Some Identities Involving 2-Variable Modified Degenerate Hermite Polynomials Arising from Differential Equations and Distribution of Their Zeros

Abstract

In this chapter, we introduce the 2-variable modified degenerate Hermite polynomials and obtain some new symmetric identities for 2-variable modified degenerate Hermite polynomials. In order to give explicit identities for 2-variable modified degenerate Hermite polynomials, differential equations arising from the generating functions of 2-variable modified degenerate Hermite polynomials are studied. Finally, we investigate the structure and symmetry of the zeros of the 2-variable modified degenerate Hermite equations.

1. Introduction

The Hermite equation is defined as

$$u^{″}\left(x\right)-2x{u}^{\prime }\left(x\right)+\left(\epsilon -1\right)u\left(x\right)=0,x\in \left[-\infty ,\infty \right],$$E1

where $\epsilon$ is unrestricted. Hermite equation is encountered in the study of quantum mechanical harmonic oscillator, where $\epsilon$ represent the energy of the oscillator. The ordinary Hermite numbers $H_n$ and Hermite polynomials $H_n\left(x\right)$ are usually defined by the generating functions

$$e^{t\left(2x-t\right)}=\sum _{n=0}^{\infty }{H}_n\left(x\right)\frac{t^n}{n!}$$E2

and

$$e^{-t^2}=\sum _{n=0}^{\infty }{H}_n\frac{t^n}{n!}.$$E3

Clearly, $H_n=H_n\left(0\right)$.

It is known that these numbers and polynomials play an important role in various fields of mathematics and physics, including number theory, combinations, special functions, and differential equations. Many interested properties about that have been studied (see [1, 2, 3, 4, 5]). The ordinary Hermite polynomials $H_n\left(x\right)$ satisfy the Hermite differential equation

$$\frac{d^{2}H\left(x\right)}{{\mathit{dx}}^2}-2x\frac{\mathit{dH}\left(x\right)}{\mathit{dx}}+2\mathit{nH}\left(x\right)=0,n=\mathrm{0,1,2},\dots .$$E4

Hence ordinary Hermite polynomials $H_n\left(x\right)$ satisfy the second-order ordinary differential equation

$$u^{″}-2x{u}^{\prime }+2\mathit{nu}=0.$$E5

We remind that the 2-variable Hermite polynomials $H_n\left(x,y\right)$ defined by the generating function (see [2])

$$\sum _{n=0}^{\infty }{H}_n\left(x,y\right)\frac{t^n}{n!}=e^{t\left(x+\mathit{yt}\right)}$$E6

are the solution of heat equation

$$\frac{\partial }{\partial y}{H}_n\left(x,y\right)=\frac{{\partial }^2}{\partial x^2}{H}_n\left(x,y\right),H_n\left(x,0\right)=x^n.$$E7

Observe that

$$H_n\left(2x,-1\right)=H_n\left(x\right).$$E8

Motivated by their importance and potential applications in certain problems of probability, combinatorics, number theory, differential equations, numerical analysis and other areas of mathematics and physics, several kinds of some special numbers and polynomials were recently studied by many authors (see [1, 2, 3, 4, 5, 6, 7, 8]). Many mathematicians have studied in the area of the degenerate Stiling, degenerate Bernoulli polynomials, degenerate Euler polynomials, degenerate Genocchi polynomials, and degenerate tangent polynomials (see [6, 7, 9]).

Recently, Hwang and Ryoo [10] proposed the 2-variable degenerate Hermite polynomials ${\mathcal{H}}_n\left(x,y,\lambda \right)$ by means of the generating function

$$\sum _{n=0}^{\infty }{\mathcal{H}}_n\left(x,y,\lambda \right)\frac{t^n}{n!}={\left(1+\lambda \right)}^{\frac{t\left(x+\mathit{yt}\right)}{\lambda }}.$$E9

Since ${\left(1+\mu \right)}^{\frac{t}{\mu }}\to e^t$ as $\mu \to 0$, it is evident that (9) reduces to (6). The 2-variable degenerate Hermite polynomials ${\mathcal{H}}_n\left(x,y,\lambda \right)$ in generating function (9) are the solution of equation

$$\begin{array}{l}\frac{\partial }{\partial y}{\mathcal{H}}_n\left(x,y,\lambda \right)=\frac{\lambda }{log\left(1+\lambda \right)}\frac{{\partial }^2}{\partial x^2}{\mathcal{H}}_n\left(x,y,\lambda \right),\\ {\mathcal{H}}_n\left(x,0,\lambda \right)={\left(\frac{log\left(1+\lambda \right)}{\lambda }\right)}^n{x}^n.\end{array}$$E10

Since $\frac{log\left(1+\lambda \right)}{\lambda }\to 1$ as $\lambda$ approaches to 0, it is apparent that (10) descends to (7).

Mathematicians have studied the differential equations arising from the generating functions of special numbers and polynomials (see [10, 11, 12, 13, 14]). Now, a new class of 2-variable modified degenerate Hermite polynomials are constructed based on the results so far. We can induce the differential equations generated from the generating function of 2-variable modified degenerate Hermite polynomials. By using the coefficients of this differential equation, we obtain explicit identities for the 2-variable modified degenerate Hermite polynomials. The rest of the paper is organized as follows. In Section 2, we construct the 2-variable modified degenerate Hermite polynomials and obtain basic properties of these polynomials. In Section 3, we give some symmetric identities for 2-variable modified degenerate Hermite polynomials. In Section 4, we derive the differential equations generated from the generating function of 2-variable modified degenerate Hermite polynomials. Using the coefficients of this differential equation, we have explicit identities for the 2-variable modified degenerate Hermite polynomials. In Section 5, we investigate the zeros of the 2-variable modified degenerate Hermite equations by using computer. Further, we observe the pattern of scattering phenomenon for the zeros of 2-variable modified degenerate Hermite equations. Our paper will finish with Section 6, where the conclusions and future directions of this work are showed.

2. Basic properties for the 2-variable modified degenerate Hermite polynomials

In this section, a new class of the 2-variable modified degenerate Hermite polynomials are considered. Furthermore, some properties of these polynomials are also obtained.

We define the 2-variable modified degenerate Hermite polynomials ${\mathrm{H}}_n\left(x,y|\mu \right)$ by means of the generating function

$$\sum _{n=0}^{\infty }{\mathrm{H}}_n\left(x,y|\mu \right)\frac{t^n}{n!}={\left(1+\mu \right)}^{\frac{\mathit{xt}}{\mu }}{e}^{{\mathit{yt}}^2}.$$E11

Since ${\left(1+\mu \right)}^{\frac{\mathit{xt}}{\mu }}\to e^{\mathit{xt}}$ as $\mu \to 0$, it is clear that (11) reduces to (6). Observe that degenerate Hermite polynomials ${\mathcal{H}}_n\left(x,y,\mu \right)$ and 2-variable modified degenerate Hermite polynomials ${\mathrm{H}}_n\left(x,y|\mu \right)$ are totally different.

