Identities Involving the Higher-Order Degenerate Type 2 ω-Daehee Polynomials

Abstract

In this paper, based on previous study of type 2 $\omega$-Daehee polynomials and some of their properties, we further introduce the generating function definition for the higher-order degenerate type 2 $\omega$-Daehee polynomials. By employing the methods of generating functions and Riordan arrays, we investigate the properties of these higher-order degenerate polynomials in depth and establish identities that relate them to certain special combinatorial sequences.

1. Introduction

Considerable interest in degenerate versions of special polynomials is driven by their role as a conceptual bridge connecting classical analysis to discrete mathematics. The degenerative limiting process uncovers novel combinatorial identities and algebraic symmetries that are obscured in the classical setting, thus providing unified perspectives and profound theoretical insights across multiple mathematical disciplines.

In recent years, the study of various extensions and degenerate versions of classical special polynomials has attracted considerable attention across number theory, combinatorics, and mathematical physics. This resurgence, partly inspired by Carlitz’s early work on degenerate versions, has been significantly advanced by Kim et al. [1,2], who systematically developed degenerate Bernoulli and Euler polynomials. Their introduction of relevant functions (e.g., the degenerate polylogarithm and polyexponential functions) provides a robust basis for further generalizations in the field [3]. Concurrently, the exploration of type 2 polynomial families, including type 2 Bernoulli, Euler, and Daehee polynomials [4,5], has revealed rich algebraic and combinatorial structures. These are frequently derived using advanced techniques such as p-adic integrals and generating function methods [6,7], highlighting the profound connections between p-adic analysis and classical polynomial sequences.

Several research streams have contributed to this rapidly expanding landscape. For example, Ma and Lim [6] investigated type 2 $\omega$-Daehee polynomials using p-adic invariant integrals on ${\mathbb{Z}}_p$, establishing several fundamental properties and symmetric identities. In parallel, Kim et al. [1] and Sharma et al. [8] developed degenerate versions of Daehee and Bernoulli polynomials, revealing their close connections with degenerate Stirling numbers of both kinds and the degenerate polylogarithm function. These works collectively underscore a persistent trend: the fusion of degenerate analogues with poly-type functions and higher-order constructions, leading to the discovery of novel identities and deeper combinatorial symmetries.

A key methodology emerging from these studies involves the use of generating functions and p-adic integral equations. For instance, the derivation of identities using weighted Stirling numbers [9] and the construction of new polynomial classes via modified polyexponential functions [10] or Riordan arrays [11] have become common themes. Additionally, the exploration of poly-Daehee polynomials within frameworks such as the Apostol–Euler basis further exemplifies the utility of generating functions in uncovering new identities and implicit summation formulae [12]. Furthermore, the exploration of symmetric identities through bosonic and fermionic p-adic integrals has connected these polynomial sequences to probabilistic interpretations, such as moments of random variables derived from Laplace distributions [4].

Building upon these developments, a clear impetus has emerged to unify and extend these concepts into a more comprehensive framework. The synthesis of higher-order, degenerate, and type 2 structures—spearheaded by researchers such as Kim and their collaborators [5,10,13,14,15]—points toward a promising direction for future inquiry. It is within this integrated context that the systematic study of higher-order degenerate type 2 $\omega$-Daehee polynomials arises as a natural and meaningful generalization. Such polynomials are anticipated to encapsulate several known polynomial families—including degenerate Daehee, type 2 Daehee, and poly-Daehee polynomials—while providing a unified platform for exploring new combinatorial identities, asymptotic behavior, and potential applications in symmetric function theory and probabilistic modeling. The investigation of higher-order degenerate type 2 $\omega$-Daehee polynomials can be naturally linked to the theory of generalized Bernoulli numbers. This connection is powerfully illustrated by the work of Leinartas and Shishkina [16], who employed these numbers to derive a multidimensional analog of the Euler-Maclaurin formula, thereby showcasing their utility and providing a valuable framework for our research.

We are now in a position to state some special numbers and polynomials, which will be used in the paper.

Let p be a fixed prime number. Throughout this paper, ${\mathbb{Z}}_p$, ${\mathbb{Q}}_p$, and ${\mathbb{C}}_p$ will denote the ring of p-adic integers, the field of p-adic rational numbers, and the completion of the algebraic closure of ${\mathbb{Q}}_p$, respectively.

For $n\ge 0$, the generalized Stirling numbers of the first kind are defined by (see [2,17])

$$\sum _{n=k}^{\infty }s(n,k;r){\displaystyle \frac{t^n}{n!}}={(1+t)}^{-r}{\displaystyle \frac{{\left(\ln (1+t)\right)}^k}{k!}},fork\in {\mathbb{Z}}^{+}\cup \left\{0\right\},$$

(1)

when $r=0$, the above expression becomes the Stirling numbers of the first kind:

$$\sum _{n=k}^{\infty }s(n,k){\displaystyle \frac{t^n}{n!}}={\displaystyle \frac{{\left(\ln (1+t)\right)}^k}{k!}},$$

and the generalized Stirling numbers of the second kind are defined by

$$\sum _{n=k}^{\infty }S(n,k;r){\displaystyle \frac{t^n}{n!}}=e^{rt}{\displaystyle \frac{{(e^t-1)}^k}{k!}},fork\in {\mathbb{Z}}^{+}\cup \left\{0\right\},$$

(2)

when $r=0$, the above expression becomes the Stirling numbers of the second kind:

$$\sum _{n=k}^{\infty }S(n,k){\displaystyle \frac{t^n}{n!}}={\displaystyle \frac{{(e^t-1)}^k}{k!}}.$$

For any nonzero $\lambda \in \mathbb{R}$, the degenerate exponential function is defined by (see [3])

$$e_{\lambda }^x\left(t\right)={(1+\lambda t)}^{{\displaystyle \frac{x}{\lambda }}},e_{\lambda }\left(t\right)={(1+\lambda t)}^{{\displaystyle \frac{1}{\lambda }}}.$$

In the limit as $\lambda \to 0$, the degenerate versions of exponential and logarithmic functions reduce to their classical counterparts. Note that

$$\underset{\lambda \to 0}{\lim }{e}_{\lambda }^x\left(t\right)=\sum _{n=0}^{\infty }{\displaystyle \frac{x^n{t}^n}{n!}}=e^{xt}.$$

The degenerate Stirling numbers of the first and second kind are defined by (see [1])

$$\sum _{n=k}^{\infty }{s}_{1,\lambda }(n,k){\displaystyle \frac{t^n}{n!}}={\displaystyle \frac{{\left(\frac{1}{\lambda }[{(1+t)}^{\lambda }-1]\right)}^k}{k!}},$$

$$\sum _{n=k}^{\infty }{S}_{2,\lambda }(n,k){\displaystyle \frac{t^n}{n!}}={\displaystyle \frac{{({(1+\lambda t)}^{\frac{1}{\lambda }}-1)}^k}{k!}}.$$

The generalized Lah numbers $L(n,k;r)$ have the following generating function (see [18])

$$\sum _{n=k}^{\infty }L(n,k;r){\displaystyle \frac{t^n}{n!}}={(1+t)}^r{\displaystyle \frac{1}{k!}}{\left({\displaystyle \frac{-t}{1+t}}\right)}^k.$$

(3)

The higher-order type 2 Bernoulli polynomials have the following generating function (see [4,5])

$$\sum _{n=0}^{\infty }{b}_n^{\left(r\right)}\left(x\right){\displaystyle \frac{t^n}{n!}}={\left({\displaystyle \frac{t}{e^t-e^{-t}}}\right)}^r{e}^{xt}.$$

The central factorial numbers of the second kind have the following generating function (see [5,6])

$$\sum _{n=k}^{\infty }T(n,k){\displaystyle \frac{t^n}{n!}}={\displaystyle \frac{1}{k!}}(e^{{\displaystyle \frac{t}{2}}}-e^{-{\displaystyle \frac{t}{2}}}{)}^k.$$

The generalized harmonic numbers have the following generating function (see [11])

$$\sum _{n=0}^{\infty }{H}_{n,h}{t}^n={\displaystyle \frac{{(-\ln (1-t))}^h}{h!(1-t)}}.$$

(4)

The two kinds of generalized Bell numbers $B(n,k)$ and $\beta (n,k)$ are defined by the generating function

$$\sum _{n=k}^{\infty }B(n,k){\displaystyle \frac{t^n}{n!}}={\displaystyle \frac{1}{k!}}(e^{e^t-1}-1{)}^k,$$

$$\sum _{n=k}^{\infty }\beta (n,k){\displaystyle \frac{t^n}{n!}}={\displaystyle \frac{1}{k!}}{\left(\ln (1+\ln (1+t))\right)}^k.$$

The present work is motivated by the premise that the synergistic combination of degeneracy, type 2 structure, and the parameter $\omega$ leads to a polynomial family with profoundly richer properties and broader applicability. The degenerate parameter $\lambda$ provides a continuous deformation that probes discrete analogues and reveals hidden combinatorial landscapes. The type 2 aspect incorporates distinct symmetries and connections to hyperbolic functions, yielding identities orthogonal to the classical theory. Finally, the parameter $\omega$ acts as a generative weight, crucial for deriving the most general symmetric identities and providing a tunable degree of freedom.

2. Methods

In this paper, let $f\left(t\right)$ be the generating function of the sequence ${\left\{f_n\right\}}_{n\ge 0}$, that is, $f\left(t\right)={\sum }_{n=0}^{\infty }{f}_n{t}^n,$ and let $\left[t^n\right]f\left(t\right)$ be the coefficient of $t^n$ in the series $f\left(t\right)$ [17].

A Riordan array is a formal power series pair $\left(g\right(t),f(t\left)\right)$, where $g\left(t\right)={\sum }_{k\ge 0}{g}_k{t}^k$, $f\left(t\right)={\sum }_{k\ge 1}{f}_k{t}^k$; if $g\left(t\right)$ is a reversible series and $f\left(t\right)$ is a delta series, then the Riordan array is called normal. The Riordan array $\left(g\right(t),f(t\left)\right)$ defines an infinite triangular matrix ${\left(d_{n,k}\right)}_{n,k\in \mathbb{C}}$ according to the following rules: $d_{n,k}=\left[t^n\right]g\left(t\right){\left(f\left(t\right)\right)}^k,$ where $g\left(t\right){\left(f\left(t\right)\right)}^k$ is called the column generating function of the Riordan matrix, and $d_{n,k}$ is called the general element of the Riordan matrix [5].

