Differential Equations Arising from the 3-Variable Hermite Polynomials and Computation of Their Zeros

Abstract

In this paper, we study differential equations arising from the generating functions of the 3-variable Hermite polynomials. We give explicit identities for the 3-variable Hermite polynomials. Finally, we investigate the zeros of the 3-variable Hermite polynomials by using computer.

1. Introduction

Many mathematicians have studied in the area of the Bernoulli numbers, Euler numbers, Genocchi numbers, and tangent numbers see [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15]. The special polynomials of two variables provided new means of analysis for the solution of a wide class of differential equations often encountered in physical problems. Most of the special function of mathematical physics and their generalization have been suggested by physical problems.

In [1], the Hermite polynomials are given by the exponential generating function

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

We can also have the generating function by using Cauchy’s integral formula to write the Hermite polynomials as

$$H_n\left(x\right)={\left(-1\right)}^n{e}^{x^2}\frac{d^n}{{\mathit{dx}}^n}{e}^{-x^2}=\frac{n!}{2\mathit{\pi i}}{\oint }_C\frac{e^{2\mathit{tx}-t^2}}{t^{n+1}}\mathit{dt}$$

with the contour encircling the origin. It follows that the Hermite polynomials also satisfy the recurrence relation

$$H_{n+1}\left(x\right)=2{\mathit{xH}}_n\left(x\right)-2{\mathit{nH}}_{n-1}\left(x\right).$$

Further, the two variables Hermite Kampé de Fériet polynomials $H_n\left(x,y\right)$ defined by the generating function (see [3])

$$\sum _{n=0}^{\infty }{H}_n\left(x,y\right)\frac{t^n}{n!}=e^{\mathit{xt}+{\mathit{yt}}^2}$$E1

are the solution of heat equation

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

We note that

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

The 3-variable Hermite polynomials $H_n\left(x,y,z\right)$ are introduced [4].

$$H_n\left(x,y,z\right)=n!\sum _{k=0}^{\left[\frac{n}{3}\right]}\frac{z^k{H}_{n-3k}\left(x,y\right)}{k!\left(n-3k\right)!}.$$

The differential equation and he generating function for $H_n\left(x,y,z\right)$ are given by

$$\left(3z\frac{{\partial }^3}{\partial x^3}+2y\frac{{\partial }^2}{\partial x^2}+x\frac{\partial }{\partial x}-n\right)H_n\left(x,y,z\right)=0$$

and

$$e^{\mathit{xt}+{\mathit{yt}}^2+{\mathit{zt}}^3}=\sum _{n=0}^{\infty }{H}_n\left(x,y,z\right)\frac{t^n}{n!},$$E2

respectively.

By (2), we get

$$\begin{array}{c}\sum _{n=0}^{\infty }{H}_n\left(x_1+x_2,y,z\right)\frac{t^n}{n!}=e^{\left(x_1+x_2\right)t+{\mathit{yt}}^2+{\mathit{zt}}^3}\\ =\sum _{n=0}^{\infty }{x}_2^n\frac{t^n}{n!}\sum _{n=0}^{\infty }{H}_n\left(x_1,y,z\right)\frac{t^n}{n!}\\ =\sum _{n=0}^{\infty }\left(\sum _{l=0}^n\left(\begin{array}{l}n\\ l\end{array}\right)H_l\left(x_1,y,z\right)x_2^{n-l}\right)\frac{t^n}{n!}.\end{array}$$E3

By comparing the coefficients on both sides of (3), we have the following theorem.

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

$$H_n\left(x_1+x_2,y,z\right)=\sum _{l=0}^n\left(\begin{array}{l}n\\ l\end{array}\right)H_l\left(x_1,y,z\right)x_2^{n-l}.$$

Applying Eq. (2), we obtain

$$\begin{array}{c}\sum _{n=0}^{\infty }{H}_n\left(x,y,z_1+z_2\right)\frac{t^n}{n!}=e^{\mathit{xt}+{\mathit{yt}}^2+\left(z_1+z_2\right)t^3}\\ =\sum _{k=0}^{\infty }{z}_2^n\frac{t^{3k}}{k!}\sum _{l=0}^{\infty }{H}_l\left(x,y,z_1\right)\frac{t^l}{l!}\\ =\sum _{n=0}^{\infty }\left(\sum _{k=0}^{\left[\frac{n}{3}\right]}\frac{H_{n-3k}\left(x,y,z_1\right)z_2^{k}n!}{k!\left(n-3k\right)!}.\right)\frac{t^n}{n!}.\end{array}$$

On equating the coefficients of the like power of $t$ in the above, we obtain the following theorem.