Now, we recall that the $\mu$-analogue of the falling factorial sequences as follows:

$${\left(x|\mu \right)}_0=1,{\left(x|\mu \right)}_n=x\left(x-\mu \right)\left(x-2\mu \right)\cdots \left(x-\left(n-1\right)\mu \right),\left(n\ge 1\right).$$E12

Note that ${lim}_{\mu \to 1}{\left(x|\mu \right)}_n=x\left(x-1\right)\left(x-2\right)\cdots \left(x-\left(n-1\right)\right)={\left(x\right)}_n,\left(n\ge 1\right).$ We also need the binomial theorem: for a variable $y$,

$$\begin{array}{c}{\left(1+\mu \right)}^{\mathit{yt}/\mu }=\sum _{m=0}^{\infty }{\left(\frac{\mathit{ty}}{\mu }\right)}_m\frac{{\mu }^m}{m!}\\ =\sum _{m=0}^{\infty }\left(\sum _{l=0}^m{S}_1\left(m,l\right){\left(\frac{\mathit{ty}}{\mu }\right)}^l\frac{{\mu }^m}{m!}\right)\\ =\sum _{l=0}^{\infty }\left(\sum _{m=l}^{\infty }{S}_1\left(m,l\right)y^l{\mu }^{m-l}\frac{l!}{m!}\right)\frac{t^l}{l!}.\end{array}$$E13

We remember that the classical Stirling numbers of the first kind $S_1\left(n,k\right)$ and the second kind $S_2\left(n,k\right)$ are defined by the relations (see [6, 7, 8, 9, 10, 11, 12, 13])

$${\left(x\right)}_n=\sum _{k=0}^n{S}_1\left(n,k\right)x^k\text{and}{x}^n=\sum _{k=0}^n{S}_2\left(n,k\right){\left(x\right)}_k,$$E14

respectively. We also have

$$\sum _{n=m}^{\infty }{S}_2\left(n,m\right)\frac{t^n}{n!}=\frac{{\left(e^t-1\right)}^m}{m!}\text{and}\sum _{n=m}^{\infty }{S}_1\left(n,m\right)\frac{t^n}{n!}=\frac{{\left(log\left(1+t\right)\right)}^m}{m!}.$$E15

As another application of the differential equation for ${\mathrm{H}}_n\left(x,y|\mu \right)$ is as follows: Note that

$$G\left(t,x,y,\mu \right)={\left(1+\mu \right)}^{\frac{\mathit{xt}}{\mu }}{e}^{{\mathit{yt}}^2}$$E16

satisfies

$$\frac{\partial G\left(t,x,y,\mu \right)}{\partial y}-{\left(\frac{log\left(1+\mu \right)}{\mu }\right)}^2\frac{{\partial }^{2}G\left(t,x,y,\mu \right)}{\partial x^2}=0.$$E17

Substitute the series in (11) for $G\left(t,x,y,\mu \right)$ to get

$$\frac{\partial }{\partial y}{\mathrm{H}}_n\left(x,y|\mu \right)={\left(\frac{\mu }{log\left(1+\mu \right)}\right)}^2\frac{{\partial }^2}{\partial x^2}{\mathrm{H}}_n\left(x,y|\mu \right).$$

Thus the 2-variable modified degenerate Hermite polynomials ${\mathrm{H}}_n\left(x,y|\mu \right)$ in generating function (11) are the solution of equation

$$\begin{array}{l}{\left(\frac{log\left(1+\mu \right)}{\mu }\right)}^2\frac{\partial }{\partial y}{\mathrm{H}}_n\left(x,y|\mu \right)-\frac{{\partial }^2}{\partial x^2}{\mathrm{H}}_n\left(x,y|\mu \right)=0,\\ {\mathrm{H}}_n\left(x,0|\mu \right)={\left(\frac{log\left(1+\mu \right)}{\mu }\right)}^n{x}^n.\end{array}$$E18

The generating function (11) is useful for deriving several properties of the 2-variable modified degenerate Hermite polynomials ${\mathrm{H}}_n\left(x,y|\mu \right)$. For example, we have the following expression for these polynomials:

Theorem 1. For any positive integer $n$, we have

$${\mathrm{H}}_n\left(x,y|\mu \right)=\sum _{k=0}^{\left[\frac{n}{2}\right]}{\left(\frac{log\left(1+\mu \right)}{\mu }\right)}^{n-2k}{x}^{n-2k}{y}^k\frac{n!}{k!\left(n-2k\right)!},$$E19

where $\left[\right]$ denotes taking the integer part.

Proof. By (11) and (13), we have

$$\begin{array}{l}\sum _{n=0}^{\infty }{\mathrm{H}}_n\left(x,y|\mu \right)\frac{t^n}{n!}={\left(1+\mu \right)}^{\frac{\mathit{xt}}{\mu }}{e}^{{\mathit{yt}}^2}\\ =\sum _{k=0}^{\infty }{y}^k\frac{t^{2k}}{k!}\sum _{l=0}^{\infty }{\left(\frac{log\left(1+\mu \right)}{\mu }\right)}^l{x}^l\frac{t^l}{l!}\\ =\sum _{n=0}^{\infty }\left(\sum _{k=0}^{\left[\frac{n}{2}\right]}{y}^k{\left(\frac{log\left(1+\mu \right)}{\mu }\right)}^{n-2k}{x}^{n-2k}\frac{n!}{k!\left(n-2k\right)!}\right)\frac{t^n}{n!}.\end{array}$$E20

By comparing the coefficients of $\frac{t^n}{n!}$, the expected result of Theorem 1 is achieved. $\square$

Since ${lim}_{\mu \to 0}\frac{log\left(1+\mu \right)}{\mu }=1,$ we get

$$H_n\left(x,y\right)=n!\sum _{k=0}^{\left[\frac{n}{2}\right]}\frac{y^k{x}^{n-2k}}{k!\left(n-2k\right)!}.$$E21

The following basic properties of the 2-variable degenerate Hermite polynomials ${\mathrm{H}}_n\left(x,y|\mu \right)$ are induced form (11). Therefore, it is enough to delete involved detail explanation.

