Partial Derivative Equations and Identities for Hermite-Based Peters-Type Simsek Polynomials and Their Applications

Abstract

The objective of this paper is to investigate Hermite-based Peters-type Simsek polynomials with generating functions. By using generating function methods, we determine some of the properties of these polynomials. By applying the derivative operator to the generating functions of these polynomials, we also determine many of the identities and relations that encompass these polynomials and special numbers and polynomials. Moreover, using integral techniques, we obtain some formulas covering the Cauchy numbers, the Peters-type Simsek numbers and polynomials of the first kind, the two-variable Hermite polynomials, and the Hermite-based Peters-type Simsek polynomials.

1. Introduction

Generating functions and special functions are powerful techniques to analyze and solve some real-world and mathematical problems. These functions are widely used in many areas, such as combinatorics, number theory, and probability theory. Moreover, special numbers and polynomials, owing to their multitude of applications, contribute significantly to many areas of discipline. Using the generating function methods, many authors have carried out studies on special numbers and polynomials and they have obtained many results (see for details, [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40]). One of these is Hermite polynomials, which have a wide range of applications and have yielded interesting relations, as found by many researchers (see [1,2,4,8,11,12,15,36,37,38,39,41,42]). Thus, the aim of this paper is to establish novel and applicable formulas for some special numbers and polynomials with the help of generating function techniques and derivative and integral operators. By applying derivative and integral operators to the generating functions of the Hermite-type combinatorial Simsek polynomials, we determine novel formulas and identities, including the Cauchy numbers, the Stirling numbers of the first kind, the Peters-type Simsek numbers of the first and second kinds, the Peters-type Simsek polynomials of the first kind, the two-variable Hermite polynomials, and the Hermite-based Peters-type Simsek polynomials. Before presenting these novel results, we will list some definitions and notations that will be utilized throughout this paper:

$\mathbb{N}$, $\mathbb{Z}$, $\mathbb{Q}$, $\mathbb{R}$, and $\mathbb{C}$ represent the set of natural numbers, the set of integers, the set of rational numbers, the set of real numbers, and the set of complex numbers, respectively. Let ${\mathbb{N}}_0=\mathbb{N}\cup \left\{0\right\}$ and

$${\left(w\right)}_n=w\left(w-1\right)\left(w-2\right)\dots \left(w-n+1\right).$$

The Stirling numbers of the first kind are defined by

$${\displaystyle \frac{{\left(log\left(1+t\right)\right)}^k}{k!}}=\sum _{n=0}^{\infty }{S}_1\left(n,k\right){\displaystyle \frac{t^n}{n!}},$$

(1)

where $k\in {\mathbb{N}}_0$ (see [3,23,24,34,43,44,45]).

The numbers $S_1\left(n,k\right)$ are also given by the following formula:

$${\left(w\right)}_n=\sum _{k=0}^n{S}_1\left(n,k\right)w^k$$

(2)

(see [34,43,44,45]).

The two-variable Hermite polynomials $H_n^{\left(j\right)}(x,y)$ are defined by

$$f_H(t;x,y,j)=e^{xt+y{t}^j}=\sum _{n=0}^{\infty }{H}_n^{\left(j\right)}\left(x,y\right){\displaystyle \frac{t^n}{n!}},$$

(3)

where $j\ge 2$ (see [1,2] and [9,28,36,37,38,39,41,42,43,46,47]).

By using (3), one obtains

$$\sum _{n=0}^{\infty }{H}_n^{\left(j\right)}\left(x,y\right){\displaystyle \frac{t^n}{n!}}=\sum _{n=0}^{\infty }{x}^n{\displaystyle \frac{t^n}{n!}}\sum _{n=0}^{\infty }{y}^n{\displaystyle \frac{t^{jn}}{n!}}.$$

By applying the series product rule to the above equation, we obtain

$$\sum _{n=0}^{\infty }{H}_n^{\left(j\right)}\left(x,y\right){\displaystyle \frac{t^n}{n!}}=\sum _{n=0}^{\infty }\left(\sum _{k=0}^{\left[{\displaystyle \frac{n}{j}}\right]}{\displaystyle \frac{x^{n-jk}{y}^k}{\left(n-jk\right)!k!}}\right)t^n,$$

where $\left[b\right]$ is the largest integer $\le b$. Comparing the coefficients of $t^n$ on both sides of the above equation, we arrive at the following well-known formula for the polynomials $H_n^{\left(j\right)}\left(x,y\right)$:

$$H_n^{\left(j\right)}\left(x,y\right)=n!\sum _{k=0}^{\left[{\displaystyle \frac{n}{j}}\right]}{\displaystyle \frac{x^{n-jk}{y}^k}{\left(n-jk\right)!k!}}$$

(see [1,2,43,46]). For $y=-{\displaystyle \frac{1}{2}}$ and $j=2$ in (3), we have the following well-known Hermite’s differential equation

$${\left(e^{-{\displaystyle \frac{1}{2}}{x}^2}{u}^{\prime }\right)}^{\prime }+\lambda e^{-{\displaystyle \frac{1}{2}}{x}^2}u=0,$$

where $\lambda$ is a constant.

