Logarithms and Stirling numbers associated with delta series

TL;DR

The paper addresses the need for a robust framework for Stirling numbers associated with a delta series $f(t)$ that preserves orthogonality and inversion. It defines the logarithm $log_{f(t)}(1+t)$ and derives a Schlomilch-type formula linking $S_1(n,k; f(t))$ and $S_2(n,k; f(t))$, enabling new representations of the logarithm in terms of $S_2(n,k; f(t))$. It further shows that $S_1(n,k; f(t))$ can be expressed via a Bernoulli-type polynomial attached to $f$ and its inverse, and that these numbers reproduce classical, probabilistic, and degenerate variants as special cases. Fifteen concrete examples demonstrate the unifying power of the approach across many combinatorial families, recovering known results and extending them. The framework provides a foundation for future work including $q$-analogues and applications to other families of special sequences.

Abstract

This paper investigates the Stirling numbers of the first and second kind associated with a delta series f (t). These numbers provide a robust framework that satisfies the orthogonality and inverse relations, often lacking in recent probabilistic Stirling and B-Stirling numbers. Key contributions include the definition and analysis of the logarithm associated with a delta series f (t). We further establish a Schlomilch-type formula, which provides an explicit connection between the two kinds of Stirling numbers. Using this formula, we derive another expression for the associated logarithm in terms of the Stirling numbers of the second kind associated with a delta series f (t). Finally, we provide fifteen concrete examples to illustrate the versatility of this framework, demonstrating how it unifies and extends several known results in combinatorial analysis.

Theorems & Definitions (11)

• Proposition 2.1
• Theorem 2.2
• Lemma 2.3
• proof
• Lemma 2.4
• proof
• Theorem 2.5
• Corollary 2.6
• Theorem 2.7
• proof