Lemma 1

([11]). Let $D=(g\left(t\right),f\left(t\right))={\left(d_{n,k}\right)}_{n,k\in \mathbb{N}}$ be a Riordan matrix, and let $h\left(t\right)={\sum }_{k\ge 0}{h}_k{t}^k$ be the occurrence function of the sequence $\left\{h_n\right\}$; then

$$\sum _{k=0}^n{d}_{n,k}{h}_k=\left[t^n\right]g\left(t\right)h\left(f\left(t\right)\right).$$

(5)

Lemma 2

([9,17]). Let {$f_n$} and {$g_n$} be two sequences whose generating functions are

$$f=\sum _{n=0}^{\infty }{f}_n{\displaystyle \frac{t^n}{n!}},g=\sum _{n=0}^{\infty }{g}_n{\displaystyle \frac{t^n}{n!}},$$

Then we have the following inversion formula:

$$f_n=\sum _{k=0}^{n}s(n,k)g_k\iff g_n=\sum _{k=0}^{n}S(n,k)f_k,$$

(6)

$$f_n=\sum _{k=0}^{n}s(n,k;r)g_k\iff g_n=\sum _{k=0}^{n}S(n,k;r)f_k,$$

(7)

$$f_n=\sum _{k=0}^n{s}_{1,\lambda }(n,k)g_k\iff g_n=\sum _{k=0}^n{S}_{2,\lambda }(n,k)f_k.$$

(8)

3. Higher-Order Degenerate Type 2 $\mathit{\omega }$-Daehee Numbers and Polynomials

Ma and Lim [1] gave the generating function of higher-order type 2 $\omega$-Daehee polynomials:

$$\sum _{n=0}^{\infty }{d}_{n,\omega }^{\left(\alpha \right)}\left(x\right){\displaystyle \frac{t^n}{n!}}={\left({\displaystyle \frac{\ln (1+t)}{{(1+t)}^{\omega }-{(1+t)}^{-\omega }}}\right)}^{\alpha }{(1+t)}^x,$$

(9)

when $x=0, d_{n,\omega }^{\left(\alpha \right)}=d_{n,\omega }^{\left(\alpha \right)}\left(0\right)$ are called type 2 $\omega$-Daehee numbers of order $\alpha$.

Motivated by (9), we define the higher-order degenerate type 2 $\omega$-Daehee polynomials.

Definition 1.

Let n be a non-negative integer, α, λ be real numbers, and $\lambda \ne 0$. The generating function of the higher-order degenerate type 2 ω-Daehee polynomial is defined as follows, for $t,x\in {\mathbb{C}}_p$ and $\omega \in \mathbb{N}$:

$$\sum _{n=0}^{\infty }{d}_{n,\omega ,\lambda }^{\left(\alpha \right)}\left(x\right){\displaystyle \frac{t^n}{n!}}={\left({\displaystyle \frac{\frac{1}{\lambda }\ln (1+\lambda \ln (1+t))}{{(1+\lambda \ln (1+t))}^{\frac{\omega }{\lambda }}-{(1+\lambda \ln (1+t))}^{-\frac{\omega }{\lambda }}}}\right)}^{\alpha }{(1+\lambda \ln (1+t))}^{{\displaystyle \frac{x}{\lambda }}},$$

(10)

when $x=0, d_{n,\omega ,\lambda }^{\left(\alpha \right)}=d_{n,\omega ,\lambda }^{\left(\alpha \right)}\left(0\right)$ are called higher-order degenerate type 2 ω-Daehee numbers of order α.

Theorem 1.

Let n be a nonnegative integer, and let $r_i\ge 0$ be integers for $i\in \left[m\right]$ with $m\ge 1$. Then the higher-order degenerate type 2 ω-Daehee polynomials satisfy the following property:

$$\begin{array}{c}{d}_{n,\omega ,\lambda }^{(r_1+r_2+\cdots +r_m)}(x_1+x_2+\cdots +x_m)\hfill \\ =\sum _{n_1+n_2+\cdots +n_m=n}\left({\displaystyle \genfrac{}{}{0pt}{}{n}{n_1,n_2,\cdots ,n_m}}\right)d_{n_1,\omega ,\lambda }^{\left(r_1\right)}\left(x_1\right)d_{n_2,\omega ,\lambda }^{\left(r_2\right)}\left(x_2\right)\cdots d_{n_m,\omega ,\lambda }^{\left(r_m\right)}\left(x_m\right).\hfill \end{array}$$

(11)

Proof. 

From the generating function (10), we obtain

$$\begin{array}{cc}\hfill & \sum _{n=0}^{\infty }{d}_{n,\omega ,\lambda }^{(r_1+r_2+\cdots +r_m)}(x_1+x_2+\cdots +x_m){\displaystyle \frac{t^n}{n!}}\hfill \\ \hfill & =\left(\sum _{n=0}^{\infty }{d}_{n,\omega ,\lambda }^{\left(r_1\right)}\left(x_1\right){\displaystyle \frac{t^n}{n!}}\right)\left(\sum _{n=0}^{\infty }{d}_{n,\omega ,\lambda }^{\left(r_2\right)}\left(x_2\right){\displaystyle \frac{t^n}{n!}}\right)\cdots \left(\sum _{n=0}^{\infty }{d}_{n,\omega ,\lambda }^{\left(r_m\right)}\left(x_m\right){\displaystyle \frac{t^n}{n!}}\right)\hfill \\ \hfill & =\sum _{n=0}^{\infty }\left(\sum _{\begin{array}{c}{n}_1+n_2+\cdots +n_m=n\end{array}}\left({\displaystyle \genfrac{}{}{0pt}{}{n}{n_1,n_2,\cdots ,n_m}}\right)d_{n_1,\omega ,\lambda }^{\left(r_1\right)}\left(x_1\right)d_{n_2,\omega ,\lambda }^{\left(r_2\right)}\left(x_2\right)\cdots d_{n_m,\omega ,\lambda }^{\left(r_m\right)}\left(x_m\right)\right){\displaystyle \frac{t^n}{n!}}.\hfill \end{array}$$

By comparing the coefficients of ${\displaystyle \frac{t^n}{n!}}$ in the first and last expressions, Theorem 1 is proved. □

Corollary 1.

When $x_i=0,i\in \left[m\right]$ in (11), we get higher-order degenerate type 2 ω-Daehee number property

$$\begin{array}{c}\hfill d_{n,\omega ,\lambda }^{(r_1+r_2+\cdots +r_m)}=\sum _{\begin{array}{c}{n}_1+n_2+\cdots +n_m=n\end{array}}\left({\displaystyle \genfrac{}{}{0pt}{}{n}{n_1,n_2,\cdots ,n_m}}\right)d_{n_1,\omega ,\lambda }^{\left(r_1\right)}{d}_{n_2,\omega ,\lambda }^{\left(r_2\right)}\cdots d_{n_m,\omega ,\lambda }^{\left(r_m\right)}.\end{array}$$

(12)

In (11), let $m=2,r_1=0,r_2=\alpha$, we have

$$\begin{array}{c}\hfill d_{n,\omega ,\lambda }^{\left(\alpha \right)}(x_1+x_2)=\sum _{l=0}^n\sum _{k=0}^l\left({\displaystyle \genfrac{}{}{0pt}{}{n}{l}}\right){\left(x_1\right)}_{k,\lambda }s(l,k)d_{n-l,\omega ,\lambda }^{\left(\alpha \right)}\left(x_2\right),\end{array}$$

where ${\left(x_1\right)}_{0,\lambda }=1,{\left(x_1\right)}_{k,\lambda }=x_1(x_1-\lambda )\cdots (x_1-(k-1)\lambda )$, for $k\ge 1$.

Theorem 2.

Let $n\ge 1$ be an integer. Then the higher-order degenerate type 2 ω-Daehee polynomials satisfy the following equality:

$$\begin{array}{c}\hfill d_{n-1,\omega ,\lambda }^{\left(\alpha \right)}(x+y)-d_{n-1,\omega ,\lambda }^{\left(\alpha \right)}\left(x\right)={\displaystyle \frac{1}{n}}\sum _{l=1}^n\sum _{k=1}^{n-l}\left({\displaystyle \genfrac{}{}{0pt}{}{n}{l}}\right)l{\left(y\right)}_{k,\lambda }s(n-l,k)d_{l-1,\omega ,\lambda }^{\left(\alpha \right)}\left(x\right).\end{array}$$

(13)

Proof. 

From the generating function (10), we obtain

$$\begin{array}{cc}\hfill & \sum _{n=1}^{\infty }n\left(d_{n-1,\omega ,\lambda }^{\left(\alpha \right)}(x+y)-d_{n-1,\omega ,\lambda }^{\left(\alpha \right)}\left(x\right)\right){\displaystyle \frac{t^n}{n!}}\hfill \\ \hfill & =t{\left({\displaystyle \frac{\frac{1}{\lambda }\ln (1+\lambda \ln (1+t))}{{(1+\lambda \ln (1+t))}^{\frac{\omega }{\lambda }}-{(1+\lambda \ln (1+t))}^{-\frac{\omega }{\lambda }}}}\right)}^{\alpha }{(1+\lambda \ln (1+t))}^{{\displaystyle \frac{x}{\lambda }}}[{(1+\lambda \ln (1+t))}^{{\displaystyle \frac{y}{\lambda }}}-1]\hfill \\ \hfill & =\sum _{n=1}^{\infty }n{d}_{n-1,\omega ,\lambda }^{\left(\alpha \right)}\left(x\right){\displaystyle \frac{t^n}{n!}}\sum _{k=1}^{\infty }{\left(y\right)}_{k,\lambda }{\displaystyle \frac{{\left(\ln (1+t)\right)}^k}{k!}}\hfill \\ \hfill & =\sum _{n=1}^{\infty }n{d}_{n-1,\omega ,\lambda }^{\left(\alpha \right)}\left(x\right){\displaystyle \frac{t^n}{n!}}\sum _{n=1}^{\infty }\sum _{k=1}^n{\left(y\right)}_{k,\lambda }s(n,k){\displaystyle \frac{t^n}{n!}}\hfill \\ \hfill & =\sum _{n=1}^{\infty }\left(\sum _{l=1}^n\sum _{k=1}^{n-l}\left({\displaystyle \genfrac{}{}{0pt}{}{n}{l}}\right)l{\left(y\right)}_{k,\lambda }s(n-l,k)d_{l-1,\omega ,\lambda }^{\left(\alpha \right)}\left(x\right)\right){\displaystyle \frac{t^n}{n!}}.\hfill \end{array}$$

By comparing the coefficients of ${\displaystyle \frac{t^n}{n!}}$ on both sides, Theorem 2 is proved. □

Corollary 2.