Theorem 2. For any positive integer $n$, we have

$$\begin{array}{l}1.{\mathrm{H}}_n\left(x,y|\mu \right)=\sum _{k=0}^{\left[\frac{n}{2}\right]}\sum _{m=n-2k}^{\infty }{y}^k{S}_1\left(m,n-2k\right)x^{n-2k}{\mu }^{m-\left(n-2k\right)}\frac{n!}{m!k!}.\\ 2.{\mathrm{H}}_n\left(x_1+x_2,y|\mu \right)=\sum _{l=0}^n\left(\begin{array}{c}n\\ l\end{array}\right){\left(\frac{log\left(1+\mu \right)}{\mu }\right)}^l{x}_2^l{\mathrm{H}}_{n-l}\left(x_1,y|\mu \right).\\ 3.{\mathrm{H}}_n\left(x_1+x_2,y|\mu \right)=\sum _{l=0}^n\left(\begin{array}{c}n\\ l\end{array}\right){\mathrm{H}}_{n-l}\left(x_1,y|\mu \right)\sum _{m=l}^{\infty }{S}_1\left(m,l\right)x_2^l{\mu }^{m-l}\frac{l!}{m!}.\\ 4.{\mathrm{H}}_n\left(x,y_1+y_2|\mu \right)=\sum _{k=0}^{\left[\frac{n}{2}\right]}{\mathrm{H}}_{n-2k}\left(x,y_1|\mu \right)\frac{y_2^{k}n!}{k!\left(n-2k\right)!}.\\ 5.{\mathrm{H}}_n\left(x_1+x_2,y_1+y_2|\mu \right)=\sum _{l=0}^n\left(\begin{array}{c}n\\ l\end{array}\right){\mathrm{H}}_l\left(x_1,y_1|\mu \right){\mathrm{H}}_{n-l}\left(x_2,y_2|\mu \right).\end{array}$$E22

3. Symmetric identities for 2-variable modified degenerate Hermite polynomials

In this section, we give some new symmetric identities for 2-variable modified degenerate Hermite polynomials. We also get some explicit formulas and properties for 2-variable modified degenerate Hermite polynomials.

Theorem 3. Let $a,b>0$ ($a\ne b$). The following identity holds true:

$$b^m{\mathrm{H}}_m\left(\mathit{ax},a^{2}y|\mu \right)=a^m{\mathrm{H}}_m\left(\mathit{bx},b^{2}y|\mu \right).$$E23

Proof. Let $a,b>0$ ($a\ne b$). We start with

$$\mathcal{G}\left(t,\mu \right)={\left(1+\mu \right)}^{\frac{\mathit{\text{abxt}}}{\mu }}{e}^{a^2{b}^2{\mathit{yt}}^2}.$$E24

Then the expression for $\mathcal{G}\left(t,\mu \right)$ is symmetric in $a$ and $b$

$$\mathcal{G}\left(t,\mu \right)=\sum _{m=0}^{\infty }{\mathrm{H}}_m\left(\mathit{ax},a^{2}y|\mu \right)\frac{{\left(\mathit{bt}\right)}^m}{m!}=\sum _{m=0}^{\infty }{b}^m{\mathrm{H}}_m\left(\mathit{ax},a^{2}y|\mu \right)\frac{t^m}{m!}.$$E25

By the similar way, we get that

$$\mathcal{G}\left(t,\mu \right)=\sum _{m=0}^{\infty }{\mathrm{H}}_m\left(\mathit{bx},b^{2}y|\mu \right)\frac{{\left(\mathit{at}\right)}^m}{m!}=\sum _{m=0}^{\infty }{a}^m{\mathrm{H}}_m\left(\mathit{bx},b^{2}y|\mu \right)\frac{t^m}{m!}.$$E26

By comparing the coefficients of $\frac{t^m}{m!}$ in last two equations, the expected result of Theorem 3 is achieved. $\square$

Again, we now use

$$\mathcal{F}\left(t,\mu \right)=\frac{\mathit{abt}{\left(1+\mu \right)}^{\frac{\mathit{\text{abxt}}}{\mu }}{e}^{a^2{b}^2{\mathit{yt}}^2}\left({\left(1+\mu \right)}^{\frac{\mathit{abt}}{\mu }}-1\right)}{\left({\left(1+\mu \right)}^{\frac{\mathit{at}}{\mu }}-1\right)\left({\left(1+\mu \right)}^{\frac{\mathit{bt}}{\mu }}-1\right)}.$$E27

For $\mu \in \mathbb{C}$, we introduce the modified degenerate Bernoulli polynomials given by the generating function

$$\sum _{n=0}^{\infty }{\beta }_n\left(x|\mu \right)\frac{t^n}{n!}=\frac{t}{{\left(1+\mu \right)}^{\frac{t}{\mu }}-1}{\left(1+\mu \right)}^{\frac{\mathit{xt}}{\mu }},\left(\mathrm{see}\left[6,7\right]\right).$$E28

When $x=0$ and ${\beta }_n\left(\mu \right)={\beta }_n\left(0|\mu \right)$ are called the modified degenerate Bernoulli numbers. Note that

$$\underset{\mu \to 0}{lim}{\beta }_n\left(\mu \right)=B_n,$$E29

where $B_n$ are called the Bernoulli numbers. The first few of them are

$$\begin{array}{l}{\beta }_0\left(x|\mu \right)=\frac{\mu }{log\left(1+\mu \right)},\\ {\beta }_1\left(x|\mu \right)=-\frac{1}{2}+x,\\ {\beta }_2\left(x|\mu \right)=\frac{log\left(1+\mu \right)}{6\mu }-\frac{xlog\left(1+\mu \right)}{\mu }+\frac{x^{2}log\left(1+\mu \right)}{\mu },\\ {\beta }_3\left(x|\mu \right)=\frac{xlog{\left(1+\mu \right)}^2}{2{\mu }^2}-\frac{3x^{2}log{\left(1+\mu \right)}^2}{2{\mu }^2}+\frac{x^{3}log{\left(1+\mu \right)}^2}{{\mu }^2},\\ {\beta }_4\left(x|\mu \right)=-\frac{log{\left(1+\mu \right)}^3}{30{\mu }^3}+\frac{x^{2}log{\left(1+\mu \right)}^3}{{\mu }^3}-\frac{2x^{3}log{\left(1+\mu \right)}^3}{{\mu }^3}+\frac{x^{4}log{\left(1+\mu \right)}^3}{{\mu }^3}.\end{array}$$E30

