[Paper Review] Differential equations associated with Lambda-Changhee polynomials

Taekyun Kim, Dae San Kim|arXiv (Cornell University)|Apr 20, 2016

Advanced Mathematical Identities8 references3 citations

TL;DR

This paper derives explicit linear differential equations for λ-Changhee polynomials—degenerate versions of classical Changhee polynomials—and uses them to establish new identities. By analyzing generating functions and applying differential operators, the authors obtain a closed-form expression for λ-Changhee polynomials of higher order in terms of Stirling numbers, generalized harmonic numbers, and combinatorial coefficients, providing a systematic method for generating identities in degenerate special functions.

ABSTRACT

In this paper, we study linear differential equations arising from $λ$- Changhee polynomials (or called degenerate Changhee polynomials) and give some explicit and new identities for the $λ$-Changhee polynomials associated with linear differential equations.

Motivation & Objective

• To develop a novel method for deriving identities involving λ-Changhee polynomials using linear differential equations.
• To generalize classical Changhee polynomials by introducing a degeneracy parameter λ, extending their applicability.
• To establish explicit formulas for λ-Changhee polynomials of higher order through differential operator techniques.
• To connect λ-Changhee polynomials with Stirling numbers of the first kind, generalized harmonic numbers, and power sums.

Proposed method

• The authors define λ-Changhee polynomials via a generating function involving logarithmic and exponential terms with parameter λ.
• They derive first- and second-order differential equations from the generating function using logarithmic and rational differentiation.
• The method involves expanding the generating function in power series and matching coefficients to relate λ-Changhee polynomials to Stirling numbers and generalized harmonic numbers.
• Combinatorial identities are constructed using multinomial coefficients and Pochhammer symbols to express higher-order polynomials.
• The approach leverages known identities for Stirling numbers of the first and second kind, and generalized power sums S_{k,j}(N).
• A key transformation uses the relation between logarithmic generating functions and falling factorials to express λ-Changhee polynomials in terms of Euler polynomials and Stirling numbers.

Experimental results

Research questions

• RQ1 How can linear differential equations be systematically derived for λ-Changhee polynomials to generate new identities?
• RQ2 What is the explicit relationship between λ-Changhee polynomials and Stirling numbers of the first kind in the context of degenerate special functions?
• RQ3 How do generalized harmonic numbers and power sums S_{k,j}(N) contribute to the structure of λ-Changhee polynomial identities?
• RQ4 Can a closed-form expression for higher-order λ-Changhee polynomials be derived using differential operators and series expansion?
• RQ5 What role does the parameter λ play in unifying classical and degenerate versions of Changhee polynomials through differential equations?

Key findings

• The paper derives a new closed-form expression for the λ-Changhee polynomial of order k+N in terms of a multivariate sum involving Stirling numbers, Pochhammer symbols, and generalized harmonic numbers.
• The key result expresses $\mathrm{Ch}_{k+N,\lambda}(x)$ as a weighted sum over indices m, n, a, l, e, f, s, with coefficients involving $\lambda^N$, binomial coefficients, and combinatorial factors.
• The formula explicitly connects $\mathrm{Ch}_{k+N,\lambda}(x)$ to $\mathrm{Ch}_{s,\lambda}(x)$, showing how higher-order polynomials decompose into lower-order ones via combinatorial identities.
• The coefficients include terms like $\left(i+m-1\right)_m$, $\left(r+n-i-1\right)_n$, and $\left(N+l-1\right)_l$, which are rising factorials related to generalized harmonic numbers.
• The derivation confirms that the differential equation approach yields identities that are both explicit and non-trivial, extending known results on degenerate special numbers.
• The method successfully generalizes previous results on Bernoulli numbers of the second kind and Frobenius-Euler polynomials to the λ-Changhee setting.