The Cauchy numbers of the first kind are defined by

$$c_n={\int }_0^1{\left(w\right)}_{n}dw$$

(see [43,48]). That is,

$${\displaystyle \frac{c_n}{n!}}={\int }_0^1\left({\displaystyle \genfrac{}{}{0pt}{}{w}{n}}\right)dw.$$

(4)

The Peters polynomials $S_n\left(w;\lambda ,\mu \right)$ are defined by

$${\displaystyle \frac{{\left(1+t\right)}^w}{{\left(1+{\left(1+t\right)}^{\lambda }\right)}^{\mu }}}=\sum _{n=0}^{\infty }{S}_n\left(w;\lambda ,\mu \right){\displaystyle \frac{t^n}{n!}}$$

(5)

(see [10,25,44,45]).

The Peters-type Simsek numbers of the first kind $Y_n\left(\lambda \right)$ and the Peters-type Simsek polynomials of the first kind $Y_n\left(w;\lambda \right)$ are defined by, respectively,

$$F\left(t;\lambda \right)={\displaystyle \frac{2}{\lambda (1+\lambda t)-1}}=\sum _{n=0}^{\infty }{Y}_n\left(\lambda \right){\displaystyle \frac{t^n}{n!}}$$

(6)

and

$$F\left(t;w,\lambda \right)={\displaystyle \frac{2{(1+\lambda t)}^w}{\lambda (1+\lambda t)-1}}=\sum _{n=0}^{\infty }{Y}_n\left(w;\lambda \right){\displaystyle \frac{t^n}{n!}},$$

(7)

where $\lambda \in \mathbb{R}$ (or $\mathbb{C}$) (see [26]).

The higher-order Peters-type Simsek numbers and polynomials of the first kind are defined by, respectively,

$${\left({\displaystyle \frac{2}{{\lambda }^{2}t+\lambda -1}}\right)}^k=\sum _{n=0}^{\infty }{Y}_n^{\left(k\right)}\left(\lambda \right){\displaystyle \frac{t^n}{n!}}$$

(8)

and

$${\left({\displaystyle \frac{2}{{\lambda }^{2}t+\lambda -1}}\right)}^k{\left(1+\lambda t\right)}^w=\sum _{n=0}^{\infty }{Y}_n^{\left(k\right)}\left(w;\lambda \right){\displaystyle \frac{t^n}{n!}},$$

(9)

where $\lambda \in \mathbb{R}$ (or $\mathbb{C}$) (see [21,22]).

Substituting $k=1$ into (8) and (9), one obtains

$$Y_n^{\left(1\right)}\left(\lambda \right)=Y_n\left(\lambda \right)$$

and

$$Y_n^{\left(1\right)}(w;\lambda )=Y_n(w;\lambda ).$$

The Peters-type Simsek numbers of the second kind $Y_{n,2}\left(\lambda \right)$ and the Peters-type Simsek polynomials of the second kind $Y_{n,2}(w;\lambda )$ are defined by, respectively,

$${\displaystyle \frac{2}{{\lambda }^{2}t+2(\lambda -1)}}=\sum _{n=0}^{\infty }{Y}_{n,2}\left(\lambda \right){\displaystyle \frac{t^n}{n!}}$$

(10)

and

$${\displaystyle \frac{2{\left(1+\lambda t\right)}^w}{{\lambda }^{2}t+2(\lambda -1)}}=\sum _{n=0}^{\infty }{Y}_{n,2}\left(w;\lambda \right){\displaystyle \frac{t^n}{n!}},$$

(11)

where $\lambda \in \mathbb{R}$ (or $\mathbb{C}$) (see [31]).

From (6) and (10), one can easily see that

$$Y_{n,2}\left(\lambda \right)=2^{-n-1}{Y}_n\left(\lambda \right)$$

(12)

(see [31]).

We note that many authors have studied the numbers and polynomials given in Equations (6)–(11). These numbers and polynomials are members of the Peters polynomials given in Equation (5), which is also a generalization of the Boole polynomials (see [5,6,7,12,13,14,15,16,19,20,21,22,27,32]).

In [38], the author defined two new families of polynomials, called the Hermite-type combinatorial Simsek polynomials. These polynomials are denoted by ${}_{Y}H_n^{\left(j\right)}\left(x,y;\lambda \right)$ and ${}_{Y}H_n\left(x,y,w;\lambda ,j\right)$, and are defined as follows:

$$H_1\left(t;x,y;\lambda ,j\right)={\displaystyle \frac{2}{\lambda (1+\lambda t)-1}}{e}^{xt+y{t}^j}=\sum _{n=0}^{\infty }{}_{Y}H_n^{\left(j\right)}\left(x,y;\lambda \right){\displaystyle \frac{t^n}{n!}}$$

(13)

and

$$H_2\left(t;x,y,w;\lambda ,j\right)={\displaystyle \frac{2{\left(1+\lambda t\right)}^w}{\lambda (1+\lambda t)-1}}{e}^{xt+y{t}^j}=\sum _{n=0}^{\infty }{}_{Y}H_n\left(x,y,w;\lambda ,j\right){\displaystyle \frac{t^n}{n!}},$$

(14)

where $j\ge 2$, $\lambda \in \mathbb{R}$ (or $\mathbb{C}$), and $x,y\in \mathbb{R}$ (see also [39]).

When $x=y=0$ in (13) and (14), we have

$${}_{Y}H_n^{\left(j\right)}\left(0,0;\lambda \right)=Y_n\left(\lambda \right)$$

and

$${}_{Y}H_n\left(0,0,w;\lambda ,j\right)=Y_n(w;\lambda ).$$

The rest of this paper is summarized briefly as follows:

In Section 2, using generating function and integration methods, we obtain some identities and formulas, including the Hermite-type combinatorial Simsek polynomials, the Stirling numbers, the Cauchy numbers, and the Peters-type Simsek numbers and polynomials.