For each integer $k\ge 0$, ${\mathrm{S}}_k\left(n\right)=0^k+1^k+2^k+\cdots +{\left(n-1\right)}^k$ is called sum of integers. A modified generalized falling factorial sum ${\sigma }_k\left(n,\mu \right)$ can be defined by the generating function

$$\sum _{k=0}^{\infty }{\sigma }_k\left(n|\mu \right)\frac{t^k}{k!}=\frac{{\left(1+\mu \right)}^{\frac{\left(n+1\right)t}{\mu }}-1}{{\left(1+\mu \right)}^{\frac{t}{\mu }}-1}.$$E31

Note that ${lim}_{\mu \to 0}{\sigma }_k\left(n|\mu \right)={\mathrm{S}}_k\left(n\right).$ From $\mathcal{F}\left(t,\mu \right)$, we get the following result:

$$\begin{array}{l}\mathcal{F}\left(t,\mu \right)=\frac{\mathit{abt}{\left(1+\mu \right)}^{\frac{\mathit{\text{abxt}}}{\mu }}{e}^{a^2{b}^2{\mathit{yt}}^2}\left({\left(1+\mu \right)}^{\frac{\mathit{abt}}{\mu }}-1\right)}{\left({\left(1+\mu \right)}^{\frac{\mathit{at}}{\mu }}-1\right)\left({\left(1+\mu \right)}^{\frac{\mathit{bt}}{\mu }}-1\right)}\\ =\frac{\mathit{abt}}{\left({\left(1+\mu \right)}^{\frac{\mathit{at}}{\mu }}-1\right)}{\left(1+\mu \right)}^{\frac{\mathit{\text{abxt}}}{\mu }}{e}^{a^2{b}^2{\mathit{yt}}^2}\frac{\left({\left(1+\mu \right)}^{\frac{\mathit{abt}}{\mu }}-1\right)}{\left({\left(1+\mu \right)}^{\frac{\mathit{bt}}{\mu }}-1\right)}\\ =b\sum _{n=0}^{\infty }{\beta }_n\left(\mu \right)\frac{{\left(\mathit{at}\right)}^n}{n!}\sum _{n=0}^{\infty }{\mathrm{H}}_n\left(\mathit{bx},b^{2}y|\mu \right)\frac{{\left(\mathit{at}\right)}^n}{n!}\sum _{n=0}^{\infty }{\sigma }_k\left(a-1|\mu \right)\frac{{\left(\mathit{bt}\right)}^n}{n!}\\ =\sum _{n=0}^{\infty }\left(\sum _{i=0}^n\sum _{m=0}^i\left(\begin{array}{c}n\\ i\end{array}\right)\left(\begin{array}{c}i\\ m\end{array}\right)a^i{b}^{n+1-i}{\beta }_m\left(\mu \right){\mathrm{H}}_{i-m}\left(\mathit{bx},b^{2}y|\mu \right){\sigma }_{n-i}\left(a-1|\mu \right)\right)\frac{t^n}{n!}.\end{array}$$E32

In a similar fashion we have

$$\begin{array}{c}\mathcal{F}\left(t,\mu \right)=\frac{\mathit{abt}}{\left({\left(1+\mu \right)}^{\frac{\mathit{bt}}{\mu }}-1\right)}{\left(1+\mu \right)}^{\frac{\mathit{\text{abxt}}}{\mu }}{e}^{a^2{b}^2{\mathit{yt}}^2}\frac{\left({\left(1+\mu \right)}^{\frac{\mathit{abt}}{\mu }}-1\right)}{\left({\left(1+\mu \right)}^{\frac{\mathit{at}}{\mu }}-1\right)}\\ =a\sum _{n=0}^{\infty }{\beta }_n\left(\mu \right)\frac{{\left(\mathit{bt}\right)}^n}{n!}\sum _{n=0}^{\infty }{\mathrm{H}}_n\left(\mathit{ax},a^{2}y|\mu \right)\frac{{\left(\mathit{bt}\right)}^n}{n!}\sum _{n=0}^{\infty }{\sigma }_k\left(b-1|\mu \right)\frac{{\left(\mathit{at}\right)}^n}{n!}\\ =\sum _{n=0}^{\infty }\left(\sum _{i=0}^n\sum _{m=0}^i\left(\begin{array}{c}n\\ i\end{array}\right)\left(\begin{array}{c}i\\ m\end{array}\right)b^i{a}^{n+1-i}{\beta }_m\left(\mu \right){\mathrm{H}}_{i-m}\left(\mathit{ax},a^{2}y|\mu \right){\sigma }_{n-i}\left(b-1|\mu \right)\right)\frac{t^n}{n!}.\end{array}$$E33

By comparing the coefficients of $\frac{t^m}{m!}$ on the right hand sides of the last two equations, we have the below theorem.

Theorem 4. Let $a,b>0$($a\ne b$). The the following identity holds true:

$$\begin{array}{l}\sum _{i=0}^n\sum _{m=0}^i\left(\begin{array}{c}n\\ i\end{array}\right)\left(\begin{array}{c}i\\ m\end{array}\right)a^i{b}^{n+1-i}{\beta }_m\left(\mu \right){\mathrm{H}}_{i-m}\left(\mathit{bx},b^{2}y|\mu \right){\sigma }_{n-i}\left(a-1|\mu \right)\\ =\sum _{i=0}^n\sum _{m=0}^i\left(\begin{array}{c}n\\ i\end{array}\right)\left(\begin{array}{c}i\\ m\end{array}\right)b^i{a}^{n+1-i}{\beta }_m\left(\mu \right){\mathrm{H}}_{i-m}\left(\mathit{ax},a^{2}y|\mu \right){\sigma }_{n-i}\left(b-1|\mu \right).\end{array}$$E34

By taking the limit as $\mu \to 0$, we have the following corollary.

