPs) and the Sheffer–Mittag-Leffler function (SMLF). We establish their generating functions, series definitions, quasi-monomial properties, and differential equations. Using Wronskian and Pascal functional matrices, we derive recursive formulas and differential recursive identities for these hybrid polynomials. Various special cases including Mittag-Leffler–Appell, Mittag-Leffler–Hermite, and Mittag-Leffler–Miller–Lee polynomials are examined. Additionally, a graphical investigation of the zero distributions of selected members of these polynomial families is presented using computational tools.