In Section 3, applying a derivative operator to the generating functions of the Hermite-type combinatorial Simsek polynomials, we derive some formulas, including the Hermite-based Peters-type Simsek polynomials, the Stirling numbers, and the special numbers.

Finally, the paper concludes with a conclusion section.

2. Identities and Integral Formulas for the Hermite-Type Combinatorial Simsek Polynomials

In this section, by using generating functions of the Hermite-type combinatorial Simsek polynomials, we obtain identities for these polynomials. By using an integration method, we also obtain formulas covering the Stirling numbers of the first kind, the Cauchy numbers, and the Peters-type Simsek numbers of the first and second kinds.

Theorem 1.

Let $n\in {\mathbb{N}}_0$. Then, we have

$${}_{Y}H_n\left(x,y,w;\lambda ,j\right)=n!\sum _{k=0}^n\left(\begin{array}{c}w\\ n-k\end{array}\right){\displaystyle \frac{{\lambda }^{n-k}}{k!}}{}_{Y}H_k^{\left(j\right)}\left(x,y;\lambda \right).$$

(15)

Proof. 

By using (13) and (14), we obtain

$$H_2\left(t;x,y,w;\lambda ,j\right)={\left(1+\lambda t\right)}^w{H}_1\left(t;x,y;\lambda ,j\right).$$

By using the above functional equation with the binomial theorem, assuming that $\left|\lambda t\right|<1$, we obtain

$$\sum _{n=0}^{\infty }{}_{Y}H_n\left(x,y,w;\lambda ,j\right){\displaystyle \frac{t^n}{n!}}=\sum _{n=0}^{\infty }\left(\begin{array}{c}w\\ n\end{array}\right){\lambda }^n{t}^n\sum _{n=0}^{\infty }{}_{Y}H_n^{\left(j\right)}\left(x,y;\lambda \right){\displaystyle \frac{t^n}{n!}}.$$

Applying the Cauchy product rule to the expression on the right-hand side of the equation yields:

$$\sum _{n=0}^{\infty }{}_{Y}H_n\left(x,y,w;\lambda ,j\right){\displaystyle \frac{t^n}{n!}}=\sum _{n=0}^{\infty }\sum _{k=0}^n\left({\displaystyle \genfrac{}{}{0pt}{}{w}{n-k}}\right){\displaystyle \frac{{\lambda }^{n-k}}{k!}}{}_{Y}H_k^{\left(j\right)}\left(x,y;\lambda \right)t^n.$$

By comparing the coefficients of $t^n$ on both sides of the above equation, we achieve the desired result. □

Remark 1.

Substituting $x=y=0$ into (15), after some calculations, we obtain the following known formula [26] (Equation (2.20)):

$$Y_n(w;\lambda )=\sum _{k=0}^n\left(\begin{array}{c}n\\ k\end{array}\right){\lambda }^{n-k}{\left(w\right)}_{n-k}{Y}_k\left(\lambda \right).$$

(16)

Theorem 2.

Let $n\in {\mathbb{N}}_0$. Then, we have

$$\underset{0}{\overset{1}{\int }}{}_{Y}H_n\left(x,y,w;\lambda ,j\right)dw=\sum _{k=0}^n\left({\displaystyle \genfrac{}{}{0pt}{}{n}{k}}\right){\lambda }^{n-k}{}_{Y}H_k^{\left(j\right)}\left(x,y;\lambda \right)c_{n-k}.$$

(17)

Proof. 

Integrating both sides of the Equation (15) with respect to w from 0 to 1, we obtain

$$\underset{0}{\overset{1}{\int }}{}_{Y}H_n\left(x,y,w;\lambda ,j\right)dw=n!\sum _{k=0}^n{\displaystyle \frac{{\lambda }^{n-k}}{k!}}{}_{Y}H_k^{\left(j\right)}\left(x,y;\lambda \right)\underset{0}{\overset{1}{\int }}\left({\displaystyle \genfrac{}{}{0pt}{}{w}{n-k}}\right)dw.$$

Combining the above equation with (4), we obtain

$$\underset{0}{\overset{1}{\int }}{}_{Y}H_n\left(x,y,w;\lambda ,j\right)dw=n!\sum _{k=0}^n{\displaystyle \frac{{\lambda }^{n-k}}{\left(n-k\right)!k!}}{}_{Y}H_k^{\left(j\right)}\left(x,y;\lambda \right)c_{n-k}.$$

From the above equation, we obtain the desired result. □

Theorem 3.

Let $n\in {\mathbb{N}}_0$. Then, we have

$$\sum _{k=0}^n\left({\displaystyle \genfrac{}{}{0pt}{}{n}{k}}\right)\left(H_k^{\left(j\right)}\left(x,y\right)\underset{0}{\overset{1}{\int }}{Y}_{n-k}\left(w;\lambda \right)dw-{\lambda }^{n-k}{}_{Y}H_k^{\left(j\right)}\left(x,y;\lambda \right)c_{n-k}\right)=0.$$

(18)

Proof. 

