<p>This study uses the umbral calculus to analyze special functions, highlighting their structure and applications. With symbolic operators, we introduce a new class of special polynomials, termed the Mittag–Leffler–Hermite polynomials, which encompass several well-known families as special cases. We examine their fundamental properties, including generating functions, symbolic representations, operational rules, and summation formulas. The significance of these polynomials is demonstrated through applications to fractional kinetic equations. The Mittag–Leffler–Hermite polynomials are distinguished by their ability to provide exact analytic solutions to fractional kinetic equations, thereby extending solvable classes of problems in fractional calculus beyond what other generalizations offer.</p>