Corollary 5. Let $a,b>0$ ($a\ne b$). The the following identity holds true:

$$\begin{array}{l}\sum _{i=0}^n\sum _{m=0}^i\left(\begin{array}{c}n\\ i\end{array}\right)\left(\begin{array}{c}i\\ m\end{array}\right)a^i{b}^{n+1-i}{\mathrm{S}}_{n-i}\left(a-1\right)B_m{H}_{i-m}\left(\mathit{bx},b^{2}y\right)\\ =\sum _{i=0}^n\sum _{m=0}^i\left(\begin{array}{c}n\\ i\end{array}\right)\left(\begin{array}{c}i\\ m\end{array}\right)b^i{a}^{n+1-i}{\mathrm{S}}_{n-i}\left(b-1\right)B_m{H}_{i-m}\left(\mathit{ax},a^{2}y\right).\end{array}$$E35

4. Differential equations associated with 2-variable modified degenerate Hermite polynomials

In this section, we construct the differential equations with coefficients $a_i\left(N,x,y,\mu \right)$ arising from the generating functions of the 2-variable modified degenerate Hermite polynomials:

$$\begin{array}{l}{\left(\frac{\partial }{\partial t}\right)}^{N}G\left(t,x,y,\mu \right)-a_0\left(N,x,y,\mu \right)G\left(t,x,y,\mu \right)-\cdots \\ -a_{2N}\left(N,x,y,\mu \right)t^{N}G\left(t,x,y,\mu \right)=0.\end{array}$$E36

By using the coefficients of this differential equation, we can get explicit identities for the 2-variable modified degenerate Hermite polynomials ${\mathrm{H}}_n\left(x,y,\mu \right)$. Recall that

$$\begin{array}{c}G=G\left(t,x,y,\mu \right)\\ ={\left(1+\mu \right)}^{\frac{\mathit{xt}}{\mu }}{e}^{{\mathit{yt}}^2}\\ =\sum _{n=0}^{\infty }{\mathrm{H}}_n\left(x,y|\mu \right)\frac{t^n}{n!},\mu ,x,t\in \mathbb{C}.\end{array}$$E37

Then, by (37), we have

$$\begin{array}{c}{G}^{\left(1\right)}=\frac{\partial }{\partial t}G\left(t,x,y,\mu \right)\\ =\frac{\partial }{\partial t}\left({\left(1+\mu \right)}^{\frac{\mathit{xt}}{\mu }}{e}^{{\mathit{yt}}^2}\right)\\ =\left(\frac{log\left(1+\mu \right)}{\mu }x+2\mathit{yt}\right)\left({\left(1+\mu \right)}^{\frac{\mathit{xt}}{\mu }}{e}^{{\mathit{yt}}^2}\right)\\ =\left(\frac{log\left(1+\mu \right)}{\mu }x+2\mathit{yt}\right)G\left(t,x,y,\mu \right),\end{array}$$E38

$$\begin{array}{c}{G}^{\left(2\right)}=\frac{\partial }{\partial t}{G}^{\left(1\right)}\left(t,x,y,\mu \right)\\ =2\mathit{yG}\left(t,x,y,\mu \right)+\left(\frac{log\left(1+\mu \right)}{\mu }x+2\mathit{yt}\right)G^{\left(1\right)}\left(t,x,y,\mu \right)\\ =\left(2y+{\left(\frac{log\left(1+\mu \right)}{\mu }\right)}^2{x}^2\right)G\left(t,x,y,\mu \right)\\ +\left(\frac{log\left(1+\mu \right)}{\mu }4\mathit{xy}\right)\mathit{tG}\left(t,x,y,\mu \right)\\ +{\left(2y\right)}^2{t}^{2}G\left(t,x,y,\mu \right).\end{array}$$E39

By continuing this process as shown in (39), we can get easily that

$$\begin{array}{c}{G}^{\left(N\right)}={\left(\frac{\partial }{\partial t}\right)}^{N}G\left(t,x,y,\mu \right)\\ =\sum _{i=0}^N{a}_i\left(N,x,y,\mu \right)t^{i}G\left(t,x,y,\mu \right),\left(N=\mathrm{0,1,2},\dots \right).\end{array}$$E40

By differentiating (40) with respect to $t$, we have

$$\begin{array}{l}{G}^{\left(N+1\right)}=\frac{\partial G^{\left(N\right)}}{\partial t}=\sum _{i=0}^N{a}_i\left(N,x,y,\mu \right)\left(i\right)t^{i-1}G\left(t,x,y,\mu \right)\\ +\sum _{i=0}^N{a}_i\left(N,x,y,\mu \right)t^i{G}^{\left(1\right)}\left(t,x,y,\mu \right)\\ =\sum _{i=0}^N\left(i\right)a_i\left(N,x,y,\mu \right)t^{i-1}G\left(t,x,y,\mu \right)\\ +\sum _{i=0}^N\left(\frac{log\left(1+\mu \right)}{\mu }x\right)a_i\left(N,x,y,\mu \right)t^{i}G\left(t,x,y,\mu \right)\\ +\sum _{i=0}^N\left(2y\right)a_i\left(N,x,y,\mu \right)t^{i+1}G\left(t,x,y,\mu \right)\\ =\sum _{i=0}^{N-1}\left(i+1\right)a_{i+1}\left(N,x,y,\mu \right)t^{i}G\left(t,x,y,\mu \right)\\ +\sum _{i=0}^N\left(\frac{log\left(1+\mu \right)}{\mu }x\right)a_i\left(N,x,y,\mu \right)t^{i}G\left(t,x,y,\mu \right)\\ +\sum _{i=1}^{N+1}\left(2y\right)a_{i-1}\left(N,x,y,\mu \right)t^{i}G\left(t,x,y,\mu \right).\end{array}$$E41

Now we replace $N$ by $N+1$ in (40). We find

$$G^{\left(N+1\right)}=\sum _{i=0}^{N+1}{a}_i\left(N+1,x,y,\mu \right)t^{i}G\left(t,x,y,\mu \right).$$E42

By comparing the coefficients on both sides of (41) and (42), we get

$$\begin{array}{l}{a}_0\left(N+1,x,y,\mu \right)=a_1\left(N,x,y,\mu \right)+\left(\frac{log\left(1+\mu \right)}{\mu }x\right)a_0\left(N,x,y,\mu \right),\\ a_N\left(N+1,x,y,\mu \right)=\left(\frac{log\left(1+\mu \right)}{\mu }x\right)a_N\left(N,x,y,\mu \right)\\ +2{\mathit{ya}}_{N-1}\left(N,x,y,\mu \right),\\ a_{N+1}\left(N+1,x,y,\mu \right)=\left(2y\right)a_N\left(N,x,y,\mu \right),\end{array}$$E43

and

$$\begin{array}{l}{a}_i\left(N+1,x,y,\mu \right)=\left(i+1\right)a_{i+1}\left(N,x,y,\mu \right)\\ +\left(\frac{log\left(1+\mu \right)}{\mu }x\right)a_i\left(N,x,y,\mu \right)\\ +\left(2y\right)a_{i-1}\left(N,x,y,\mu \right),\left(1\le i\le N-1\right).\end{array}$$E44

In addition, by (37), we have

$$G\left(t,x,y,\mu \right)=G^{\left(0\right)}\left(t,x,y,\mu \right)=a_0\left(0,x,y,\mu \right)G\left(t,x,y,\mu \right).$$E45