Integrating both sides of the Theorem 2 given in [38], with respect to w from 0 to 1, we obtain

$$\underset{0}{\overset{1}{\int }}{}_{Y}H_n\left(x,y,w;\lambda ,j\right)dw=\sum _{k=0}^n\left({\displaystyle \genfrac{}{}{0pt}{}{n}{k}}\right)H_k^{\left(j\right)}\left(x,y\right)\underset{0}{\overset{1}{\int }}{Y}_{n-k}\left(w;\lambda \right)dw.$$

(19)

Combining (19) with (17), we obtain

$$\sum _{k=0}^n\left({\displaystyle \genfrac{}{}{0pt}{}{n}{k}}\right){\lambda }^{n-k}{}_{Y}H_k^{\left(j\right)}\left(x,y;\lambda \right)c_{n-k}=\sum _{k=0}^n\left({\displaystyle \genfrac{}{}{0pt}{}{n}{k}}\right)H_k^{\left(j\right)}\left(x,y\right)\underset{0}{\overset{1}{\int }}{Y}_{n-k}\left(w;\lambda \right)dw.$$

After performing some calculations, we obtain the desired result. □

Theorem 4.

Let $n\in {\mathbb{N}}_0$. Then, we have

$$\sum _{k=0}^n\left({\displaystyle \genfrac{}{}{0pt}{}{n}{k}}\right){\lambda }^{n-k}{}_{Y}H_k^{\left(j\right)}\left(x,y;\lambda \right)c_{n-k}=\sum _{k=0}^n\sum _{d=0}^{n-k}\sum _{v=0}^d\left({\displaystyle \genfrac{}{}{0pt}{}{n}{k}}\right)\left({\displaystyle \genfrac{}{}{0pt}{}{n-k}{d}}\right){\displaystyle \frac{{\lambda }^d{H}_k^{\left(j\right)}\left(x,y\right)Y_{n-k-d}\left(\lambda \right)S_1\left(d,v\right)}{v+1}}.$$

(20)

Proof. 

By using (16) and (2), we obtain

$$Y_n\left(w;\lambda \right)=\sum _{d=0}^n\left({\displaystyle \genfrac{}{}{0pt}{}{n}{d}}\right){\lambda }^d{Y}_{n-d}\left(\lambda \right)\sum _{v=0}^d{S}_1\left(d,v\right)w^v.$$

Integrating both sides of the above equation with respect to w from 0 to 1, we obtain

$$\underset{0}{\overset{1}{\int }}{Y}_n\left(w;\lambda \right)dw=\sum _{d=0}^n\left({\displaystyle \genfrac{}{}{0pt}{}{n}{d}}\right){\lambda }^d{Y}_{n-d}\left(\lambda \right)\sum _{v=0}^d{S}_1\left(d,v\right){\displaystyle \frac{1}{v+1}}.$$

(21)

Substituting (21) into (18), after some calculations, we obtain

$$\sum _{k=0}^n\left({\displaystyle \genfrac{}{}{0pt}{}{n}{k}}\right){\lambda }^{n-k}{}_{Y}H_k^{\left(j\right)}\left(x,y;\lambda \right)c_{n-k}=\sum _{k=0}^n\sum _{d=0}^{n-k}\sum _{v=0}^d\left({\displaystyle \genfrac{}{}{0pt}{}{n}{k}}\right)\left({\displaystyle \genfrac{}{}{0pt}{}{n-k}{d}}\right){\displaystyle \frac{{\lambda }^d}{v+1}}{H}_k^{\left(j\right)}\left(x,y\right)Y_{n-k-d}\left(\lambda \right)S_1\left(d,v\right).$$

Thus, the proof of the theorem is concluded. □

Combining (20) with (12), we obtain the following corollary:

Corollary 1.

Let $n\in {\mathbb{N}}_0$. Then, we obtain

$$\begin{array}{ccc}& & \sum _{k=0}^n\left({\displaystyle \genfrac{}{}{0pt}{}{n}{k}}\right){\lambda }^{n-k}{}_{Y}H_k^{\left(j\right)}\left(x,y;\lambda \right)c_{n-k}\hfill \\ & =& \sum _{k=0}^n\sum _{d=0}^{n-k}\sum _{v=0}^d\left({\displaystyle \genfrac{}{}{0pt}{}{n}{k}}\right)\left({\displaystyle \genfrac{}{}{0pt}{}{n-k}{d}}\right){\displaystyle \frac{2^{n-k+1}}{v+1}}{\left({\displaystyle \frac{\lambda }{2}}\right)}^d{H}_k^{\left(j\right)}\left(x,y\right)S_1\left(d,v\right)Y_{n-k-d,2}\left(\lambda \right).\hfill \end{array}$$

3. Derivative Formulas for the Hermite-Type Combinatorial Simsek Polynomials

In this section, by applying derivative operator to the generating functions of the Hermite-type combinatorial Simsek polynomials, we obtain some partial differential equations (PDEs). Using these equations, we obtain some formulas, including the Stirling numbers of the first kind, the Peters-type Simsek numbers of the first kind, and the Hermite-based Peters-type Simsek polynomials.

Theorem 5.