In (13), set $y=\lambda$, we obtain

$$\begin{array}{cc}\hfill & d_{n-1,\omega ,\lambda }^{\left(\alpha \right)}(x+\lambda )-d_{n-1,\omega ,\lambda }^{\left(\alpha \right)}\left(x\right)\hfill \\ \hfill & ={\displaystyle \frac{1}{n}}\sum _{l=1}^n\sum _{k=1}^{n-l}\left({\displaystyle \genfrac{}{}{0pt}{}{n}{l}}\right)l{\left(\lambda \right)}_{k,\lambda }s(n-l,k)d_{l-1,\omega ,\lambda }^{\left(\alpha \right)}\left(x\right)\hfill \\ \hfill & ={\displaystyle \frac{1}{n}}\sum _{l=1}^n\sum _{k=0}^{n-l}\left({\displaystyle \genfrac{}{}{0pt}{}{n}{l}}\right){(-1)}^{n-l-1}l(n-l-1)!{\omega }^{k-\alpha }{\lambda }^{m-k+1}s(m,k)s(l-1,m)b_k^{\left(\alpha \right)}\left({\displaystyle \frac{x}{\omega }}\right).\hfill \end{array}$$

Corollary 3.

In (13), set $\alpha =1,x=-{\displaystyle \frac{\omega }{\lambda }},y={\displaystyle \frac{2\omega }{\lambda }}$, and when $n\ge 2$, we obtain

$$\begin{array}{cc}\hfill d_{n-1,\omega ,\lambda }\left({\displaystyle \frac{\omega }{\lambda }}\right)-d_{n-1,\omega ,\lambda }(-{\displaystyle \frac{\omega }{\lambda }})& ={\displaystyle \frac{1}{n}}\sum _{l=1}^n\sum _{k=1}^{n-l}\left({\displaystyle \genfrac{}{}{0pt}{}{n}{l}}\right)l{\left({\displaystyle \frac{2\omega }{\lambda }}\right)}_{k,\lambda }s(n-l,k)d_{l-1,\omega ,\lambda }(-{\displaystyle \frac{\omega }{\lambda }})\hfill \\ \hfill & =\sum _{l=1}^{n-1}{(-\lambda )}^{l-1}(l-1)!s(n-1,l).\hfill \end{array}$$

In this section, we first presented the definition of the higher-order degenerate type 2 $\omega$-Daehee polynomials. Subsequently, starting from this definition, we proceeded to establish and prove several inherent properties of these polynomials.

4. Some Identities of Higher-Order Degenerate Type 2 $\mathit{\omega }$-Daehee Polynomials and Numbers with Special Sequences

Theorem 3.

Let n be a nonnegative integer, the following identity holds:

$$\begin{array}{c}\hfill \sum _{k=0}^n{\lambda }^{k-\alpha }s(n,k;r)d_{k,{\displaystyle \frac{\omega }{\lambda }}}^{\left(\alpha \right)}\left({\displaystyle \frac{x}{\lambda }}\right)=\sum _{k=0}^n\left({\displaystyle \genfrac{}{}{0pt}{}{n}{k}}\right){(-r)}_{n-k}{d}_{k,\omega ,\lambda }^{\left(\alpha \right)}\left(x\right).\end{array}$$

(14)

Proof. 

By Lemma 1 and (1), we obtain the Riordan matrix

$$\begin{array}{c}\hfill \Re \left({\displaystyle \frac{k!}{n!}}{\lambda }^{k}s(n,k;r)\right)=\left({(1+\lambda t)}^{-r},\lambda \ln (1+t)\right),\end{array}$$

and

$$\begin{array}{cc}\hfill & n!\sum _{k=0}^n{\displaystyle \frac{k!}{n!}}{\lambda }^{k}s(n,k;r){\displaystyle \frac{d_{k,\frac{\omega }{\lambda }}^{\left(\alpha \right)}\left(\frac{x}{\lambda }\right)}{k!}}\hfill \\ \hfill & =n!\left[t^n\right]{(1+t)}^{-r}\left[{\left({\displaystyle \frac{\ln (1+y)}{{(1+y)}^{\frac{\omega }{\lambda }}-{(1+y)}^{-\frac{\omega }{\lambda }}}}\right)}^{\alpha }{(1+y)}^{{\displaystyle \frac{x}{\lambda }}}|y=\lambda \ln (1+t)\right]\hfill \\ \hfill & =n!\left[t^n\right]{\lambda }^{\alpha }{(1+t)}^{-r}{\left({\displaystyle \frac{\frac{1}{\lambda }\ln (1+\lambda \ln (1+t))}{{(1+\lambda \ln (1+t))}^{\frac{\omega }{\lambda }}-{(1+\lambda \ln (1+t))}^{-\frac{\omega }{\lambda }}}}\right)}^{\alpha }{(1+\lambda \ln (1+t))}^{{\displaystyle \frac{x}{\lambda }}}\hfill \\ \hfill & =n!\left[t^n\right]{\lambda }^{\alpha }\sum _{n=0}^{\infty }{(-r)}_n{\displaystyle \frac{t^n}{n!}}\sum _{n=0}^{\infty }{d}_{n,\omega ,\lambda }^{\left(\alpha \right)}\left(x\right){\displaystyle \frac{t^n}{n!}}\hfill \\ \hfill & =n!\left[t^n\right]{\lambda }^{\alpha }\sum _{n=0}^{\infty }\sum _{k=0}^n\left({\displaystyle \genfrac{}{}{0pt}{}{n}{k}}\right){(-r)}_{n-k}{d}_{k,\omega ,\lambda }^{\left(\alpha \right)}\left(x\right){\displaystyle \frac{t^n}{n!}}\hfill \\ \hfill & ={\lambda }^{\alpha }\sum _{k=0}^n\left({\displaystyle \genfrac{}{}{0pt}{}{n}{k}}\right){(-r)}_{n-k}{d}_{k,\omega ,\lambda }^{\left(\alpha \right)}\left(x\right).\hfill \end{array}$$

By simplifying both sides of the above equation, Theorem 3 is proved. □

Corollary 4.

When $r=0$ in (14), the following equation holds:

$$\begin{array}{c}\hfill \sum _{k=0}^n{\lambda }^{k-\alpha }s(n,k)d_{k,{\displaystyle \frac{\omega }{\lambda }}}^{\left(\alpha \right)}\left({\displaystyle \frac{x}{\lambda }}\right)=d_{k,\omega ,\lambda }^{\left(\alpha \right)}\left(x\right).\end{array}$$

By the inversion relation (7) and (14), the following equation holds:

$$\begin{array}{c}\hfill d_{n,{\displaystyle \frac{\omega }{\lambda }}}^{\left(\alpha \right)}\left({\displaystyle \frac{x}{\lambda }}\right)={\lambda }^{\alpha -n}\sum _{k=0}^n\sum _{j=0}^k\left({\displaystyle \genfrac{}{}{0pt}{}{k}{j}}\right){(-r)}^{k-j}S(n,k;r)d_{j,\omega ,\lambda }^{\left(\alpha \right)}\left(x\right).\end{array}$$

Theorem 4.

Let n be a nonnegative integer, the following identity holds:

$$\begin{array}{cc}\hfill \sum _{k=0}^{n}S(n,k;r)d_{k,\omega ,\lambda }^{\left(\alpha \right)}\left(x\right)& =\sum _{k=0}^n\left({\displaystyle \genfrac{}{}{0pt}{}{n}{k}}\right){\lambda }^{k-\alpha }{r}^{n-k}{d}_{k,{\displaystyle \frac{\omega }{\lambda }}}^{\left(\alpha \right)}\left({\displaystyle \frac{x}{\lambda }}\right)\hfill \\ \hfill & =\sum _{k=0}^n\sum _{j=0}^n\left({\displaystyle \genfrac{}{}{0pt}{}{n}{k}}\right){\omega }^{j-\alpha }{\lambda }^{k-j}{r}^{n-k}s(k,j)b_j^{\left(\alpha \right)}\left({\displaystyle \frac{x}{\omega }}\right).\hfill \end{array}$$

(15)

Proof. 

By Lemma 1 and (2), we obtain the Riordan matrix

$$\begin{array}{cc}\hfill \Re \left({\displaystyle \frac{k!}{n!}}S(n,k;r)\right)& =\left(e^{rt},e^t-1\right),\hfill \end{array}$$

so we can obtain

$$\begin{array}{cc}\hfill & n!\sum _{k=0}^n{\displaystyle \frac{k!}{n!}}S(n,k;r){\displaystyle \frac{d_{k,\omega ,\lambda }^{\left(\alpha \right)}\left(x\right)}{k!}}\hfill \\ \hfill & =n!\left[t^n\right]e^{rt}\left[{\left({\displaystyle \frac{\frac{1}{\lambda }\ln (1+\lambda \ln (1+y))}{{(1+\lambda \ln (1+y))}^{\frac{\omega }{\lambda }}-{(1+\lambda \ln (1+y))}^{-\frac{\omega }{\lambda }}}}\right)}^{\alpha }{(1+\lambda \ln (1+y))}^{{\displaystyle \frac{x}{\lambda }}}|y=e^t-1\right]\hfill \\ \hfill & =n!\left[t^n\right]e^{rt}{\left({\displaystyle \frac{\frac{1}{\lambda }\ln (1+\lambda t)}{{(1+\lambda t)}^{\frac{\omega }{\lambda }}-{(1+\lambda t)}^{-\frac{\omega }{\lambda }}}}\right)}^{\alpha }{(1+\lambda t)}^{{\displaystyle \frac{x}{\lambda }}}\hfill \\ \hfill & =n!\left[t^n\right]{\lambda }^{-\alpha }\sum _{n=0}^{\infty }{r}^n{\displaystyle \frac{t^n}{n!}}\sum _{n=0}^{\infty }{\lambda }^n{d}_{n,{\displaystyle \frac{\omega }{\lambda }}}^{\left(\alpha \right)}\left({\displaystyle \frac{x}{\lambda }}\right){\displaystyle \frac{t^n}{n!}}\hfill \\ \hfill & =n!\left[t^n\right]\sum _{n=0}^{\infty }\sum _{k=0}^n\left({\displaystyle \genfrac{}{}{0pt}{}{n}{k}}\right){\lambda }^{k-\alpha }{r}^{n-k}{d}_{k,{\displaystyle \frac{\omega }{\lambda }}}^{\left(\alpha \right)}\left({\displaystyle \frac{x}{\lambda }}\right){\displaystyle \frac{t^n}{n!}}\hfill \\ \hfill & =\sum _{k=0}^n\left({\displaystyle \genfrac{}{}{0pt}{}{n}{k}}\right){\lambda }^{k-\alpha }{r}^{n-k}{d}_{k,{\displaystyle \frac{\omega }{\lambda }}}^{\left(\alpha \right)}\left({\displaystyle \frac{x}{\lambda }}\right).\hfill \end{array}$$

