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Study on Morgan-Voyce type polynomials with Euler-Seidel algorithm

This paper introduces two new families of Morgan-Voyce type polynomials and utilizes the Euler-Seidel matrix method to establish their structural properties, derive exponential generating functions, and express specific binomial-type sums of Bell polynomials as finite sums involving these new families.

Original authors: Taekyun Kim, Dae san Kim

Published 2026-07-16

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CC0 1.0

Original authors: Taekyun Kim, Dae san Kim

Original paper dedicated to the public domain under CC0 1.0 (http://creativecommons.org/publicdomain/zero/1.0/). ✨ 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 detective trying to solve a mystery hidden inside a giant, chaotic library of numbers. In the world of mathematics, this library is called "combinatorics," a branch of math that studies how things can be arranged, counted, and grouped. Sometimes, the patterns in this library are so complex that they look like a tangled ball of yarn. To untangle them, mathematicians use special tools. One famous tool is the Euler-Seidel matrix, which acts like a magical sorting machine. You feed it a list of numbers, and it crunches them together in a specific way to reveal a new, hidden list that often holds the secret to the original puzzle. Another set of tools are polynomials, which are just fancy algebraic expressions (like $x^2 + 2x + 1$ ) that can model all sorts of shapes and patterns. Recently, mathematicians have been playing with "degenerate" versions of these polynomials. Think of "degenerate" not as "broken," but as "twisted" or "warped" by a special knob (a parameter called $\lambda$ ). Turning this knob changes the shape of the polynomials, revealing new, strange, and beautiful patterns that the normal versions hide. The big question is: if we take these twisted, degenerate polynomials and run them through our magical sorting machine, what new secrets will we find?

This paper is the story of two mathematicians, Taekyun Kim and Dae San Kim, who decided to mix these three ingredients: the twisted polynomials, the magical sorting machine, and a specific family of number patterns called Morgan-Voyce polynomials (which originally came from studying electrical circuits but turned out to be great for counting things). They didn't just look at the existing patterns; they invented two brand new families of Morgan-Voyce polynomials, which they named $M_n(x)$ and $N_n(x)$ . They then created three more related families, $K_n(x)$ , $J_n(x)$ , and $L_n(x)$ , to see how they fit together.

The authors' main discovery is that these new polynomials are like keys that unlock specific doors in the library. They proved that three different, complicated sums involving Bell polynomials (which are used to count how many ways you can split a group of items into smaller groups) are actually equal to simple, finite sums of these new polynomials. In other words, they found a shortcut. Instead of doing a massive, messy calculation, you can just use their new formulas to get the answer instantly.

They also took these new polynomials and twisted them with that special "degenerate" knob to create $K_{n,\lambda}(x)$ , $J_{n,\lambda}(x)$ , and $L_{n,\lambda}(x)$ . They showed that even in this twisted, degenerate world, the magical sorting machine (the degenerate Euler-Seidel matrix) still works perfectly. They mapped out exactly how these twisted polynomials relate to each other and wrote down the exact rules (called generating functions) that describe how they grow. The paper doesn't just suggest these things might be true; the authors provide rigorous mathematical proofs, meaning these connections are solid facts, not just guesses. They didn't find a way to fix a broken engine or cure a disease, but they did expand the map of the mathematical library, showing that if you know how to use these new keys, you can unlock patterns that were previously invisible.

Technical Summary: Study on Morgan–Voyce Type Polynomials with Euler–Seidel Algorithm

Problem Statement
This paper addresses the need to bridge three distinct mathematical domains: the theory of degenerate special polynomials and numbers, the Euler–Seidel matrix method, and the classical Morgan–Voyce polynomials. While degenerate variants of Bernoulli, Euler, and Stirling numbers have been extensively studied using generating functions and $p$ -adic analysis, and while the Euler–Seidel matrix has been applied to various sequences, the intersection of these methods with Morgan–Voyce polynomials remains underexplored. The primary objective is to introduce new families of Morgan–Voyce type polynomials, characterize their structural properties, and utilize both standard and degenerate Euler–Seidel matrices to derive novel algebraic identities involving Bell polynomials.

Methodology
The authors employ a combination of generating function techniques, combinatorial identities, and matrix algorithms.