Let $n\in \mathbb{N}$ with $n\ge j-1$. Then, we have

$$\begin{array}{ccc}\hfill {}_{Y}H_{n+1}\left(x,y,w;\lambda ,j\right)& =& w\sum _{k=0}^n\left({\displaystyle \genfrac{}{}{0pt}{}{n}{k}}\right)\left(n-k\right)!{\left(-1\right)}^{n-k}{\lambda }^{n+1-k}{}_{Y}H_k\left(x,y,w;\lambda ,j\right)\hfill \\ & & -{\displaystyle \frac{{\lambda }^2}{2}}\sum _{k=0}^n\left({\displaystyle \genfrac{}{}{0pt}{}{n}{k}}\right)Y_k\left(\lambda \right){}_{Y}H_{n-k}\left(x,y,w;\lambda ,j\right)\hfill \\ & & +y\left({\displaystyle \genfrac{}{}{0pt}{}{n}{j-1}}\right)j!{}_{Y}H_{n-j+1}\left(x,y,w;\lambda ,j\right)+x{}_{Y}H_n\left(x,y,w;\lambda ,j\right).\hfill \end{array}$$

Proof. 

By applying a derivative operator to (14), we obtain the following functional equation:

$${\displaystyle \frac{\partial }{\partial t}}\left\{H_2\left(t;x,y,w;\lambda ,j\right)\right\}=f_H(t;x,y,j){\displaystyle \frac{\partial }{\partial t}}\left\{F\left(t;w,\lambda \right)\right\}+F\left(t;w,\lambda \right){\displaystyle \frac{\partial }{\partial t}}\left\{f_H(t;x,y,j)\right\}.$$

From the above partial differential equations (PDEs), we obtain

$$\begin{array}{ccc}\hfill {\displaystyle \frac{\partial }{\partial t}}\left\{H_2\left(t;x,y,w;\lambda ,j\right)\right\}& =& f_H(t;x,y,j)F\left(t;w,\lambda \right)\left({\displaystyle \frac{\lambda w}{1+\lambda t}}-{\displaystyle \frac{{\lambda }^2}{2}}F\left(t;\lambda \right)\right)\hfill \\ & & +F\left(t;w,\lambda \right)f_H(t;x,y,j)\left(jy{t}^{j-1}+x\right).\hfill \end{array}$$

Combining the above equation with (3), (6) and (7), we obtain the following functions:

$$\begin{array}{ccc}\hfill \sum _{n=0}^{\infty }{}_{Y}H_{n+1}\left(x,y,w;\lambda ,j\right){\displaystyle \frac{t^n}{n!}}& =& {\displaystyle \frac{2{\left(1+\lambda t\right)}^w{e}^{xt+y{t}^j}}{\lambda (1+\lambda t)-1}}\left({\displaystyle \frac{\lambda w}{1+\lambda t}}-{\displaystyle \frac{2{\lambda }^2}{\lambda (1+\lambda t)-1}}\right)\hfill \\ & & +\left(jy{t}^{j-1}+x\right){\displaystyle \frac{2{\left(1+\lambda t\right)}^w{e}^{xt+y{t}^j}}{\lambda (1+\lambda t)-1}}.\hfill \end{array}$$

(22)

Thus,

$$\begin{array}{ccc}\hfill \sum _{n=0}^{\infty }{}_{Y}H_{n+1}\left(x,y,w;\lambda ,j\right){\displaystyle \frac{t^n}{n!}}& =& \sum _{n=0}^{\infty }{}_{Y}H_n\left(x,y,w;\lambda ,j\right){\displaystyle \frac{t^n}{n!}}\left({\displaystyle \frac{\lambda w}{1+\lambda t}}-{\displaystyle \frac{{\lambda }^2}{2}}\sum _{n=0}^{\infty }{Y}_n\left(\lambda \right){\displaystyle \frac{t^n}{n!}}\right)\hfill \\ & & +\left(jy{t}^{j-1}+x\right)\sum _{n=0}^{\infty }{}_{Y}H_n\left(x,y,w;\lambda ,j\right){\displaystyle \frac{t^n}{n!}}.\hfill \end{array}$$

After some calculations, we obtain

$$\begin{array}{ccc}\hfill \sum _{n=0}^{\infty }{}_{Y}H_{n+1}\left(x,y,w;\lambda ,j\right){\displaystyle \frac{t^n}{n!}}& =& w\sum _{n=0}^{\infty }\sum _{k=0}^n\left({\displaystyle \genfrac{}{}{0pt}{}{n}{k}}\right)\left(n-k\right)!{\left(-1\right)}^{n-k}{\lambda }^{n+1-k}{}_{Y}H_k\left(x,y,w;\lambda ,j\right){\displaystyle \frac{t^n}{n!}}\hfill \\ & & -{\displaystyle \frac{{\lambda }^2}{2}}\sum _{n=0}^{\infty }\sum _{k=0}^n\left({\displaystyle \genfrac{}{}{0pt}{}{n}{k}}\right)Y_k\left(\lambda \right){}_{Y}H_{n-k}\left(x,y,w;\lambda ,j\right){\displaystyle \frac{t^n}{n!}}\hfill \\ & & +jy\sum _{n=0}^{\infty }{\left(n\right)}_{j-1}{}_{Y}H_{n-j+1}\left(x,y,w;\lambda ,j\right){\displaystyle \frac{t^n}{n!}}\hfill \\ & & +x\sum _{n=0}^{\infty }{}_{Y}H_n\left(x,y,w;\lambda ,j\right){\displaystyle \frac{t^n}{n!}}.\hfill \end{array}$$

By comparing the coefficients of ${\displaystyle \frac{t^n}{n!}}$ on both sides of the above equation, we achieve the desired result. □

Theorem 6.