From the above proof process, we get

$$\begin{array}{cc}\hfill n!\sum _{k=0}^n{\displaystyle \frac{k!}{n!}}S(n,k;r){\displaystyle \frac{d_{k,\omega ,\lambda }^{\left(\alpha \right)}\left(x\right)}{k!}}& =n!\left[t^n\right]e^{rt}{\left({\displaystyle \frac{\frac{1}{\lambda }\ln (1+\lambda t)}{{(1+\lambda t)}^{\frac{\omega }{\lambda }}-{(1+\lambda t)}^{-\frac{\omega }{\lambda }}}}\right)}^{\alpha }{(1+\lambda t)}^{{\displaystyle \frac{x}{\lambda }}}\hfill \\ \hfill & =n!\left[t^n\right]{\omega }^{-\alpha }\sum _{n=0}^{\infty }{r}^n{\displaystyle \frac{t^n}{n!}}\sum _{j=0}^{\infty }{\left({\displaystyle \frac{\omega }{\lambda }}\right)}^j{b}_j^{\left(\alpha \right)}\left({\displaystyle \frac{x}{\omega }}\right){\displaystyle \frac{{\left[\ln (1+\lambda t)\right]}^j}{j!}}\hfill \\ \hfill & =n!\left[t^n\right]\sum _{n=0}^{\infty }\sum _{k=0}^n\sum _{j=0}^k\left({\displaystyle \genfrac{}{}{0pt}{}{n}{k}}\right){\omega }^{j-\alpha }{\lambda }^{k-j}{r}^{n-k}s(k,j)b_j^{\left(\alpha \right)}\left({\displaystyle \frac{x}{\omega }}\right){\displaystyle \frac{t^n}{n!}}\hfill \\ \hfill & =\sum _{k=0}^n\sum _{j=0}^k\left({\displaystyle \genfrac{}{}{0pt}{}{n}{k}}\right){\omega }^{j-\alpha }{\lambda }^{k-j}{r}^{n-k}s(k,j)b_j^{\left(\alpha \right)}\left({\displaystyle \frac{x}{\omega }}\right).\hfill \end{array}$$

Thus, the proof is complete. □

Corollary 5.

When $r=0$ in (15), the following equation holds:

$$\begin{array}{c}\hfill {\lambda }^{n-\alpha }{d}_{n,{\displaystyle \frac{\omega }{\lambda }}}^{\left(\alpha \right)}\left({\displaystyle \frac{x}{\lambda }}\right)=\sum _{k=0}^{n}S(n,k)d_{k,\omega ,\lambda }^{\left(\alpha \right)}\left(x\right)=\sum _{k=0}^n{\omega }^{k-\alpha }{\lambda }^{n-k}s(n,k)b_k^{\left(\alpha \right)}\left({\displaystyle \frac{x}{\omega }}\right).\end{array}$$

The identity establishes the equivalence between two combinatorial decompositions of a scaled generating function: one employing set partitions, and the other utilizing the cycle structure of permutations, governed by Stirling numbers of the second and first kind, respectively.

It is noteworthy that a key corollary of our main result (Corollary 5) subsumes the important finding of Minyoung Ma (2019) (see [6]). Specifically, by setting the parameters $\lambda =1$ and $\alpha =1$ in Corollary 5, we directly recover Theorem 1 of [6]. This connection not only validates the correctness of our more general framework but also highlights how it extends the existing theory.

Theorem 5.

Let n be a nonnegative integer, the following equality holds:

$$\begin{array}{cc}\hfill & \sum _{m=0}^n\sum _{k=0}^m{s}_{1,\lambda }(n,m)S(m,k;r)d_{k,\omega ,\lambda }^{\left(\alpha \right)}\left(x\right)=\sum _{m=0}^n\sum _{k=0}^m\left({\displaystyle \genfrac{}{}{0pt}{}{n}{m}}\right)r^k{s}_{1,\lambda }(m,k)d_{n-m,\omega }^{\left(\alpha \right)}\left(x\right).\hfill \end{array}$$

(16)

Proof. 

From the proof of Theorem 4, we obtain

$$\begin{array}{cc}\hfill & \sum _{n=0}^{\infty }\sum _{k=0}^{n}S(n,k;r)d_{k,\omega ,\lambda }^{\left(\alpha \right)}\left(x\right){\displaystyle \frac{t^n}{n!}}=e^{rt}{\left({\displaystyle \frac{\frac{1}{\lambda }\ln (1+\lambda t)}{{(1+\lambda t)}^{\frac{\omega }{\lambda }}-{(1+\lambda t)}^{-\frac{\omega }{\lambda }}}}\right)}^{\alpha }{(1+\lambda t)}^{{\displaystyle \frac{x}{\lambda }}}.\hfill \end{array}$$

Thus, we obtain the Riordan matrix

$$\begin{array}{cc}\hfill & \Re \left({\displaystyle \frac{k!}{n!}}{s}_{1,\lambda }(n,k)\right)=\left(1,{\displaystyle \frac{1}{\lambda }}[{(1+t)}^{\lambda }-1]\right),\hfill \end{array}$$

and we get

$$\begin{array}{cc}\hfill & n!\sum _{m=0}^n{\displaystyle \frac{m!}{n!}}{s}_{1,\lambda }(n,m){\displaystyle \frac{1}{m!}}\sum _{k=0}^{m}S(m,k;r)d_{k,\omega ,\lambda }^{\left(\alpha \right)}\left(x\right)\hfill \\ \hfill & =n!\left[t^n\right]\left[e^{ry}{\left({\displaystyle \frac{\frac{1}{\lambda }\ln (1+\lambda y)}{{(1+\lambda y)}^{\frac{\omega }{\lambda }}-{(1+\lambda y)}^{-\frac{\omega }{\lambda }}}}\right)}^{\alpha }{(1+\lambda y)}^{{\displaystyle \frac{x}{\lambda }}}|y={\displaystyle \frac{1}{\lambda }}[{(1+t)}^{\lambda }-1]\right]\hfill \\ \hfill & =n!\left[t^n\right]e^{r·{\displaystyle \frac{1}{\lambda }}[{(1+t)}^{\lambda }-1]}{\left({\displaystyle \frac{\ln (1+t)}{{(1+t)}^{\omega }-{(1+t)}^{-\omega }}}\right)}^{\alpha }{(1+t)}^x\hfill \\ \hfill & =n!\left[t^n\right]\sum _{n=0}^{\infty }\sum _{k=0}^n{r}^k{s}_{1,\lambda }(n,k){\displaystyle \frac{t^n}{n!}}\sum _{n=0}^{\infty }{d}_{n,\omega }^{\left(\alpha \right)}\left(x\right){\displaystyle \frac{t^n}{n!}}\hfill \\ \hfill & =n!\left[t^n\right]\sum _{n=0}^{\infty }\sum _{m=0}^n\sum _{k=0}^m\left({\displaystyle \genfrac{}{}{0pt}{}{n}{m}}\right)r^k{s}_{1,\lambda }(m,k)d_{n-m,\omega }^{\left(\alpha \right)}\left(x\right){\displaystyle \frac{t^n}{n!}}\hfill \\ \hfill & =\sum _{m=0}^n\sum _{k=0}^m\left({\displaystyle \genfrac{}{}{0pt}{}{n}{m}}\right)r^k{s}_{1,\lambda }(m,k)d_{n-m,\omega }^{\left(\alpha \right)}\left(x\right).\hfill \end{array}$$

Thus, the proof is complete. □

Corollary 6.

When $r=0$ in (16), the following equation holds:

$$\begin{array}{c}\hfill d_{n,\omega }^{\left(\alpha \right)}\left(x\right)=\sum _{m=0}^n\sum _{k=0}^m{s}_{1,\lambda }(n,m)S(m,k)d_{k,\omega ,\lambda }^{\left(\alpha \right)}\left(x\right).\end{array}$$

By the inverse relations (7), (8) and (16), the following equalities hold:

$$\begin{array}{c}\hfill \sum _{j=0}^{n}S(n,j;r)d_{j,\omega ,\lambda }^{\left(\alpha \right)}\left(x\right)=\sum _{m=0}^n\sum _{l=0}^m\sum _{k=0}^l\left({\displaystyle \genfrac{}{}{0pt}{}{m}{l}}\right)r^k{s}_{1,\lambda }(l,k)S_{2,\lambda }(n,m)d_{m-l,\omega }^{\left(\alpha \right)}\left(x\right),\end{array}$$

and

$$\begin{array}{c}\hfill d_{n,\omega ,\lambda }^{\left(\alpha \right)}\left(x\right)=\sum _{j=0}^n\sum _{m=0}^j\sum _{l=0}^m\sum _{k=0}^l\left({\displaystyle \genfrac{}{}{0pt}{}{m}{l}}\right)r^k{s}_{1,\lambda }(l,k)s(n,j;r)S_{2,\lambda }(n,m)d_{m-l,\omega }^{\left(\alpha \right)}\left(x\right).\end{array}$$

Theorem 6.

Let n be a nonnegative integer, the following equality holds:

$$\begin{array}{cc}\hfill \sum _{k=0}^{n}L(n,k;r)d_{k,\omega ,\lambda }^{\left(\alpha \right)}\left(x\right)& =\sum _{k=0}^n\left({\displaystyle \genfrac{}{}{0pt}{}{n}{k}}\right){\left(r\right)}_{n-k}{d}_{k,\omega ,-\lambda }^{\left(\alpha \right)}(-x)\hfill \\ \hfill & =\sum _{k=0}^n\sum _{l=0}^k\sum _{j=0}^l\left({\displaystyle \genfrac{}{}{0pt}{}{n}{k}}\right){(-1)}^l{\left(r\right)}_{n-k}{\omega }^{j-\alpha }{\lambda }^{l-j}s(l,j)s(k,l)b_j^{\left(\alpha \right)}\left({\displaystyle \frac{x}{\omega }}\right).\hfill \end{array}$$

(17)

Proof. 

