Bell Transforms of Arithmetic Functions: Euler Products, Congruences, and Polynomial Sequences

Abstract

We present a unified algebraic framework utilizing the formal Bell transform to bridge the Dirichlet convolution of arithmetic functions with the combinatorial structure of infinite Euler-type products. By analyzing the logarithmic derivative of exponential generating functions, we establish explicit mappings between Bell exponents and Möbius inversions. We apply this framework to derive exact vanishing properties and congruence inheritances for classical sequences, including Ramanujan's tau function and prime-colored partitions. Furthermore, we demonstrate that the inverse Bell transform seamlessly recovers classical partition recurrences and provides a discrete combinatorial engine for generating special polynomial families, including classical Appell and Sheffer sequences.