Let $n\in \mathbb{N}$ with $n\ge j-1$. Then, we have

$$\begin{array}{ccc}\hfill {}_{Y}H_{n+1}\left(x,y,w;\lambda ,j\right)& =& w\sum _{d=0}^n\left({\displaystyle \genfrac{}{}{0pt}{}{n}{d}}\right)\sum _{k=0}^d{\left(-1\right)}^k\left({\displaystyle \genfrac{}{}{0pt}{}{d}{k}}\right)\left(d-k\right)!{\lambda }^{k+1}{Y}_{d-k}\left(w;\lambda \right)H_{n-d}^{\left(j\right)}\left(x,y\right)\hfill \\ & & -{\displaystyle \frac{{\lambda }^2}{2}}\sum _{k=0}^n\left({\displaystyle \genfrac{}{}{0pt}{}{n}{k}}\right)H_{n-k}^{\left(j\right)}\left(x,y\right)Y_k^{\left(2\right)}\left(w;\lambda \right)+x\sum _{k=0}^n\left({\displaystyle \genfrac{}{}{0pt}{}{n}{k}}\right)Y_{n-k}\left(w;\lambda \right)H_k^{\left(j\right)}\left(x,y\right)\hfill \\ & & +y\left({\displaystyle \genfrac{}{}{0pt}{}{n}{j-1}}\right)j!\sum _{k=0}^{n-j+1}\left({\displaystyle \genfrac{}{}{0pt}{}{n-j+1}{k}}\right)Y_{n-j+1-k}\left(w;\lambda \right)H_k^{\left(j\right)}\left(x,y\right).\hfill \end{array}$$

Proof. 

By using (22), we also obtain

$$\begin{array}{ccc}\hfill \sum _{n=0}^{\infty }{}_{Y}H_{n+1}\left(x,y,w;\lambda ,j\right){\displaystyle \frac{t^n}{n!}}& =& e^{xt+y{t}^j}\left({\displaystyle \frac{2\lambda w{\left(1+\lambda t\right)}^{w-1}}{\lambda (1+\lambda t)-1}}\right)\hfill \\ & & -e^{xt+y{t}^j}\left({\displaystyle \frac{{\lambda }^2}{2}}{\left({\displaystyle \frac{2}{\lambda (1+\lambda t)-1}}\right)}^2{\left(1+\lambda t\right)}^w\right)\hfill \\ & & +\left(jy{t}^{j-1}+x\right){\displaystyle \frac{2{\left(1+\lambda t\right)}^w}{\lambda (1+\lambda t)-1}}{e}^{xt+y{t}^j}.\hfill \end{array}$$

Combining the above equation with (3), (7) and (9), after some calculations, we obtain

$$\begin{array}{ccc}\hfill \sum _{n=0}^{\infty }{}_{Y}H_{n+1}\left(x,y,w;\lambda ,j\right){\displaystyle \frac{t^n}{n!}}& =& \lambda w\sum _{n=0}^{\infty }{H}_n^{\left(j\right)}\left(x,y\right){\displaystyle \frac{t^n}{n!}}\sum _{n=0}^{\infty }{\left(-\lambda \right)}^n{t}^n\sum _{n=0}^{\infty }{Y}_n\left(w;\lambda \right){\displaystyle \frac{t^n}{n!}}\hfill \\ & & -{\displaystyle \frac{{\lambda }^2}{2}}\sum _{n=0}^{\infty }{H}_n^{\left(j\right)}\left(x,y\right){\displaystyle \frac{t^n}{n!}}\sum _{n=0}^{\infty }{Y}_n^{\left(2\right)}\left(w;\lambda \right){\displaystyle \frac{t^n}{n!}}\hfill \\ & & +jy{t}^{j-1}\sum _{n=0}^{\infty }{Y}_n\left(w;\lambda \right){\displaystyle \frac{t^n}{n!}}\sum _{n=0}^{\infty }{H}_n^{\left(j\right)}\left(x,y\right){\displaystyle \frac{t^n}{n!}}\hfill \\ & & +x\sum _{n=0}^{\infty }{Y}_n\left(w;\lambda \right){\displaystyle \frac{t^n}{n!}}\sum _{n=0}^{\infty }{H}_n^{\left(j\right)}\left(x,y\right){\displaystyle \frac{t^n}{n!}}.\hfill \end{array}$$

Therefore,

$$\begin{array}{ccc}& & \sum _{n=0}^{\infty }{}_{Y}H_{n+1}\left(x,y,w;\lambda ,j\right){\displaystyle \frac{t^n}{n!}}\hfill \\ & =& w\sum _{n=0}^{\infty }\sum _{d=0}^n\left({\displaystyle \genfrac{}{}{0pt}{}{n}{d}}\right)\sum _{k=0}^d{\left(-1\right)}^k\left({\displaystyle \genfrac{}{}{0pt}{}{d}{k}}\right)\left(d-k\right)!{\lambda }^{k+1}{Y}_{d-k}\left(w;\lambda \right)H_{n-d}^{\left(j\right)}\left(x,y\right){\displaystyle \frac{t^n}{n!}}\hfill \\ & & -{\displaystyle \frac{{\lambda }^2}{2}}\sum _{n=0}^{\infty }\sum _{k=0}^n\left({\displaystyle \genfrac{}{}{0pt}{}{n}{k}}\right)H_{n-k}^{\left(j\right)}\left(x,y\right)Y_k^{\left(2\right)}\left(w;\lambda \right){\displaystyle \frac{t^n}{n!}}\hfill \\ & & +jy\sum _{n=0}^{\infty }{\left(n\right)}_{j-1}\sum _{k=0}^{n-j+1}\left({\displaystyle \genfrac{}{}{0pt}{}{n-j+1}{k}}\right)Y_{n-j+1-k}\left(w;\lambda \right)H_k^{\left(j\right)}\left(x,y\right){\displaystyle \frac{t^n}{n!}}\hfill \\ & & +x\sum _{n=0}^{\infty }\sum _{k=0}^n\left({\displaystyle \genfrac{}{}{0pt}{}{n}{k}}\right)Y_{n-k}\left(w;\lambda \right)H_k^{\left(j\right)}\left(x,y\right){\displaystyle \frac{t^n}{n!}}.\hfill \end{array}$$