By Lemma 1 and (3), we obtain the Riordan matrix

$$\begin{array}{c}\hfill \Re \left({\displaystyle \frac{k!}{n!}}L(n,k;r)\right)=\left({(1+t)}^r,{\displaystyle \frac{-t}{1+t}}\right),\end{array}$$

so we can obtain

$$\begin{array}{cc}\hfill & n!\sum _{k=0}^n{\displaystyle \frac{k!}{n!}}L(n,k;r){\displaystyle \frac{d_{k,\omega ,\lambda }^{\left(\alpha \right)}\left(x\right)}{k!}}\hfill \\ \hfill & =n!\left[t^n\right]{(1+t)}^r\left[{\left({\displaystyle \frac{\frac{1}{\lambda }\ln (1+\lambda \ln (1+y))}{{(1+\lambda \ln (1+y))}^{\frac{\omega }{\lambda }}-{(1+\lambda \ln (1+y))}^{-\frac{\omega }{\lambda }}}}\right)}^{\alpha }{(1+\lambda \ln (1+y))}^{{\displaystyle \frac{x}{\lambda }}}|y={\displaystyle \frac{-t}{1+t}}\right]\hfill \\ \hfill & =n!\left[t^n\right]{(1+t)}^r{\left({\displaystyle \frac{\frac{1}{\lambda }\ln (1-\lambda \ln (1+t))}{{(1-\lambda \ln (1+t))}^{\frac{\omega }{\lambda }}-{(1-\lambda \ln (1+t))}^{-\frac{\omega }{\lambda }}}}\right)}^{\alpha }{(1-\lambda \ln (1+t))}^{{\displaystyle \frac{x}{\lambda }}}\hfill \\ \hfill & =n!\left[t^n\right]\sum _{n=0}^{\infty }{\left(r\right)}_n{\displaystyle \frac{t^n}{n!}}\sum _{n=0}^{\infty }{d}_{n,\omega ,-\lambda }^{\left(\alpha \right)}(-x){\displaystyle \frac{t^n}{n!}}\hfill \\ \hfill & =n!\left[t^n\right]\sum _{n=0}^{\infty }\sum _{k=0}^n\left({\displaystyle \genfrac{}{}{0pt}{}{n}{k}}\right){\left(r\right)}_{n-k}{d}_{k,\omega ,-\lambda }^{\left(\alpha \right)}(-x){\displaystyle \frac{t^n}{n!}}\hfill \\ \hfill & =\sum _{k=0}^n\left({\displaystyle \genfrac{}{}{0pt}{}{n}{k}}\right){\left(r\right)}_{n-k}{d}_{k,\omega ,-\lambda }^{\left(\alpha \right)}(-x).\hfill \end{array}$$

From the above proof process, we obtain

$$\begin{array}{cc}\hfill & n!\sum _{k=0}^n{\displaystyle \frac{k!}{n!}}L(n,k;r){\displaystyle \frac{d_{k,\omega ,\lambda }^{\left(\alpha \right)}\left(x\right)}{k!}}\hfill \\ \hfill & =n!\left[t^n\right]{\omega }^{-\alpha }{(1+t)}^r{\left({\displaystyle \frac{\frac{\omega }{\lambda }\ln (1-\lambda \ln (1+t))}{e^{\frac{\omega }{\lambda }\ln (1-\lambda \ln (1+t))}-e^{-\frac{\omega }{\lambda }\ln (1-\lambda \ln (1+t))}}}\right)}^{\alpha }{e}^{{\displaystyle \frac{x}{\omega }}·{\displaystyle \frac{\omega }{\lambda }}\ln (1-\lambda \ln (1+t))}\hfill \\ \hfill & =n!\left[t^n\right]\sum _{n=0}^{\infty }{\left(r\right)}_n{\displaystyle \frac{t^n}{n!}}\sum _{n=0}^{\infty }\sum _{l=0}^n\sum _{j=0}^l{(-1)}^l{\omega }^{j-\alpha }{\lambda }^{l-j}s(l,j)s(n,l)b_j^{\left(\alpha \right)}({\displaystyle \frac{x}{\omega }}){\displaystyle \frac{t^n}{n!}}\hfill \\ \hfill & =n!\left[t^n\right]\sum _{n=0}^{\infty }\sum _{k=0}^n\sum _{l=0}^k\sum _{j=0}^l\left({\displaystyle \genfrac{}{}{0pt}{}{n}{k}}\right){(-1)}^l{\left(r\right)}_{n-k}{\omega }^{j-\alpha }{\lambda }^{l-j}s(l,j)s(k,l)b_j^{\left(\alpha \right)}({\displaystyle \frac{x}{\omega }}){\displaystyle \frac{t^n}{n!}}\hfill \\ \hfill & =\sum _{k=0}^n\sum _{l=0}^k\sum _{j=0}^l\left({\displaystyle \genfrac{}{}{0pt}{}{n}{k}}\right){(-1)}^l{\left(r\right)}_{n-k}{\omega }^{j-\alpha }{\lambda }^{l-j}s(l,j)s(k,l)b_j^{\left(\alpha \right)}({\displaystyle \frac{x}{\omega }}).\hfill \end{array}$$

Thus, the proof is complete. □

Corollary 7.

When $r=0$ in (17), the following equation holds:

$$\begin{array}{c}\hfill d_{n,\omega ,-\lambda }^{\left(\alpha \right)}(-x)=\sum _{k=0}^{n}L(n,k)d_{k,\omega ,\lambda }^{\left(\alpha \right)}\left(x\right)=\sum _{k=0}^n\sum _{j=0}^k{(-1)}^k{\omega }^{j-\alpha }{\lambda }^{k-j}s(k,j)s(n,k)b_j^{\left(\alpha \right)}\left({\displaystyle \frac{x}{\omega }}\right).\end{array}$$

(18)

By the inversion relation (6) and (18), the following equation holds:

$$\begin{array}{cc}\hfill \sum _{k=0}^{n}S(n,k)d_{k,\omega ,-\lambda }^{\left(\alpha \right)}(-x)& =\sum _{k=0}^n\sum _{j=0}^{k}S(n,k)L(k,j)d_{j,\omega ,\lambda }^{\left(\alpha \right)}\left(x\right)\hfill \\ & =\sum _{k=0}^n{(-1)}^n{\omega }^{k-\alpha }{\lambda }^{n-k}s(n,k)b_k^{\left(\alpha \right)}\left({\displaystyle \frac{x}{\omega }}\right).\hfill \end{array}$$

Theorem 6 reveals a fundamental combinatorial equivalence: the sum of generalized Lah numbers and higher-order degenerate type 2 $\omega$-Daehee polynomials can be transformed into a triple sum involving Stirling numbers of the first kind and Bernoulli polynomials. This identity underscores a deep connection between partition-based enumerations and permutation cycles, emphasizing invariance under parameter shifts and sign changes.

Theorem 7.

Let n be a nonnegative integer, the following equation holds:

$$\begin{array}{cc}\hfill & \sum _{k=0}^n{\lambda }^{k-\alpha }{H}_{n,r}{d}_{k,{\displaystyle \frac{\omega }{\lambda }}}^{\left(\alpha \right)}\left({\displaystyle \frac{x}{\lambda }}\right)=\sum _{k=0}^n{\displaystyle \frac{d_{k,\omega ,\lambda }^{\left(\alpha \right)}\left(x\right)}{k!}}.\hfill \end{array}$$

Proof. 

By Lemma 1 and (4), we obtain the Riordan matrix

$$\begin{array}{c}\hfill \Re \left({(-1)}^{n-r}r!{\lambda }^r{H}_{n,k}\right)=\left({\displaystyle \frac{1}{1+t}},\lambda \ln (1+t)\right),\end{array}$$

so we can obtain

$$\begin{array}{cc}\hfill & \sum _{k=0}^n{(-1)}^{n-k}k!{\lambda }^k{H}_{n,k}{\displaystyle \frac{d_{k,\frac{\omega }{\lambda }}^{\left(\alpha \right)}\left(\frac{x}{\lambda }\right)}{k!}}\hfill \\ \hfill & =\left[t^n\right]{\displaystyle \frac{1}{1+t}}\left[{\left({\displaystyle \frac{\ln (1+y)}{{(1+y)}^{\frac{\omega }{\lambda }}-{(1+y)}^{-\frac{\omega }{\lambda }}}}\right)}^{\alpha }{(1+y)}^{{\displaystyle \frac{x}{\lambda }}}|y=\lambda \ln (1+t)\right]\hfill \\ \hfill & =\left[t^n\right]{\lambda }^{\alpha }{\displaystyle \frac{1}{1+t}}{\left({\displaystyle \frac{\frac{1}{\lambda }\ln (1+\lambda \ln (1+t))}{{(1+\lambda \ln (1+t))}^{\frac{\omega }{\lambda }}-{(1+\lambda \ln (1+t))}^{-\frac{\omega }{\lambda }}}}\right)}^{\alpha }{(1+\lambda \ln (1+t))}^{{\displaystyle \frac{x}{\lambda }}}\hfill \\ \hfill & =\left[t^n\right]{\lambda }^{\alpha }\sum _{n=0}^{\infty }{(-1)}^n{t}^n\sum _{n=0}^{\infty }{d}_{n,\omega ,\lambda }^{\left(\alpha \right)}\left(x\right){\displaystyle \frac{t^n}{n!}}\hfill \\ \hfill & =\sum _{k=0}^n{(-1)}^{n-k}{\lambda }^{\alpha }{\displaystyle \frac{d_{k,\omega ,\lambda }^{\left(\alpha \right)}\left(x\right)}{k!}}.\hfill \end{array}$$

By simplifying both sides of the above equation, Theorem 7 is proved. □

Theorem 8.

Let n be a nonnegative integer, the following equation holds:

$$\begin{array}{c}\hfill \sum _{k=0}^{n}s(n,k;r)d_{k,\omega ,\lambda }^{\left(\alpha \right)}\left(x\right)=\sum _{k=0}^n\sum _{j=0}^k\sum _{i=0}^j\left({\displaystyle \genfrac{}{}{0pt}{}{n}{k}}\right){\omega }^{i-\alpha }{\lambda }^{j-i}{(-r)}_{n-k}s(j,i)\beta (k,j)b_i^{\left(\alpha \right)}\left({\displaystyle \frac{x}{\omega }}\right).\end{array}$$

(19)

Proof. 

