UMBRAL CALCULUS AND SHEFFER THEORY: A UNIFIED ALGEBRAIC FRAMEWORK

Abstract

This study presents a concise development of the modern umbral calculus emphasizing algebraic foundations, operator duality, and Sheffer sequences. Rigorous proofs are provided for foundational theorems and three new results: (1) an operator orthogonality criterion for Sheffer sequences, (2) a formal q-transfer formula with a worked q-Hermite example, and (3) a combinatorial encoding of umbral shifts yielding generalized Stirling numbers. The presentation follows the normalization conventions of Roman (1982) and is self-contained.