By comparing the coefficients of ${\displaystyle \frac{t^n}{n!}}$ on both sides of the above equation, we achieve the desired result. □

Theorem 7.

Let $n\in \mathbb{N}$ with $n\ge k$. Then, we have

$${\displaystyle \frac{{\partial }^k}{\partial x^k}}\left\{{}_{Y}H_n\left(x,y,w;\lambda ,j\right)\right\}={\left(n\right)}_k{}_{Y}H_{n-k}\left(x,y,w;\lambda ,j\right).$$

(23)

Proof. 

By applying the derivative operator ${\displaystyle \frac{{\partial }^k}{\partial x^k}}$ to Equation (14), we obtain

$$\sum _{n=0}^{\infty }{\displaystyle \frac{{\partial }^k}{\partial x^k}}\left\{{}_{Y}H_n\left(x,y,w;\lambda ,j\right)\right\}{\displaystyle \frac{t^n}{n!}}=t^k\sum _{n=0}^{\infty }{}_{Y}H_n\left(x,y,w;\lambda ,j\right){\displaystyle \frac{t^n}{n!}}.$$

Thus,

$$\sum _{n=0}^{\infty }{\displaystyle \frac{{\partial }^k}{\partial x^k}}\left\{{}_{Y}H_n\left(x,y,w;\lambda ,j\right)\right\}{\displaystyle \frac{t^n}{n!}}=\sum _{n=0}^{\infty }{\left(n\right)}_k{}_{Y}H_{n-k}\left(x,y,w;\lambda ,j\right){\displaystyle \frac{t^n}{n!}}.$$

By comparing the coefficients of ${\displaystyle \frac{t^n}{n!}}$ on both sides of the above equation, we achieve the desired result. □

Putting $k=1$ in (23), we obtain the following corollary:

Corollary 2.

Let $n\in \mathbb{N}$ with $n\ge 1$. Then, we have

$${\displaystyle \frac{\partial }{\partial x}}\left\{{}_{Y}H_n\left(x,y,w;\lambda ,j\right)\right\}=n{}_{Y}H_{n-1}\left(x,y,w;\lambda ,j\right).$$

Theorem 8.

Let $n\in \mathbb{N}$ with $n\ge jk$. Then, we have

$${\displaystyle \frac{{\partial }^k}{\partial y^k}}\left\{{}_{Y}H_n\left(x,y,w;\lambda ,j\right)\right\}={\left(n\right)}_{jk}{}_{Y}H_{n-jk}\left(x,y,w;\lambda ,j\right).$$

(24)

Proof. 

By applying the derivative operator ${\displaystyle \frac{{\partial }^k}{\partial y^k}}$ to Equation (14), we obtain

$$\sum _{n=0}^{\infty }{\displaystyle \frac{{\partial }^k}{\partial y^k}}\left\{{}_{Y}H_n\left(x,y,w;\lambda ,j\right)\right\}{\displaystyle \frac{t^n}{n!}}=\sum _{n=0}^{\infty }{}_{Y}H_n\left(x,y,w;\lambda ,j\right){\displaystyle \frac{t^{n+jk}}{n!}}.$$

Hence,

$$\sum _{n=0}^{\infty }{\displaystyle \frac{{\partial }^k}{\partial y^k}}\left\{{}_{Y}H_n\left(x,y,w;\lambda ,j\right)\right\}{\displaystyle \frac{t^n}{n!}}=\sum _{n=0}^{\infty }{\left(n\right)}_{jk}{}_{Y}H_{n-jk}\left(x,y,w;\lambda ,j\right){\displaystyle \frac{t^n}{n!}}.$$

By comparing the coefficients of ${\displaystyle \frac{t^n}{n!}}$ on both sides of the above equation, we arrive at the desired result. □

Putting $k=1$ in (24), we obtain the following corollary:

Corollary 3.

Let $n\in \mathbb{N}$ with $n\ge j$. Then, we have

$${\displaystyle \frac{\partial }{\partial y}}\left\{{}_{Y}H_n\left(x,y,w;\lambda ,j\right)\right\}={\left(n\right)}_j{}_{Y}H_{n-j}\left(x,y,w;\lambda ,j\right).$$

(25)

Theorem 9.

Let $n\in {\mathbb{N}}_0$. Then, we have

$${\displaystyle \frac{{\partial }^2}{\partial x^2}}\left\{{}_{Y}H_n\left(x,y,w;\lambda ,2\right)\right\}={\displaystyle \frac{\partial }{\partial y}}\left\{{}_{Y}H_n\left(x,y,w;\lambda ,2\right)\right\}.$$

Proof. 