By Lemma 1, we obtain

$$\begin{array}{cc}\hfill & n!\sum _{k=0}^n{\displaystyle \frac{k!}{n!}}s(n,k;r){\displaystyle \frac{d_{k,\omega ,\lambda }^{\left(\alpha \right)}\left(x\right)}{k!}}\hfill \\ \hfill & =n!\left[t^n\right]{\omega }^{-\alpha }\sum _{n=0}^{\infty }{(-r)}_n{\displaystyle \frac{t^n}{n!}}\sum _{i=0}^{\infty }{\left({\displaystyle \frac{\omega }{\lambda }}\right)}^i{b}_i^{\left(\alpha \right)}\left({\displaystyle \frac{x}{\omega }}\right)\sum _{j=i}^{\infty }{\lambda }^{j}s(j,i)\sum _{k=j}^{\infty }\beta (k,j){\displaystyle \frac{t^k}{k!}}\hfill \\ \hfill & =\sum _{k=0}^n\sum _{j=0}^k\sum _{i=0}^j\left({\displaystyle \genfrac{}{}{0pt}{}{n}{k}}\right){\omega }^{i-\alpha }{\lambda }^{j-i}{(-r)}_{n-k}s(j,i)\beta (k,j)b_i^{\left(\alpha \right)}\left({\displaystyle \frac{x}{\omega }}\right).\hfill \end{array}$$

By simplifying both sides of the above equation, Theorem 8 is proved. □

Corollary 8.

When $r=0$ in (19), the following equation holds:

$$\begin{array}{c}\hfill \sum _{k=0}^{n}s(n,k)d_{k,\omega ,\lambda }^{\left(\alpha \right)}\left(x\right)=\sum _{k=0}^n\sum _{j=0}^k{\omega }^{j-\alpha }{\lambda }^{k-j}s(k,j)\beta (n,k)b_j^{\left(\alpha \right)}\left({\displaystyle \frac{x}{\omega }}\right).\end{array}$$

By the inversion relation (7) and (19), the following equation holds:

$$\begin{array}{c}\hfill d_{n,\omega ,\lambda }^{\left(\alpha \right)}\left(x\right)=\sum _{m=0}^n\sum _{k=0}^m\sum _{j=0}^k\sum _{i=0}^j\left({\displaystyle \genfrac{}{}{0pt}{}{m}{k}}\right){\omega }^{i-\alpha }{\lambda }^{j-i}{(-r)}_{m-k}s(j,i)\beta (k,j)S(n,m;r)b_i^{\left(\alpha \right)}\left({\displaystyle \frac{x}{\omega }}\right).\end{array}$$

Theorem 9.

Let n be a nonnegative integer, the following equation holds:

$$\begin{array}{c}\hfill \sum _{m=0}^n\sum _{k=0}^{m}s(m,k;r)B(n,m)d_{k,\omega ,\lambda }^{\left(\alpha \right)}\left(x\right)=\sum _{m=0}^n\sum _{k=0}^m\left({\displaystyle \genfrac{}{}{0pt}{}{n}{m}}\right){(-r)}^k{\lambda }^{n-m-\alpha }S(m,k)d_{n-m,{\displaystyle \frac{\omega }{\lambda }}}^{\left(\alpha \right)}\left({\displaystyle \frac{x}{\lambda }}\right).\end{array}$$

Proof. 

By Lemma 1 and the proof process of Theorem 8, we obtain

$$\begin{array}{cc}\hfill & n!\sum _{m=0}^n{\displaystyle \frac{m!}{n!}}B(n,m){\displaystyle \frac{1}{m!}}\sum _{k=0}^{m}s(m,k;r)d_{k,\omega ,\lambda }^{\left(\alpha \right)}\left(x\right)\hfill \\ \hfill & =n!\left[t^n\right][{(1+y)}^{-r}\left({\displaystyle \frac{\frac{1}{\lambda }\ln (1+\lambda \ln (1+\ln (1+y)))}{{(1+\lambda \ln (1+\ln (1+y)))}^{-\frac{\omega }{\lambda }}-{(1+\lambda \ln (1+\ln (1+y)))}^{\frac{\omega }{\lambda }}}}\right)\hfill & \\ & ·{(1+\lambda \ln (1+\ln (1+y)))}^{{\displaystyle \frac{x}{\lambda }}}|y=e^{e^t-1}-1]\hfill \\ \hfill & =n!\left[t^n\right]e^{-r(e^t-1)}{\left({\displaystyle \frac{\frac{1}{\lambda }\ln (1+\lambda t)}{{(1+\lambda t)}^{\frac{\omega }{\lambda }}-{(1+\lambda t)}^{-\frac{\omega }{\lambda }}}}\right)}^{\alpha }{(1+\lambda t)}^{{\displaystyle \frac{x}{\lambda }}}\hfill \\ \hfill & =n!\left[t^n\right]{\lambda }^{-\alpha }\sum _{n=0}^{\infty }\sum _{k=0}^n{(-r)}^{k}S(n,k){\displaystyle \frac{t^n}{n!}}\sum _{n=0}^{\infty }{\lambda }^n{d}_{n,{\displaystyle \frac{\omega }{\lambda }}}^{\left(\alpha \right)}\left({\displaystyle \frac{x}{\lambda }}\right){\displaystyle \frac{t^n}{n!}}\hfill \\ \hfill & =\sum _{m=0}^n\sum _{k=0}^m\left({\displaystyle \genfrac{}{}{0pt}{}{n}{m}}\right){(-r)}^k{\lambda }^{n-m-\alpha }S(m,k)d_{n-m,{\displaystyle \frac{\omega }{\lambda }}}^{\left(\alpha \right)}\left({\displaystyle \frac{x}{\lambda }}\right).\hfill \end{array}$$

By simplifying both sides of the above equation, Theorem 9 is proved. □

Theorem 10.

Let n be a nonnegative integer, the following equation holds:

$$\begin{array}{c}\hfill {\left(2\omega \right)}^{n+k}T(n+k,k)=\left({\displaystyle \genfrac{}{}{0pt}{}{n+k}{k}}\right)\sum _{m=0}^n\sum _{l=0}^m{\lambda }^{n-m}S(m,l)S(n,m)d_{l,\omega ,\lambda }^{(-k)}.\end{array}$$

Proof. 

Let $\alpha =-k,x=0$ in the generating function of the higher-order degenerate type 2 $\omega$-Daehee polynomials, and set $t=e^{{\displaystyle \frac{1}{\lambda }}(e^{{\displaystyle \frac{\lambda t}{2}}}-1)}-1$, then we have

$$\begin{array}{cc}\hfill \sum _{l=0}^{\infty }{d}_{l,\omega ,\lambda }^{(-k)}{\displaystyle \frac{(e^{\frac{1}{\lambda }(e^{\frac{\lambda t}{2}}-1)}-1{)}^l}{l!}}& ={\left({\displaystyle \frac{2}{t}}\right)}^k{\left(e^{{\displaystyle \frac{\omega t}{2}}}-e^{-{\displaystyle \frac{\omega t}{2}}}\right)}^k\hfill \\ \hfill & ={\displaystyle \frac{2^{k}k!}{t^k}}\sum _{l=k}^{\infty }{\omega }^{l}T(l,k){\displaystyle \frac{t^l}{l!}}\hfill \\ \hfill & =2^{k}k!\sum _{n=0}^{\infty }{\omega }^{n+k}T(n+k,k){\displaystyle \frac{t^n}{(n+k)!}}\hfill \\ \hfill & =2^k\sum _{n=0}^{\infty }{\omega }^{n+k}T(n+k,k){\displaystyle \frac{1}{\left(\genfrac{}{}{0pt}{}{n+k}{k}\right)}}{\displaystyle \frac{t^n}{n!}}.\hfill \end{array}$$

The left-hand side of the equation equals

$$\begin{array}{cc}\hfill \sum _{l=0}^{\infty }{d}_{l,\omega ,\lambda }^{(-k)}{\displaystyle \frac{(e^{\frac{1}{\lambda }(e^{\frac{\lambda t}{2}}-1)}-1{)}^l}{l!}}& =\sum _{l=0}^{\infty }{d}_{l,\omega ,\lambda }^{(-k)}\sum _{m=l}^{\infty }S(m,l){\lambda }^{-m}\sum _{n=m}^{\infty }S(n,m){\left({\displaystyle \frac{\lambda }{2}}\right)}^n{\displaystyle \frac{t^n}{n!}}\hfill \\ \hfill & =\sum _{n=0}^{\infty }{\displaystyle \frac{1}{2^n}}\sum _{m=0}^n\sum _{l=0}^m{\lambda }^{n-m}S(m,l)S(n,m)d_{l,\omega ,\lambda }^{(-k)}{\displaystyle \frac{t^n}{n!}}.\hfill \end{array}$$

Equating the coefficients of ${\displaystyle \frac{t^n}{n!}}$ in the above two expressions yields the desired result. □

Theorem 10 reveals a combinatorial identity where the central factorial numbers $T(n+k,k)$, scaled by ${\left(2\omega \right)}^{n+k}$, are expressed as a double sum involving binomial coefficients, Stirling numbers of the second kind, and higher-order degenerate type 2 $\omega$-Daehee polynomials. This demonstrates a fundamental link between enumeration problems for factorial numbers and those involving set partitions and polynomial sequences, emphasizing invariance under parameter transformations.

The primary focus of this section was to explore and reveal the relationships linking the higher-order degenerate type 2 $\omega$-Daehee polynomials with a family of significant special polynomials and numbers. These interconnected entities included the generalized Stirling numbers, higher-order type 2 Bernoulli polynomials, degenerate Stirling numbers, generalized Lah numbers, generalized Bell numbers, and central factorial numbers of the second kind.