By (45), we get

$$a_0\left(0,x,y,\mu \right)=1.$$E46

It is not difficult to show that

$$\begin{array}{l}\frac{xlog\left(1+\mu \right)}{\mu }G\left(t,x,y,\mu \right)+2\mathit{ytG}\left(t,x,y,\mu \right)\\ =G^{\left(1\right)}\left(t,x,y,\mu \right)\\ =\sum _{i=0}^1{a}_i\left(1,x,y,\mu \right)t^{i}G\left(t,x,y,\mu \right)\\ =a_0\left(1,x,y,\mu \right)G\left(t,x,y,\mu \right)+a_1\left(1,x,y,\mu \right)\mathit{tG}\left(t,x,y,\mu \right).\end{array}$$E47

Thus, by (38) and (47), we also get

$$a_0\left(1,x,y,\mu \right)=\frac{xlog\left(1+\mu \right)}{\mu },a_1\left(1,x,y,\mu \right)=2y.$$E48

From (43) and (44), we note that

$$\begin{array}{l}{a}_0\left(N+1,x,y,\mu \right)=a_1\left(N,x,y,\mu \right)+\frac{xlog\left(1+\mu \right)}{\mu }{a}_0\left(N,x,y,\mu \right),\\ a_0\left(N,x,y,\mu \right)=a_1\left(N-1,x,y,\mu \right)+\frac{xlog\left(1+\mu \right)}{\mu }{a}_0\left(N-1,x,y,\mu \right),\\ \dots \\ a_0\left(N+1,x,y,\mu \right)=\sum _{i=0}^N{\left(\frac{xlog\left(1+\mu \right)}{\mu }\right)}^i{a}_1\left(N-i,x,y,\mu \right)\\ +{\left(\frac{log\left(1+\mu \right)}{\mu }\right)}^{N+1}{x}^{N+1},\end{array}$$E49

$$\begin{array}{l}{a}_N\left(N+1,x,y,\mu \right)=\frac{xlog\left(1+\mu \right)}{\mu }{a}_N\left(N,x,y,\mu \right)\\ +\left(2y\right)a_{N-1}\left(N,x,y,\mu \right),\\ a_{N-1}\left(N,x,y,\mu \right)=\frac{xlog\left(1+\mu \right)}{\mu }{a}_{N-1}\left(N-1,x,y,\mu \right)\\ +\left(2y\right)a_{N-2}\left(N-1,x,y,\mu \right),\dots \\ a_N\left(N+1,x,y,\mu \right)=\left(N+1\right)x{\left(2y\right)}^N\left(\frac{log\left(1+\mu \right)}{\mu }\right),\end{array}$$E50

and

$$\begin{array}{l}{a}_{N+1}\left(N+1,x,y,\mu \right)=\left(2y\right)a_N\left(N,x,y,\mu \right),\\ a_N\left(N,x,y,\mu \right)=\left(2y\right)a_{N-1}\left(N-1,x,y,\mu \right),\dots \\ a_{N+1}\left(N+1,x,y,\mu \right)={\left(2y\right)}^{N+1}.\end{array}$$E51

For $i=1$ in (44), we have

$$\begin{array}{l}{a}_1\left(N+1,x,y,\mu \right)=2\sum _{k=0}^N{\left(\frac{xlog\left(1+\mu \right)}{\mu }\right)}^k{a}_2\left(N-k,x,y,\mu \right)\\ +2y\sum _{k=0}^N{\left(\frac{xlog\left(1+\mu \right)}{\mu }\right)}^k{a}_0\left(N-k,x,y,\mu \right),\end{array}$$E52

Continuing this process, we can deduce that, for $1\le i\le N-1,$

$$\begin{array}{l}{a}_i\left(N+1,x,y,\mu \right)=\left(i+1\right)\sum _{k=0}^N{\left(\frac{xlog\left(1+\mu \right)}{\mu }\right)}^k{a}_{i+1}\left(N-k,x,y,\mu \right)\\ +2y\sum _{k=0}^N{\left(\frac{xlog\left(1+\mu \right)}{\mu }\right)}^k{a}_{i-1}\left(N-k,x,y,\mu \right).\end{array}$$E53

Note that, from (37)–(53), here the matrix $a_i{\left(j,x,y,\mu \right)}_{0\le i,j\le N+1}$ is given by

$$\left(\begin{array}{ccccc}1& \frac{xlog\left(1+\mu \right)}{\mu }& 2y+{\left(\frac{log\left(1+\mu \right)}{\mu }\right)}^2{x}^2& \cdots & \cdot \\ & & & & \\ 0& 2y& \left(\frac{log\left(1+\mu \right)}{\mu }\right)4\mathit{xy}& \cdots & \cdot \\ & & & & \\ 0& 0& {\left(2y\right)}^2& \cdots & \cdot \\ & & & & \\ ⋮& ⋮& ⋮& \ddots & \cdot \\ & & & & \\ 0& 0& 0& \cdots & {\left(2y\right)}^{N+1}\end{array}\right)$$E54

Therefore, from (37)–(53), we obtain the following theorem.

Theorem 5. For $N=\mathrm{0,1,2},\dots ,$ the differential equation

$${\left(\frac{\partial }{\partial t}\right)}^{N}G\left(t,x,y,\mu \right)-\left(\sum _{i=0}^N{a}_i\left(N,x,y,\mu \right)t^i\right)G\left(t,x,y,\mu \right)=0$$E55

has a solution

$$G=G\left(t,x,y,\mu \right)={\left(1+\mu \right)}^{\frac{\mathit{xt}}{\mu }}{e}^{{\mathit{yt}}^2},$$E56

where

$$\begin{array}{l}{a}_0\left(N+1,x,y,\mu \right)=\sum _{i=0}^N{\left(\frac{xlog\left(1+\mu \right)}{\mu }\right)}^i{a}_1\left(N-i,x,y,\mu \right)\\ +{\left(\frac{log\left(1+\mu \right)}{\mu }\right)}^{N+1}{x}^{N+1},\\ a_N\left(N+1,x,y,\mu \right)=\left(N+1\right){\left(2y\right)}^N\left(\frac{xlog\left(1+\mu \right)}{\mu }\right),\\ a_{N+1}\left(N+1,x,y,\mu \right)={\left(2y\right)}^{N+1},\\ a_i\left(N+1,x,y,\mu \right)=\left(i+1\right)\sum _{k=0}^N{\left(\frac{xlog\left(1+\mu \right)}{\mu }\right)}^k{a}_{i+1}\left(N-k,x,y,\mu \right)\\ +2y\sum _{k=0}^N{\left(\frac{xlog\left(1+\mu \right)}{\mu }\right)}^k{a}_{i-1}\left(N-k,x,y,\mu \right),\left(1\le i\le N-1\right).\end{array}$$E57

Here is a plot of the surface for this solution. In the left picture of Figure 1, we choose $-2\le x\le 2,-1\le t\le 1,\mu =1/10,$ and $y=0.1$. In the right picture of Figure 1, we choose $-2\le y\le 2,-1\le t\le 1,\mu =1/10,$ and $x=0.1$.