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

$$H_n\left(x,y,z_1+z_2\right)=n!\sum _{k=0}^{\left[\frac{n}{3}\right]}\frac{H_{n-3k}\left(x,y,z_1\right)z_2^k}{k!\left(n-3k\right)!}.$$

Also, the 3-variable Hermite polynomials $H_n\left(x,y,z\right)$ satisfy the following relations

$$\frac{\partial }{\partial y}{H}_n\left(x,y,z\right)=\frac{{\partial }^2}{\partial x^2}{H}_n\left(x,y,z\right),$$

and

$$\frac{\partial }{\partial z}{H}_n\left(x,y,z\right)=\frac{{\partial }^3}{\partial x^3}{H}_n\left(x,y,z\right).$$

The following elementary properties of the 3-variable Hermite polynomials $H_n\left(x,y,z\right)$ are readily derived form (2). We, therefore, choose to omit the details involved.

Theorem 3. For any positive integer $n$, we have

1. $H_n\left(2x,-1,0\right)=H_n\left(x\right).$

2. $H_n\left(x,y_1+y_2,z\right)=n!\sum _{k=0}^{\left[\frac{n}{2}\right]}\frac{H_{n-2k}\left(x,y_1,z\right)y_2^k}{k!\left(n-2k\right)!}.$

3. $H_n\left(x,y,z\right)=\sum _{l=0}^n\left(\begin{array}{l}n\\ l\end{array}\right)H_l\left(x\right)H_{n-l}\left(-x,y+1,z\right).$

Theorem 4. For any positive integer $n$, we have

1. $H_n\left(x_1+x_2,y_1+y_2,z\right)=\sum _{l=0}^n\left(\begin{array}{l}n\\ l\end{array}\right)H_l\left(x_1,y_1,z\right)H_{n-l}\left(x_2,y_2\right).$

2. $H_n\left(x_1+x_2,y_1+y_2,z_1+z_2\right)=\sum _{l=0}^n\left(\begin{array}{l}n\\ l\end{array}\right)H_l\left(x_1,y_1,z\right)H_{n-l}\left(x_2,y_2,z_2\right).$

The 3-variable Hermite polynomials can be determined explicitly. A few of them are

$$\begin{array}{l}{H}_0\left(x,y,z\right)=1,\\ H_1\left(x,y,z\right)=x,\\ H_2\left(x,y,z\right)=x^2+2y,\\ H_3\left(x,y,z\right)=x^3+6\mathit{xy}+6z,\\ H_4\left(x,y,z\right)=x^4+12x^{2}y+12y^2+24\mathit{xz},\\ H_5\left(x,y,z\right)=x^5+20x^{3}y+60{\mathit{xy}}^2+60x^{2}z+120\mathit{yz},\\ H_6\left(x,y,z\right)=x^6+30x^{4}y+180x^2{y}^2+120y^3+120x^{3}z+720\mathit{xyz}+360z^2,\\ H_7\left(x,y,z\right)=x^7+42x^{5}y+420x^3{y}^2+840{\mathit{xy}}^3+210x^{4}z+2520x^2\mathit{yz}+2520y^{2}z+2520{\mathit{xz}}^2,\\ H_8\left(x,y,z\right)=x^8+56x^{6}y+840x^4{y}^2+3360x^2{y}^3+1680y^4+336x^{5}z+6720x^3\mathit{yz}\\ +20160{\mathit{xy}}^{2}z+10080x^2{z}^2+20160{\mathit{yz}}^2.\\ H_9\left(x,y,z\right)=x^9+72x^{7}y+1512x^5{y}^2+10080x^3{y}^3+15120{\mathit{xy}}^4+504x^{6}z+15120x^4\mathit{yz}\\ +90720x^2{y}^{2}z+60480y^{3}z+30240x^3{z}^2+181440{\mathit{xyz}}^2+60480z^3,\\ H_{10}\left(x,y,z\right)=x^{10}+90x^{8}y+2520x^6{y}^2+25200x^4{y}^3+75600x^2{y}^4+30240y^5+720x^{7}z\\ +30240x^5\mathit{yz}+302400x^3{y}^{2}z+604800{\mathit{xy}}^{3}z+75600x^4{z}^2\\ +907200x^2{\mathit{yz}}^2+907200y^2{z}^2+604800{\mathit{xz}}^3.\end{array}$$

Recently, many mathematicians have studied the differential equations arising from the generating functions of special polynomials (see [7, 8, 12, 16, 17, 18, 19]). In this paper, we study differential equations arising from the generating functions of the 3-variable Hermite polynomials. We give explicit identities for the 3-variable Hermite polynomials. In addition, we investigate the zeros of the 3-variable Hermite polynomials using numerical methods. Using computer, a realistic study for the zeros of the 3-variable Hermite polynomials is very interesting. Finally, we observe an interesting phenomenon of ‘scattering’ of the zeros of the 3-variable Hermite polynomials.