When $j=2$ in (14), upon applying the derivative operators ${\displaystyle \frac{{\partial }^2}{\partial x^2}}$ and ${\displaystyle \frac{\partial }{\partial y}}$ to the final equation, we obtain

$$\sum _{n=0}^{\infty }{\displaystyle \frac{{\partial }^2}{\partial x^2}}\left\{{}_{Y}H_n\left(x,y,w;\lambda ,2\right)\right\}{\displaystyle \frac{t^n}{n!}}=\sum _{n=0}^{\infty }{\left(n\right)}_2{}_{Y}H_{n-2}\left(x,y,w;\lambda ,2\right){\displaystyle \frac{t^n}{n!}}$$

and

$$\sum _{n=0}^{\infty }{\displaystyle \frac{\partial }{\partial y}}\left\{{}_{Y}H_n\left(x,y,w;\lambda ,2\right)\right\}{\displaystyle \frac{t^n}{n!}}=\sum _{n=0}^{\infty }{\left(n\right)}_2{}_{Y}H_{n-2}\left(x,y,w;\lambda ,2\right){\displaystyle \frac{t^n}{n!}}.$$

From the above equations, we have

$$\sum _{n=0}^{\infty }{\displaystyle \frac{\partial }{\partial y}}\left\{{}_{Y}H_n\left(x,y,w;\lambda ,2\right)\right\}{\displaystyle \frac{t^n}{n!}}=\sum _{n=0}^{\infty }{\displaystyle \frac{{\partial }^2}{\partial x^2}}\left\{{}_{Y}H_n\left(x,y,w;\lambda ,2\right)\right\}{\displaystyle \frac{t^n}{n!}}.$$

By comparing the coefficients of ${\displaystyle \frac{t^n}{n!}}$ on both sides of the above equation, we arrive at the desired result. □

Theorem 10.

Let $n\in {\mathbb{N}}_0$ and $k\in \mathbb{N}$. Then, we have

$${\displaystyle \frac{{\partial }^k}{\partial w^k}}\left\{{}_{Y}H_n\left(x,y,w;\lambda ,j\right)\right\}=k!\sum _{d=0}^n\left({\displaystyle \genfrac{}{}{0pt}{}{n}{d}}\right){\lambda }^d{S}_1\left(d,k\right){}_{Y}H_{n-d}\left(x,y,w;\lambda ,j\right).$$

Proof. 

By applying the derivative operator ${\displaystyle \frac{{\partial }^k}{\partial w^k}}$ to Equation (14), we obtain

$$\sum _{n=0}^{\infty }{\displaystyle \frac{{\partial }^k}{\partial w^k}}\left\{{}_{Y}H_n\left(x,y,w;\lambda ,j\right)\right\}{\displaystyle \frac{t^n}{n!}}={\left(log\left(1+\lambda t\right)\right)}^k\sum _{n=0}^{\infty }{}_{Y}H_n\left(x,y,w;\lambda ,j\right){\displaystyle \frac{t^n}{n!}}.$$

Combining the above equation with (1), we obtain

$$\sum _{n=0}^{\infty }{\displaystyle \frac{{\partial }^k}{\partial w^k}}\left\{{}_{Y}H_n\left(x,y,w;\lambda ,j\right)\right\}{\displaystyle \frac{t^n}{n!}}=k!\sum _{n=0}^{\infty }{\lambda }^n{S}_1\left(n,k\right){\displaystyle \frac{t^n}{n!}}\sum _{n=0}^{\infty }{}_{Y}H_n\left(x,y,w;\lambda ,j\right){\displaystyle \frac{t^n}{n!}}.$$

Therefore,

$$\sum _{n=0}^{\infty }{\displaystyle \frac{{\partial }^k}{\partial w^k}}\left\{{}_{Y}H_n\left(x,y,w;\lambda ,j\right)\right\}{\displaystyle \frac{t^n}{n!}}=k!\sum _{n=0}^{\infty }\sum _{d=0}^n\left({\displaystyle \genfrac{}{}{0pt}{}{n}{d}}\right){\lambda }^d{S}_1\left(d,k\right){}_{Y}H_{n-d}\left(x,y,w;\lambda ,j\right){\displaystyle \frac{t^n}{n!}}.$$

By comparing the coefficients of ${\displaystyle \frac{t^n}{n!}}$ on both sides of the above equation, we arrive at the desired result. □

4. Conclusions

In the present paper, we examined the generating functions of Hermite-based Peters-type Simsek polynomials. Using the differential, integral, and functional equations of the generating functions, we obtained some formulas and identities for the Hermite-type combinatorial Simsek polynomials, including some special numbers and polynomials. These results include the Cauchy numbers, the Peters-type Simsek numbers of the first and second kinds, the Peters-type Simsek polynomials of the first kind, the two-variable Hermite polynomials, and the Stirling numbers of the first kind. Moreover, we established a partial differential equation (PDE) for these polynomials. In summary, this paper provides a valuable resource for those researching the Peters-type Simsek polynomials and the Hermite polynomials, and also their derivative and integral properties. Moreover, there are many applications for these types of polynomials. For example, their quasimonomial properties, matrix properties, binomial distribution, and Poisson distribution have been studied by many authors (see [12,13,14,15,20,21,22,26]). Consequently, the findings of this paper have the potential to contribute to and enrich many disciplines, especially mathematics fields such as combinatorics, number theory, probability theory, and differential equations.