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Logarithms and Stirling numbers associated with delta series

This paper establishes a unified framework for Stirling numbers of the first and second kind associated with a delta series, defining an associated logarithm and deriving a Schlomilch-type formula that connects these numbers while providing fifteen examples to demonstrate the theory's ability to generalize and unify existing combinatorial results.

Original authors: Dae san Kim, Taekyun Kim

Published 2026-02-03

📖 4 min read🧠 Deep dive

CC BY 4.0

Original authors: Dae san Kim, Taekyun Kim

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are a master architect working with a special set of building blocks. In the world of mathematics, these blocks are called Stirling numbers. For a long time, mathematicians had two main types of these blocks: "First Kind" and "Second Kind." They were incredibly useful for rearranging things, like shuffling a deck of cards or counting how many ways you can group items.

However, in recent years, mathematicians started inventing new, more complex versions of these blocks to handle tricky problems involving probability and "degenerate" (or slightly broken) versions of standard math. The problem? These new blocks didn't fit together perfectly. They lacked a crucial feature called orthogonality, which is like a perfect lock-and-key mechanism. Without it, you couldn't easily reverse your calculations or switch between the two types of blocks reliably.

The Paper's Big Idea: A Universal Adapter

Dae San Kim and Taekyun Kim, the authors of this paper, decided to build a universal adapter. They introduced a new way to look at these numbers by associating them with something called a Delta Series.

Think of a Delta Series as a "control knob" or a "master key."

If you turn the knob to position "A," you get the classic, old-school Stirling numbers.
If you turn it to position "B," you get the "degenerate" versions.
If you turn it to position "C," you get the "probabilistic" versions.

By defining their Stirling numbers based on this single "Delta Series" knob, the authors created a single framework where all these different versions of Stirling numbers finally play nice together. They restored the perfect "lock-and-key" relationship (orthogonality) that was missing in the newer, probabilistic versions.

The "Logarithm" Connection

The paper also introduces a new concept called the Logarithm Associated with a Delta Series.

To understand this, imagine you have a machine that transforms numbers.

The "First Kind" Stirling numbers are like the machine's instruction manual for taking a complex shape and breaking it down into simple pieces.
The "Second Kind" numbers are the manual for taking simple pieces and building them back up into a complex shape.
The Delta Series is the specific blueprint of the machine you are using.

The authors discovered a special formula (called a Schlömilch-type formula) that acts like a translator. It allows you to look at the "building" instructions (Second Kind) and instantly figure out the "breaking down" instructions (First Kind) without having to start from scratch. This translator also reveals a hidden "Logarithm" function specific to that machine's blueprint.

The "Fifteen Examples" Tour

To prove their new system works, the authors didn't just talk about theory; they took a tour of fifteen different mathematical neighborhoods.

Imagine walking through a town where every house represents a different known mathematical problem (like Bernoulli polynomials, Lah numbers, or Bell polynomials).

In the past, you had to use a different set of tools for every house.
With this new "Delta Series" framework, the authors walked into all fifteen houses and showed that their single set of tools could open every door, solve every puzzle, and explain the history of that house.

They showed that their new method doesn't just invent new math; it unifies everything that was already known, making it all fit under one roof.

The Bottom Line

In simple terms, this paper says: "We found a way to organize all the different types of Stirling numbers into one neat, consistent system. We fixed the broken parts of the newer versions so they work perfectly with the old ones. We also found a new formula that lets you switch between the two types of numbers instantly, and we proved it works for fifteen different famous math problems."

It's less about discovering a new planet and more about drawing a better map that shows how all the existing countries are actually connected by a single, smooth highway.

Technical Summary: Logarithms and Stirling Numbers Associated with Delta Series

Problem Statement
Recent developments in combinatorial analysis have introduced probabilistic Stirling numbers and B-Stirling numbers to generalize classical results. However, as noted in the introduction, these formulations often suffer from structural deficiencies, specifically a lack of standard orthogonality and inverse relations necessary for robust inversion formulas. Furthermore, certain definitions fail to recover classical Stirling numbers in deterministic limits (e.g., when a random variable $Y=1$ ). The paper addresses the need for a unified framework that restores these fundamental algebraic properties while extending the scope of Stirling numbers to include degenerate and probabilistic variants.