2. Differential equations associated with the 3-variable Hermite polynomials

In this section, we study differential equations arising from the generating functions of the 3-variable Hermite polynomials.

Let

$$F=F\left(t,x,y,z\right)=e^{\mathit{xt}+{\mathit{yt}}^2+{\mathit{zt}}^3}=\sum _{n=0}^{\infty }{H}_n\left(x,y,z\right)\frac{t^n}{n!},x,y,z,t\in \mathrm{ℂ}.$$E4

Then, by (4), we have

$$\begin{array}{c}{F}^{\left(1\right)}=\frac{\partial }{\partial t}F\left(t,x,y,z\right)=\frac{\partial }{\partial t}\left(e^{\mathit{xt}+{\mathit{yt}}^2+{\mathit{zt}}^3}\right)=e^{\mathit{xt}+{\mathit{yt}}^2+{\mathit{zt}}^3}\left(x+2\mathit{yt}+3{\mathit{zt}}^2\right)\\ =\left(x+2\mathit{yt}+3{\mathit{zt}}^2\right)F\left(t,x,y,z\right),\end{array}$$E5

$$\begin{array}{c}{F}^{\left(2\right)}=\frac{\partial }{\partial t}{F}^{\left(1\right)}\left(t,x,y,z\right)=\left(2y+6\mathit{zt}\right)F\left(t,x,y,z\right)+\left(x+2\mathit{yt}+3{\mathit{zt}}^2\right)F^{\left(1\right)}\left(t,x,y,z\right)\\ =\left(\left(x^2+2y\right)+\left(6z+4\mathit{xy}\right)t+\left(4y^2+6\mathit{xz}\right)t^2+\left(12\mathit{yz}\right)t^3+\left(9z^2\right)t^4\right)F\left(t,x,y,z\right).\end{array}$$E6

Continuing this process, we can guess that

$$F^{\left(N\right)}={\left(\frac{\partial }{\partial t}\right)}^{N}F\left(t,x,y,z\right)=\sum _{i=0}^{2N}{a}_i\left(N,x,y,z\right)t^{i}F\left(t,x,y,z\right),\left(N=\mathrm{0,1,2},\dots \right).$$E7

Differentiating (7) with respect to $t$, we have

$$\begin{array}{c}{F}^{\left(N+1\right)}=\frac{\partial F^{\left(N\right)}}{\partial t}=\sum _{i=0}^{2N}{a}_i\left(N,x,y,z\right){\mathit{it}}^{i-1}F\left(t,x,y,z\right)+\sum _{i=0}^{2N}{a}_i\left(N,x,y,z\right)t^i{F}^{\left(1\right)}\left(t,x,y,z\right)\\ =\sum _{i=0}^{2N}{a}_i\left(N,x,y,z\right){\mathit{it}}^{i-1}F\left(t,x,y,z\right)+\sum _{i=0}^{2N}{a}_i\left(N,x,y,z\right)t^i\left(x+2\mathit{yt}+3{\mathit{zt}}^2\right)F\left(t,x,y,z\right)\\ =\sum _{i=0}^{2N}i{a}_i\left(N,x,y,z\right)t^{i-1}F\left(t,x,y,z\right)+\sum _{i=0}^{2N}x{a}_i\left(N,x,y,z\right)t^{i}F\left(t,x,y,z\right)\\ +\sum _{i=0}^{2N}2{\mathit{ya}}_i\left(N,x,y,z\right)t^{i+1}F\left(t,x,y,z\right)+\sum _{i=0}^{2N}3{\mathit{za}}_i\left(N,x,y,z\right)t^{i+2}F\left(t,x,y,z\right)\\ =\sum _{i=0}^{2N-1}\left(i+1\right)a_{i+1}\left(N,x,y,z\right)t^{i}F\left(t,x,y,z\right)+\sum _{i=0}^{2N}x{a}_i\left(N,x,y,z\right)t^{i}F\left(t,x,y,z\right)\\ +\sum _{i=1}^{2N+1}2{\mathit{ya}}_{i-1}\left(N,x,y,z\right)t^{i}F\left(t,x,y,z\right)+\sum _{i=2}^{2N+2}3{\mathit{za}}_{i-2}\left(N,x,y,z\right)t^{i}F\left(t,x,y,z\right)\end{array}$$