Figure 1.

The surface for the solution $G\left(t,x,y,\mu \right)$.

Making $N$-times derivative for (10) with respect to $t$, we have

$${\left(\frac{\partial }{\partial t}\right)}^{N}G\left(t,x,y,\mu \right)=\sum _{m=0}^{\infty }{\mathrm{H}}_{m+N}\left(x,y,\mu \right)\frac{t^m}{m!}.$$E58

By (58) and Theorem 5, we have

$$\begin{array}{l}{a}_0\left(N,x,y,\mu \right)G\left(t,x,y,\mu \right)\\ +a_1\left(N,x,y,\mu \right)\mathit{tG}\left(t,x,y,\mu \right)\\ +\cdots \\ +a_N\left(N,x,y,\mu \right)t^{N}G\left(t,x,y,\mu \right)\\ =\sum _{m=0}^{\infty }{\mathrm{H}}_{m+N}\left(x,y,\mu \right)\frac{t^m}{m!}.\end{array}$$E59

Hence we have the following theorem.

Theorem 6. For $N=\mathrm{0,1,2},\dots ,$ we get

$${\mathrm{H}}_{m+N}\left(x,y,\mu \right)=\sum _{i=0}^m\frac{{\mathrm{H}}_{m-i}\left(x\right)a_i\left(N,x,y,\mu \right)m!}{\left(m-i\right)!}.$$E60

If we take $m=0$ in (60), then we have the below corollary.

Corollary 7. For $N=\mathrm{0,1,2},\dots ,$ we have

$${\mathrm{H}}_N\left(x,y,\mu \right)=a_0\left(N,x,y,\mu \right),$$E61

where

$$\begin{array}{l}{a}_0\left(0,x,y,\mu \right)=1,\\ a_0\left(N+1,x,y,\mu \right)=\sum _{i=0}^N{\left(\frac{xlog\left(1+\mu \right)}{\mu }\right)}^i{a}_1\left(N-i,x,y,\mu \right)\\ +{\left(\frac{log\left(1+\mu \right)}{\mu }\right)}^{N+1}{x}^{N+1}.\end{array}$$E62

The first few of them are

$$\begin{array}{l}{\mathrm{H}}_0\left(x,y,\mu \right)=1,\\ {\mathrm{H}}_1\left(x,y,\mu \right)=\frac{log\left(1+\mu \right)}{\mu }x,\\ {\mathrm{H}}_2\left(x,y,\mu \right)=2y+\frac{{\left(log\left(1+\mu \right)\right)}^2}{{\mu }^2}{x}^2,\\ {\mathrm{H}}_3\left(x,y,\mu \right)=6\mathit{xy}\frac{log\left(1+\mu \right)}{\mu }+\frac{{\left(log\left(1+\mu \right)\right)}^3}{{\mu }^3}{x}^3,\\ {\mathrm{H}}_4\left(x,y,\mu \right)=12y^2+12x^{2}y\frac{{\left(log\left(1+\mu \right)\right)}^2}{{\mu }^2}+\frac{{\left(log\left(1+\mu \right)\right)}^4}{{\mu }^4}{x}^4,\\ {\mathrm{H}}_5\left(x,y,\mu \right)=60{\mathit{xy}}^2\frac{log\left(1+\mu \right)}{\mu }+20x^{3}y\frac{{\left(log\left(1+\mu \right)\right)}^3}{{\mu }^3}+\frac{{\left(log\left(1+\mu \right)\right)}^5}{{\mu }^5}{x}^5.\end{array}$$E63

5. Zeros of the 2-variable modified degenerate Hermite polynomials

This section shows the benefits of supporting theoretical prediction through numerical experiments and finding new interesting pattern of the zeros of the 2-variable modified degenerate Hermite equations ${\mathrm{H}}_n\left(x,y|\mu \right)=0$. By using computer, the 2-variable modified degenerate Hermite polynomials ${\mathrm{H}}_n\left(x,y|\mu \right)$ can be determined explicitly. We investigate the zeros of the 2-variable modified degenerate Hermite equations ${\mathrm{H}}_n\left(x,y|\mu \right)=0$. The zeros of the ${\mathrm{H}}_n\left(x,y|\mu \right)=0$ for $n=30,y=3,-3,3+i,-3-i,\mu =1/2,$ and $x\in \mathbb{C}$ are displayed in Figure 2. In the top-left picture of Figure 2, we choose $n=30$ and $y=3$. In the top-right picture of Figure 2, we choose $n=30$ and $y=-3$. In the bottom-left picture of Figure 2, we choose $n=30$ and $y=-3+i$ . In the bottom-right picture of Figure 2, we choose $n=30$ and $y=-3-i$.

Figure 2.

Zeros of ${\mathrm{H}}_n\left(x,y|\mu \right)=0$.

Stacks of zeros of the 2-variable modified degenerate Hermite equations ${\mathrm{H}}_n\left(x,y|\mu \right)=0$ for $1\le n\le 50,\mu =1/2$ from a 3-D structure are presented in Figure 3. In the top-left picture of Figure 3, we choose $y=3$. In the top-right picture of Figure 3, we choose $y=-3$. In the bottom-left picture of Figure 3, we choose $y=-3+i$. In the bottom-right picture of Figure 3, we choose $y=-3-i$.

Figure 3.

Stacks of zeros of ${\mathrm{H}}_n\left(x,y|\mu \right)=0,1\le n\le 50$.

Our numerical results for approximate solutions of real zeros of the 2-variable modified degenerate Hermite equations ${\mathrm{H}}_n\left(x,y|\mu \right)=0$ are displayed (Tables 1 and 2).

	$y=3,\mu =1/2$		$y=-3,\mu =1/2$
Degree $n$	Real zeros	Complex zeros	Real zeros	Complex zeros
1	1	0	1	0
2	0	2	2	0
3	1	2	3	0
4	0	4	4	0
5	1	4	5	0
6	0	6	6	0
7	1	6	7	0
8	0	8	8	0
9	1	8	9	0
10	0	10	10	0

Table 1.

Numbers of real and complex zeros of ${\mathrm{H}}_n\left(x,y|\mu \right)=0$.