5. Symmetric Identities for Higher-Order Degenerate Type 2 $\mathit{\omega }$-Daehee Polynomials

To study symmetric identities related to higher-order degenerate type 2 $\omega$-Daehee polynomials, by setting $\alpha =\omega =1$ in the higher-order degenerate type 2 $\omega$-Daehee polynomials (10), we obtain:

$$\begin{array}{c}\hfill {\displaystyle \frac{\frac{1}{\lambda }\ln (1+\lambda \ln (1+t))}{{(1+\lambda \ln (1+t))}^{\frac{1}{\lambda }}-{(1+\lambda \ln (1+t))}^{-\frac{1}{\lambda }}}}{(1+\lambda \ln (1+t))}^{{\displaystyle \frac{x}{\lambda }}}=\sum _{n=0}^{\infty }{d}_{n,1,\lambda }\left(x\right){\displaystyle \frac{t^n}{n!}}.\end{array}$$

Let n be a nonnegative integer and $N\ge 1$ be an integer. From the above, we observe:

$$\begin{array}{cc}\hfill & {\displaystyle \frac{{(1+\lambda \ln (1+t))}^{\frac{1}{\lambda }}\ln (1+\lambda \ln (1+t))}{\lambda }}\sum _{n=0}^{\infty }\sum _{l=0}^{N-1}{2}^n{\left(l\right)}_{n,{\displaystyle \frac{\lambda }{2}}}{\displaystyle \frac{{\left(\ln (1+t)\right)}^n}{n!}}\hfill \\ \hfill & ={\displaystyle \frac{{(1+\lambda \ln (1+t))}^{\frac{1}{\lambda }}\ln (1+\lambda \ln (1+t))}{\lambda }}\sum _{l=0}^{N-1}{(1+\lambda \ln (1+t))}^{{\displaystyle \frac{2l}{\lambda }}}\hfill \\ \hfill & ={\displaystyle \frac{{(1+\lambda \ln (1+t))}^{\frac{1}{\lambda }}\ln (1+\lambda \ln (1+t))}{\lambda }}{\displaystyle \frac{{(1+\lambda \ln (1+t))}^{\frac{2N}{\lambda }}-1}{{(1+\lambda \ln (1+t))}^{\frac{2}{\lambda }}-1}}\hfill \\ \hfill & ={\lambda }^{n-1}\sum _{n=0}^{\infty }\left[d_{n,{\displaystyle \frac{1}{\lambda }}}\left({\displaystyle \frac{2N}{\lambda }}\right)-d_{n,{\displaystyle \frac{1}{\lambda }}}\right]{\displaystyle \frac{{\left(\ln (1+t)\right)}^n}{n!}}.\hfill \end{array}$$

Rearranging, we get:

$$\begin{array}{c}\hfill \sum _{l=0}^{N-1}{\left(l\right)}_{n,{\displaystyle \frac{\lambda }{2}}}={\left({\displaystyle \frac{\lambda }{2}}\right)}^n{\displaystyle \frac{d_{n,\frac{1}{\lambda }}\left(\frac{2N}{\lambda }\right)-d_{n,\frac{1}{\lambda }}}{{(1+\lambda \ln (1+t))}^{\frac{1}{\lambda }}\ln (1+\lambda \ln (1+t))}}.\end{array}$$

Theorem 11.

Let n be a nonnegative integer, ${\omega }_1$, ${\omega }_2$, $m\in \mathbb{N}$, the following equation holds:

$$\begin{array}{c}\hfill \sum _{j=0}^n\sum _{i=0}^j\sum _{k=0}^i\left({\displaystyle \genfrac{}{}{0pt}{}{n}{j}}\right)\left({\displaystyle \genfrac{}{}{0pt}{}{j}{i}}\right)d_{n-j,{\omega }_2,\lambda }^{(m-1)}\left({\omega }_1{\omega }_{2}y\right){\left(2{\omega }_2\right)}^{k}s(i,k)T_{k,{\displaystyle \frac{\lambda }{2{\omega }_2}}}({\omega }_1-1\mid l)d_{j-i,{\omega }_1,\lambda }^{\left(m\right)}({\omega }_1{\omega }_{2}x+{\omega }_2)\\ \hfill =\sum _{j=0}^n\sum _{i=0}^j\sum _{k=0}^i\left({\displaystyle \genfrac{}{}{0pt}{}{n}{j}}\right)\left({\displaystyle \genfrac{}{}{0pt}{}{j}{i}}\right)d_{n-j,{\omega }_1,\lambda }^{(m-1)}\left({\omega }_1{\omega }_{2}y\right){\left(2{\omega }_1\right)}^{k}s(i,k)T_{k,{\displaystyle \frac{\lambda }{2{\omega }_1}}}({\omega }_2-1\mid l)d_{j-i,{\omega }_2,\lambda }^{\left(m\right)}({\omega }_1{\omega }_{2}x+{\omega }_1).\end{array}$$

(20)

Proof. 

Let $T_{k,\lambda }\left(n\right|l)={\sum }_{l=0}^n{\left(l\right)}_{k,\lambda }$, where n is a nonnegative integer, and ${\omega }_1,{\omega }_2,m\in \mathbb{N}$. We consider the following functional equation derived from the generating function for the m-order degenerate type 2 $\omega$-Daehee polynomials:

$$\begin{array}{cc}\hfill & I({\omega }_1,{\omega }_2)\hfill \\ \hfill & ={\left({\displaystyle \frac{\frac{1}{\lambda }\ln (1+\lambda \ln (1+t))}{{(1+\lambda \ln (1+t))}^{\frac{2{\omega }_1}{\lambda }}-1}}\right)}^m{(1+\lambda \ln (1+t))}^{{\displaystyle \frac{{\omega }_1{\omega }_{2}x+m{\omega }_1}{\lambda }}}·{\displaystyle \frac{\lambda \left[{(1+\lambda \ln (1+t))}^{\frac{2{\omega }_1{\omega }_2}{\lambda }}-1\right]}{\ln (1+\lambda \ln (1+t\left)\right)}}\hfill \\ \hfill & \times {\left({\displaystyle \frac{\frac{1}{\lambda }\ln (1+\lambda \ln (1+t))}{{(1+\lambda \ln (1+t))}^{\frac{2{\omega }_2}{\lambda }}-1}}\right)}^m{(1+\lambda \ln (1+t))}^{{\displaystyle \frac{{\omega }_1{\omega }_{2}y+m{\omega }_2}{\lambda }}}\hfill \\ \hfill & =\sum _{n=0}^{\infty }{d}_{n,{\omega }_1,\lambda }^{\left(m\right)}({\omega }_1{\omega }_{2}x+{\omega }_2){\displaystyle \frac{t^n}{n!}}\sum _{l=0}^{{\omega }_1-1}{(1+\lambda \ln (1+t))}^{{\displaystyle \frac{2{\omega }_{2}l}{\lambda }}}\sum _{n=0}^{\infty }{d}_{n,{\omega }_2,\lambda }^{(m-1)}\left({\omega }_1{\omega }_{2}y\right){\displaystyle \frac{t^n}{n!}}\hfill \\ \hfill & =\sum _{n=0}^{\infty }{d}_{n,{\omega }_1,\lambda }^{\left(m\right)}({\omega }_1{\omega }_{2}x+{\omega }_2){\displaystyle \frac{t^n}{n!}}\sum _{n=0}^{\infty }\sum _{k=0}^{n}s(n,k){\left(2{\omega }_2\right)}^k\sum _{l=0}^{{\omega }_1-1}{\left(l\right)}_{k,{\displaystyle \frac{\lambda }{2{\omega }_2}}}{\displaystyle \frac{t^n}{n!}}\sum _{n=0}^{\infty }{d}_{n,{\omega }_2,\lambda }^{(m-1)}\left({\omega }_1{\omega }_{2}y\right){\displaystyle \frac{t^n}{n!}}\hfill \\ \hfill & =\sum _{n=0}^{\infty }{d}_{n,{\omega }_1,\lambda }^{\left(m\right)}({\omega }_1{\omega }_{2}x+{\omega }_2){\displaystyle \frac{t^n}{n!}}\sum _{n=0}^{\infty }\sum _{k=0}^{n}s(n,k){\left(2{\omega }_2\right)}^k{T}_{k,{\displaystyle \frac{\lambda }{2{\omega }_2}}}({\omega }_1-1\mid l)\sum _{n=0}^{\infty }{d}_{n,{\omega }_2,\lambda }^{(m-1)}\left({\omega }_1{\omega }_{2}y\right){\displaystyle \frac{t^n}{n!}}\hfill \\ \hfill & =\sum _{n=0}^{\infty }\sum _{i=0}^n\left({\displaystyle \genfrac{}{}{0pt}{}{n}{i}}\right)d_{n-i,{\omega }_1,\lambda }^{\left(m\right)}({\omega }_1{\omega }_{2}x+{\omega }_2)\sum _{k=0}^{i}s(i,k){\left(2{\omega }_2\right)}^k{T}_{k,{\displaystyle \frac{\lambda }{2{\omega }_2}}}({\omega }_1-1\mid l){\displaystyle \frac{t^n}{n!}}\sum _{n=0}^{\infty }{d}_{n,{\omega }_2,\lambda }^{(m-1)}\left({\omega }_1{\omega }_{2}y\right){\displaystyle \frac{t^n}{n!}}\hfill \\ \hfill & =\sum _{n=0}^{\infty }\sum _{j=0}^n\sum _{i=0}^j\sum _{k=0}^i\left({\displaystyle \genfrac{}{}{0pt}{}{n}{j}}\right)\left({\displaystyle \genfrac{}{}{0pt}{}{j}{i}}\right)d_{n-j,{\omega }_2,\lambda }^{(m-1)}\left({\omega }_1{\omega }_{2}y\right){\left(2{\omega }_2\right)}^{k}s(i,k)T_{k,{\displaystyle \frac{\lambda }{2{\omega }_2}}}({\omega }_1-1\mid l)d_{j-i,{\omega }_1,\lambda }^{\left(m\right)}({\omega }_1{\omega }_{2}x+{\omega }_2){\displaystyle \frac{t^n}{n!}}.\hfill \end{array}$$

Similarly, by the symmetry of ${\omega }_1$ and ${\omega }_2$ in $I({\omega }_1,{\omega }_2)$, we obtain

$$\begin{array}{cc}\hfill & I({\omega }_1,{\omega }_2)\hfill \\ & =\sum _{n=0}^{\infty }\sum _{j=0}^n\sum _{i=0}^j\sum _{k=0}^i\left({\displaystyle \genfrac{}{}{0pt}{}{n}{j}}\right)\left({\displaystyle \genfrac{}{}{0pt}{}{j}{i}}\right)d_{n-j,{\omega }_1,\lambda }^{(m-1)}\left({\omega }_1{\omega }_{2}y\right){\left(2{\omega }_1\right)}^{k}s(i,k)T_{k,{\displaystyle \frac{\lambda }{2{\omega }_1}}}({\omega }_2-1\mid l)d_{j-i,{\omega }_2,\lambda }^{\left(m\right)}({\omega }_1{\omega }_{2}x+{\omega }_1){\displaystyle \frac{t^n}{n!}}.\hfill \end{array}$$

Comparing the coefficients of ${\displaystyle \frac{t^n}{n!}}$ on both sides proves the theorem. □

Corollary 9.