Definitions and Preliminaries: The study begins by defining degenerate exponentials ( $e_{\lambda}(t)$ ), degenerate logarithms ( $\log_{\lambda} t$ ), degenerate Stirling numbers of the first and second kind, and Bell polynomials ( $\phi_n(x)$ ). It also reviews the standard Euler–Seidel matrix recurrence ( $a_{k,n} = a_{k-1,n} + a_{k-1,n+1}$ ) and its degenerate counterpart ( $a_{k,n}(\lambda) = (1-(k-n)\lambda)a_{k-1,n} + a_{k-1,n+1}$ ).
Construction of New Polynomials: Two new Morgan–Voyce type polynomials, $M_n(x)$ and $N_n(x)$ , are defined via specific recurrence relations. From these, three additional polynomial families— $K_n(x)$ , $J_n(x)$ , and $L_n(x)$ —are derived.
Euler–Seidel Application: The authors construct Euler–Seidel matrices using the initial sequences of these new polynomials. By applying the Seidel transformation rules, they relate the initial sequences to the final sequences (the first column of the matrix).
Generating Functions and Identities: Exponential generating functions are derived for all new polynomial families. By substituting specific transformations (e.g., $t \to 1-e^{-t}$ ) into these generating functions and comparing coefficients with known expansions of Bell polynomials and Stirling numbers, the authors establish finite sum identities.
Degenerate Extension: The methodology is extended to the degenerate domain by introducing parameters $\lambda$ , defining $K_{n,\lambda}(x)$ , $J_{n,\lambda}(x)$ , and $L_{n,\lambda}(x)$ , and analyzing their properties within the degenerate Euler–Seidel framework.

Key Contributions and Results

New Polynomial Families: The paper introduces $M_n(x)$ and $N_n(x)$ with explicit formulas involving binomial coefficients:
$M_n(x) = n! \sum_{k=0}^n \binom{n+k+1}{n-k} x^k, \quad N_n(x) = n! \sum_{k=0}^n \binom{n+k}{n-k} x^k$
It further defines $K_n(x)$ , $J_n(x)$ , and $L_n(x)$ as variations of these, providing their explicit forms and recurrence relations.
Binomial-Type Identities for Bell Polynomials: A central result is the derivation of three distinct identities expressing binomial-type sums of Bell polynomials as finite sums of the new polynomials:
1. A sum involving $K_n(x)$ is shown to equal a convolution of Bell polynomials (Theorem 3.4).
2. A sum involving $J_n(x)$ is equated to a different convolution of Bell polynomials (Theorem 3.6).
3. A sum involving $L_n(x)$ is derived as a finite sum of Bell polynomials (Theorem 3.7).
  Additionally, relationships between the new polynomials themselves are established, such as $K_n(x) = n! \sum_{k=0}^n \frac{J_k(x)}{k!}$ (Theorem 3.5).
Degenerate Variants: The paper successfully constructs degenerate versions $K_{n,\lambda}(x)$ , $J_{n,\lambda}(x)$ , and $L_{n,\lambda}(x)$ . It proves analogous structural relations (Theorems 4.1 and 4.2) and derives their exponential generating functions, which involve degenerate exponentials $e_{\lambda}(t)$ .
Matrix Frameworks: Explicit Euler–Seidel and degenerate Euler–Seidel matrices are constructed for the initial sequences of $M_n(x)$ , $N_n(x)$ , $K_n(x)$ , $J_n(x)$ , and $L_n(x)$ . The paper provides the explicit relationship between the initial sequence $a_{0,n}$ and the final sequence $a_{n,0}$ for both standard and degenerate cases.

Significance and Claims
The authors claim that their work yields "novel algebraic identities" and "expands the application of matrix methods in combinatorial analysis." By integrating the Euler–Seidel algorithm with degenerate polynomial frameworks, the study reveals hidden algebraic structures connecting Morgan–Voyce polynomials, Bell polynomials, and degenerate special numbers. The paper posits that this integration serves as a "fertile ground" for discovering such structures in analytic number theory and combinatorics. The results are presented as a systematic characterization of these new families, offering explicit formulas and recurrence relations that were previously unavailable. The authors do not propose experimental applications but suggest that future research could extend these methodologies to other classical special polynomials or explore their $p$ -adic properties.

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