Methodology
The authors, Dae San Kim and Taekyun Kim, establish a theoretical framework based on delta series. A delta series is defined as a formal power series $f(t)$ such that $f(0)=0$ and $f'(0) \neq 0$ . The methodology proceeds as follows:

Association with Polynomial Sequences: For a delta series $f(t)$ , let $\bar{f}(t)$ be its compositional inverse. The authors define a sequence of polynomials $\{p_n(x)\}$ via the generating function $\sum_{n=0}^\infty p_n(x) \frac{t^n}{n!} = e^{x\bar{f}(t)}$ .
Definition of Generalized Stirling Numbers:
- Second Kind ( $S_2(n,k; f(t))$ ): Defined by the generating function $\frac{1}{k!}(e^{\bar{f}(t)} - 1)^k = \sum_{n=k}^\infty S_2(n,k; f(t)) \frac{t^n}{n!}$ .
- First Kind ( $S_1(n,k; f(t))$ ): Defined by the generating function $\frac{1}{k!}(\bar{e}_{f(t)}(t))^k = \sum_{n=k}^\infty S_1(n,k; f(t)) \frac{t^n}{n!}$ , where $\bar{e}_{f(t)}(t)$ is the compositional inverse of $e^{\bar{f}(t)} - 1$ .
Introduction of Associated Logarithms: The paper defines a "logarithm associated with $f(t)$ " as $\log_{f(t)}(1+t) = f(\log(1+t))$ . This function serves as the compositional inverse of $e^{\bar{f}(t)} - 1$ .
Derivation of Identities: Using Lagrange inversion formulas and properties of partial Bell polynomials, the authors derive explicit expressions relating these generalized numbers to Bernoulli polynomials associated with $f(t)$ and to each other.

Key Contributions and Results

Restoration of Orthogonality and Inverse Relations: The paper proves that the defined numbers $S_1(n,k; f(t))$ and $S_2(n,k; f(t))$ satisfy the fundamental orthogonality and inverse relations (Proposition 2.1). Specifically, $\sum_{k=l}^n S_2(n,k; f(t))S_1(k,l; f(t)) = \delta_{n,l}$ , ensuring they form a consistent pair for inversion formulas, a property often missing in recent probabilistic variants.
Schlömilch-type Formula: A central result is the derivation of a Schlömilch-type formula (Theorem 2.7) that provides an explicit connection between the two kinds of Stirling numbers:
$S_1(n,k; f(t)) = \sum_{j=0}^{n-k} \binom{n+j-1}{n+j-k} \binom{2n-k}{n-k-j} (-1)^j (\bar{f}'(0))^{-n-j} S_2(n-k+j, j; f(t))$
Explicit Expression for the Associated Logarithm: Utilizing the Schlömilch-type formula, the authors derive a new expression for the associated logarithm $\log_{f(t)}(1+t)$ entirely in terms of the Stirling numbers of the second kind (Theorem 2.8):
$\log_{f(t)}(1+t) = \sum_{n=1}^\infty \left( \sum_{j=0}^{n-1} \binom{2n-1}{n-1-j} (-1)^j (\bar{f}'(0))^{-n-j} S_2(n-1+j, j; f(t)) \right) \frac{t^n}{n!}$
Unification of Known Results: The framework is shown to unify various existing number systems. The paper demonstrates that:
- Classical Stirling numbers arise when $f(t)=t$ .
- Degenerate Stirling numbers arise when $f(t) = \frac{1}{\lambda}(e^{\lambda t}-1)$ .
- Probabilistic Stirling numbers (associated with a random variable $Y$ ) correspond to the delta series where $\bar{f}(t) = \log \mathbb{E}[e^{Y_\lambda(t)}]$ .
- B-Stirling numbers correspond to the case where the generating function $B(t) = e^{\bar{f}(t)}$ .

Significance
The paper claims that this delta series approach provides a "robust framework" that resolves the structural inconsistencies found in recent probabilistic and B-Stirling number definitions. By grounding the definitions in the compositional inverse of a delta series, the authors ensure that the resulting numbers inherently satisfy orthogonality and inverse relations.

The significance lies in the versatility of the framework, illustrated by fifteen concrete examples (Section 3) covering:

Probabilistic and degenerate Stirling numbers.
Lah numbers and Lah-Bell polynomials.
Central factorials and central Bell polynomials.
Mittag-Leffler and Laguerre polynomials.

The authors conclude that this methodology not only recovers known results but extends them, offering a consistent and computationally efficient tool for combinatorial analysis. The paper suggests that future work could apply this approach to other special sequences or extend the results to $q$ -analogues.

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