Degree $n$	$x$
1	$0$
2	$-3.0206,3.0206$
3	$-5.2318,0,5.2318$
4	$-7.0513,-2.2412,2.2412,7.0513$
5	$-8.6297,-4.0948,0,4.0948,8.6297$
6	$-10.041,-5.7064,-1.8628,1.8628,5.7064,10.041$
7	$-11.329,-7.1490,-3.4870,0,3.4870,7.1490,11.329$
8	$-12.519,-8.4652,-4.9433,-1.6283,1.6283,4.9433,8.4652,12.519$

Table 2.

Approximate solutions of ${\mathrm{H}}_n\left(x,y|\mu \right)=0,x\in \mathbb{R}$.

We observed a remarkable regular structure of the complex roots of the 2-variable modified degenerate Hermite equations ${\mathrm{H}}_n\left(x,y|\mu \right)=0$ and also hope to verify same kind of regular structure of the complex roots of the 2-variable modified degenerate Hermite equations ${\mathrm{H}}_n\left(x,y|\mu \right)=0$ (Table 1).

Plot of real zeros of the 2-variable modified degenerate Hermite equations ${\mathrm{H}}_n\left(x,y|\mu \right)=0$ for $1\le n\le 50,\mu =1/2$ structure are presented in Figure 4. In the top-left picture of Figure 4, we choose $y=3$. In the top-right picture of Figure 4, we choose $y=-3$. In the bottom-left picture of Figure 4, we choose $y=-3+i$. In the bottom-right picture of Figure 4, we choose $y=-3-i$.

Figure 4.

Real zeros of ${\mathrm{H}}_n\left(x,y|\mu \right)=0\text{for}1\le n\le 50$, $\mu =\frac{1}{2}$.

Next, we calculated an approximate solution satisfying ${\mathrm{H}}_n\left(x,y|\mu \right)=0,x\in \mathbb{C}$. The results are given in Table 2. In Table 2, we choose $y=-3$ and $\mu =1/2$.

6. Conclusions

In this chapter, we constructed the 2-variable modified degenerate Hermite polynomials and got some new symmetric identities for 2-variable modified degenerate Hermite polynomials. We constructed differential equations arising from the generating function of the 2-variable modified degenerate Hermite polynomials ${\mathrm{H}}_n\left(x,y|\mu \right)$. We also investigated the symmetry of the zeros of the 2-variable modified degenerate Hermite equations ${\mathrm{H}}_n\left(x,y|\mu \right)=0$ for various variables $x$ and $y$. As a result, we found that the distribution of the zeros of 2-variable modified degenerate Hermite equations ${\mathrm{H}}_n\left(x,y|\mu \right)=0$ is very regular pattern. So, we make the following series of conjectures with numerical experiments:

Let us use the following notations. $R_{{\mathrm{H}}_n\left(x,y|\mu \right)}$ denotes the number of real zeros of ${\mathrm{H}}_n\left(x,y|\mu \right)=0$ lying on the real plane $\mathit{Im}\left(x\right)=0$ and $C_{{\mathrm{H}}_n\left(x,y|\mu \right)}$ denotes the number of complex zeros of ${\mathrm{H}}_n\left(x,y|\mu \right)=0$. Since $n$ is the degree of the polynomial ${\mathrm{H}}_n\left(x,y|\mu \right)$, we have $R_{{\mathrm{H}}_n\left(x,y|\mu \right)}=n-C_{{\mathrm{H}}_n\left(x,y|\mu \right)}$.

We can see a good regular pattern of the complex roots of the 2-variable modified degenerate Hermite equations ${\mathrm{H}}_n\left(x,y,\mu \right)=0$ for $y$ and $\mu$. Therefore, the following conjecture is possible.

Conjecture 1. Let $n$ be odd positive integer. For $a>0$ or $a\in \mathbb{C}\backslash \left\{a|a<0\right\}$, prove or disprove that

$$R_{{\mathrm{H}}_n\left(x,a,\mu \right)}=1,C_{{\mathrm{H}}_n\left(x,a,\mu \right)}=2\left[\frac{n}{2}\right],$$E64

where $\mathbb{C}$ is the set of complex numbers.

Conjecture 2. Let $n$ be odd positive integer and $a\in \mathbb{C}$. Prove or disprove that

$${\mathrm{H}}_n\left(0,a,\mu \right)=0.$$E65

As a result of investigating more $y$ and $\mu$ variables, it is still unknown whether the conjecture 1 and conjecture 2 is true or false for all variables $y$ and $\mu$.

We observe that solutions of the 2-variable modified degenerate Hermite equations ${\mathrm{H}}_n\left(x,y,\mu \right)=0$ has not $\mathit{Re}\left(x\right)=b$ reflection symmetry for $b\in \mathbb{R}$. It is expected that solutions of the 2-variable modified degenerate Hermite equations ${\mathrm{H}}_n\left(x,y,\mu \right)=0$, has not $\mathit{Re}\left(x\right)=b$ reflection symmetry (see Figures 2–4).

Conjecture 3. Prove that the zeros of ${\mathrm{H}}_n\left(x,a,\mu \right)=0,a\in \mathbb{R}$, has $\mathit{Im}\left(x\right)=0$ reflection symmetry analytic complex functions. Prove that the zeros of ${\mathrm{H}}_n\left(x,a,\mu \right)=0,a<0,a\in \mathbb{C}\backslash \mathbb{R}$, has not $\mathit{Im}\left(x\right)=0$ reflection symmetry analytic complex functions.

Finally, we consider the more general problems. How many zeros does ${\mathrm{H}}_n\left(x,y,\mu \right)$ have? We are not able to decide if ${\mathrm{H}}_n\left(x,y,\mu \right)=0$ has $n$ distinct solutions. We would like to know the number of complex zeros $C_{{\mathrm{H}}_n\left(x,y,\mu \right)}$ of ${\mathrm{H}}_n\left(x,y,\mu \right)=0.$

Conjecture 4. For $a\in \mathbb{C},$ prove or disprove that ${\mathrm{H}}_n\left(x,a,\mu \right)=0$ has $n$ distinct solutions.

As a result of investigating more $n$ variables, it is still unknown whether the conjecture is true or false for all variables $n$ (see Tables 1 and 2).

We expect that research in these directions will make a new approach using the numerical method related to the research of the 2-variable modified degenerate Hermite equations ${\mathrm{H}}_n\left(x,y,\mu \right)=0$ which appear in applied mathematics and mathematical physics.

Additional information

Mathematics Subject Classification: 05A19, 11B83, 34A30, 65L99