Let $y=0$ and $m=1$ in (20), then the equation holds:

$$\begin{array}{cc}\hfill & \sum _{i=0}^n\left({\displaystyle \genfrac{}{}{0pt}{}{n}{i}}\right)d_{n-i,{\omega }_1,\lambda }^{\left(m\right)}({\omega }_1{\omega }_{2}x+{\omega }_2)\sum _{k=0}^{i}s(i,k){\left(2{\omega }_2\right)}^k{T}_{k,{\displaystyle \frac{\lambda }{2{\omega }_2}}}({\omega }_1-1\mid l)\hfill \\ \hfill & =\sum _{i=0}^n\left({\displaystyle \genfrac{}{}{0pt}{}{n}{i}}\right)d_{n-i,{\omega }_2,\lambda }^{\left(m\right)}({\omega }_1{\omega }_{2}x+{\omega }_1)\sum _{k=0}^{i}s(i,k){\left(2{\omega }_1\right)}^k{T}_{k,{\displaystyle \frac{\lambda }{2{\omega }_1}}}({\omega }_2-1\mid l).\hfill \end{array}$$

(21)

Corollary 10.

Let ${\omega }_2=1$ in (21), then the following equation holds:

$$\begin{array}{c}\hfill \sum _{i=0}^n{\lambda }^{i-1}s(n,i)d_{i,{\displaystyle \frac{1}{\lambda }}}^{\left(m\right)}\left({\displaystyle \frac{{\omega }_{1}x+{\omega }_1}{\lambda }}\right)=\sum _{i=0}^n\sum _{k=0}^i\left({\displaystyle \genfrac{}{}{0pt}{}{n}{i}}\right)d_{n-i,{\omega }_1,\lambda }^{\left(m\right)}({\omega }_{1}x+1)s(i,k)2^k{T}_{k,{\displaystyle \frac{\lambda }{2}}}({\omega }_1-1\mid l).\end{array}$$

And from the inversion relation (6), we obtain

$$\begin{array}{c}\hfill d_{n,{\displaystyle \frac{1}{\lambda }}}^{\left(m\right)}\left({\displaystyle \frac{{\omega }_{1}x+{\omega }_1}{\lambda }}\right)={\lambda }^{1-n}\sum _{j=0}^n\sum _{i=0}^j\sum _{k=0}^i\left({\displaystyle \genfrac{}{}{0pt}{}{j}{i}}\right)d_{j-i,{\omega }_1,\lambda }^{\left(m\right)}({\omega }_{1}x+1)s(i,k)S(n,j)2^k{T}_{k,{\displaystyle \frac{\lambda }{2}}}({\omega }_1-1\mid l).\end{array}$$

Theorem 12.

Let n be a nonnegative integer, ${\omega }_1,{\omega }_2,m\in \mathbb{N}$, then the following equation holds:

$$\begin{array}{cc}\hfill & \sum _{k=0}^n\left({\displaystyle \genfrac{}{}{0pt}{}{n}{k}}\right)\sum _{l=0}^{{\omega }_1-1}{d}_{k,{\omega }_1,\lambda }^{\left(m\right)}({\omega }_1{\omega }_{2}x+2{\omega }_{2}l)d_{n-k,{\omega }_2,\lambda }^{(m-1)}({\omega }_1{\omega }_{2}y+{\omega }_2)\hfill \\ \hfill & =\sum _{k=0}^n\left({\displaystyle \genfrac{}{}{0pt}{}{n}{k}}\right)\sum _{l=0}^{{\omega }_2-1}{d}_{k,{\omega }_2,\lambda }^{\left(m\right)}({\omega }_1{\omega }_{2}x+2{\omega }_{1}l)d_{n-k,{\omega }_1,\lambda }^{(m-1)}({\omega }_1{\omega }_{2}y+{\omega }_1).\hfill \end{array}$$

(22)

Proof. 

From the proof process of Theorem 11, we obtain

$$\begin{array}{cc}\hfill & I({\omega }_1,{\omega }_2)\hfill \\ \hfill & ={\left({\displaystyle \frac{\frac{1}{\lambda }\ln (1+\lambda \ln (1+t))}{{(1+\lambda \ln (1+t))}^{\frac{2{\omega }_1}{\lambda }}-1}}\right)}^m{(1+\lambda \ln (1+t))}^{{\displaystyle \frac{{\omega }_1{\omega }_{2}x+m{\omega }_1}{\lambda }}}·{\displaystyle \frac{\lambda \left[{(1+\lambda \ln (1+t))}^{\frac{2{\omega }_1{\omega }_2}{\lambda }}-1\right]}{\ln (1+\lambda \ln (1+t\left)\right)}}\hfill \\ \hfill & \times {\left({\displaystyle \frac{\frac{1}{\lambda }\ln (1+\lambda \ln (1+t))}{{(1+\lambda \ln (1+t))}^{\frac{2{\omega }_2}{\lambda }}-1}}\right)}^m{(1+\lambda \ln (1+t))}^{{\displaystyle \frac{{\omega }_1{\omega }_{2}y+m{\omega }_2}{\lambda }}}\hfill \\ \hfill & ={\left({\displaystyle \frac{\frac{1}{\lambda }\ln (1+\lambda \ln (1+t))}{{(1+\lambda \ln (1+t))}^{\frac{2{\omega }_1}{\lambda }}-1}}\right)}^m{(1+\lambda \ln (1+t))}^{{\displaystyle \frac{{\omega }_1{\omega }_{2}x+m{\omega }_1}{\lambda }}}\sum _{l=0}^{{\omega }_1-1}{(1+\lambda \ln (1+t))}^{{\displaystyle \frac{2{\omega }_{2}l}{\lambda }}}\hfill \\ \hfill & \times {\left({\displaystyle \frac{\frac{1}{\lambda }\ln (1+\lambda \ln (1+t))}{{(1+\lambda \ln (1+t))}^{\frac{2{\omega }_2}{\lambda }}-1}}\right)}^{m-1}{(1+\lambda \ln (1+t))}^{{\displaystyle \frac{{\omega }_1{\omega }_{2}y+m{\omega }_2}{\lambda }}}\hfill \\ \hfill & =\sum _{n=0}^{\infty }\sum _{l=0}^{{\omega }_1-1}{d}_{n,{\omega }_1,\lambda }^{\left(m\right)}({\omega }_1{\omega }_{2}x+2{\omega }_{2}l){\displaystyle \frac{t^n}{n!}}\sum _{n=0}^{\infty }{d}_{n,{\omega }_2,\lambda }^{(m-1)}({\omega }_1{\omega }_{2}y+{\omega }_2){\displaystyle \frac{t^n}{n!}}\hfill \\ \hfill & =\sum _{n=0}^{\infty }\sum _{k=0}^n\left({\displaystyle \genfrac{}{}{0pt}{}{n}{k}}\right)\sum _{l=0}^{{\omega }_1-1}{d}_{k,{\omega }_1,\lambda }^{\left(m\right)}({\omega }_1{\omega }_{2}x+2{\omega }_{2}l)d_{n-k,{\omega }_2,\lambda }^{(m-1)}({\omega }_1{\omega }_{2}y+{\omega }_2){\displaystyle \frac{t^n}{n!}}.\hfill \end{array}$$

Similarly, by the symmetry of ${\omega }_1$ and ${\omega }_2$ in $I({\omega }_1,{\omega }_2)$, we obtain

$$\begin{array}{c}\hfill I({\omega }_1,{\omega }_2)=\sum _{n=0}^{\infty }\sum _{k=0}^n\left({\displaystyle \genfrac{}{}{0pt}{}{n}{k}}\right)\sum _{l=0}^{{\omega }_2-1}{d}_{k,{\omega }_2,\lambda }^{\left(m\right)}({\omega }_1{\omega }_{2}x+2{\omega }_{1}l)d_{n-k,{\omega }_1,\lambda }^{(m-1)}({\omega }_1{\omega }_{2}y+{\omega }_1){\displaystyle \frac{t^n}{n!}}.\end{array}$$

Comparing the coefficients of ${\displaystyle \frac{t^n}{n!}}$ on both sides proves the theorem. □

Corollary 11.

Let $y=0,m=1$ in (22), then the following equation holds:

$$\begin{array}{cc}\hfill & \sum _{m=0}^n\sum _{k=0}^m\sum _{l=0}^{{\omega }_1-1}\left({\displaystyle \genfrac{}{}{0pt}{}{n}{m}}\right){\left({\omega }_2\right)}_{k,\lambda }s(m,k)d_{n-m,{\omega }_1,\lambda }({\omega }_1{\omega }_{2}x+2{\omega }_{2}l)\hfill \\ \hfill & =\sum _{m=0}^n\sum _{k=0}^m\sum _{l=0}^{{\omega }_2-1}\left({\displaystyle \genfrac{}{}{0pt}{}{n}{m}}\right){\left({\omega }_1\right)}_{k,\lambda }s(m,k)d_{n-m,{\omega }_2,\lambda }({\omega }_1{\omega }_{2}x+2{\omega }_{1}l).\hfill \end{array}$$

One fundamental characteristic of these results lies in the fact that their coefficients admit straightforward combinatorial interpretations in the context of enumerating discrete structures. Specifically, the identities presented in Corollary 5 and Theorem 6 unveil an equinumerous relationship between the enumerations of set partitions, permutation cycles, and ordered partitions—with this relationship mediated by the Stirling numbers and Lah numbers. Furthermore, Theorem 10 establishes a connection between the enumeration of symmetric structures (counted via the central factorial numbers) and sum expressions involving set partitions as well as the higher-order degenerate type 2 $\omega$-Daehee polynomials.

6. Conclusions

Building on previous work, this study systematically investigated the higher-order degenerate type 2 $\omega$-Daehee polynomials and numbers using generating functions and Riordan arrays. We derived explicit expressions and established various identities connecting these polynomials with generalized Stirling numbers, higher-order type 2 Bernoulli polynomials, degenerate Stirling numbers, generalized Lah numbers, generalized Bell numbers, and central factorial numbers of the second kind. Furthermore, symmetric identities for these polynomials were obtained via functional relations from their generating function, which highlighted broader combinatorial connections.

Based on our research findings and the intrinsic relationships with generalized Stirling and Bell numbers, this work may offer novel analytical tools for investigating weighted set partitions, coloring problems of signed graphs, and other complex combinatorial configurations. The derived results—including generating functions, explicit formulae, and profound connections with other polynomial families—hold promising potential for applications in the refined enumeration of combinatorial structures.