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

TL;DR

This paper introduces a unified algebraic framework using Bell transforms to connect arithmetic functions, Euler products, and polynomial sequences, revealing new properties and recurrences.

Contribution

It develops a novel Bell transform-based approach to analyze arithmetic functions, Euler products, and polynomial sequences, unifying various classical results.

Findings

Derived explicit vanishing properties for Ramanujan's tau function.

Established congruence inheritance for classical sequences.

Revealed that inverse Bell transform recovers classical recurrences.

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 M\"obius 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.