Hence we have

$$\begin{array}{c}{F}^{\left(N+1\right)}=\sum _{i=0}^{2N-1}\left(i+1\right)a_{i+1}\left(N,x,y,z\right)t^{i}F\left(t,x,y,z\right)\\ +\sum _{i=0}^{2N}{\mathit{xa}}_i\left(N,x,y,z\right)t^{i}F\left(t,x,y,z\right)\\ +\sum _{i=1}^{2N+1}2{\mathit{ya}}_{i-1}\left(N,x,y,z\right)t^{i}F\left(t,x,y,z\right)\\ +\sum _{i=2}^{2N+2}3{\mathit{za}}_{i-2}\left(N,x,y,z\right)t^{i}F\left(t,x,y,z\right).\end{array}$$E8

Now replacing $N$ by $N+1$ in (7), we find

$$F^{\left(N+1\right)}=\sum _{i=0}^{2N+2}{a}_i\left(N+1,x,y,z\right)t^{i}F\left(t,x,y,z\right).$$E9

Comparing the coefficients on both sides of (8) and (9), we obtain

$$\begin{array}{l}{a}_0\left(N+1,x,y,z\right)=a_1\left(N,x,y,z\right)+{\mathit{xa}}_0\left(N,x,y,z\right),\\ a_1\left(N+1,x,y,z\right)=2a_2\left(N,x,y,z\right)+{\mathit{xa}}_1\left(N,x,y,z\right)+2{\mathit{ya}}_0\left(N,x,y,z\right),\\ a_{2N}\left(N+1,x,y,z\right)={\mathit{xa}}_{2N}\left(N,x,y,z\right)+2{\mathit{ya}}_{2N-1}\left(N,x,y,z\right)+3{\mathit{za}}_{2N-2}\left(N,x,y,z\right),\\ a_{2N+1}\left(N+1,x,y,z\right)=2{\mathit{ya}}_{2N}\left(N,x,y,z\right)+3{\mathit{za}}_{2N-1}\left(N,x,y,z\right),\\ a_{2N+2}\left(N+1,x,y,z\right)=3{\mathit{za}}_{2N}\left(N,x,y,z\right),\end{array}$$E10

and

$$\begin{array}{l}{a}_i\left(N+1,x,y,z\right)=\left(i+1\right)a_{i+1}\left(N,x,y,z\right)+{\mathit{xa}}_i\left(N,x,y,z\right)\\ +2{\mathit{ya}}_{i-1}\left(N,x,y,z\right)+3{\mathit{za}}_{i-2}\left(N,x,y,z\right),\left(2\le i\le 2N-1\right).\end{array}$$E11

In addition, by (7), we have

$$F\left(t,x,y,z\right)=F^{\left(0\right)}\left(t,x,y,z\right)=a_0\left(0,x,y,z\right)F\left(t,x,y,z\right),$$E12

which gives

$$a_0\left(0,x,y,z\right)=1.$$E13

It is not difficult to show that

$$\begin{array}{l}\mathit{xF}\left(t,x,y\right)+2\mathit{ytF}\left(t,x,y,z\right)+3{\mathit{zt}}^{2}F\left(t,x,y,z\right)\\ =F^{\left(1\right)}\left(t,x,y,z\right)\\ =\sum _{i=0}^2{a}_i\left(1,x,y,z\right)F\left(t,x,y,z\right)\\ =\left(a_0\left(1,x,y,z\right)+a_1(1,x,y,z)t+a_2(1,x,y,z)t^2\right)F\left(t,x,y,z\right).\end{array}$$E14

Thus, by (14), we also find

$$a_0\left(1,x,y,z\right)=x,a_1\left(1,x,y,z\right)=2y,a_2\left(1,x,y,z\right)=3z.$$E15

From (10), we note that

$$\begin{array}{l}{a}_0\left(N+1,x,y,z\right)=a_1\left(N,x,y,z\right)+{\mathit{xa}}_0\left(N,x,y,z\right),\\ a_0\left(N,x,y,z\right)=a_1\left(N-1,x,y,z\right)+{\mathit{xa}}_0\left(N-1,x,y,z\right),\dots \\ a_0\left(N+1,x,y,z\right)=\sum _{i=0}^N{x}^i{a}_1\left(N-i,x,y,z\right)+x^{N+1},\end{array}$$E16

and

$$\begin{array}{l}{a}_{2N+2}\left(N+1,x,y,z\right)=3{\mathit{za}}_{2N}\left(N,x,y,z\right),\\ a_{2N}\left(N,x,y,z\right)=3{\mathit{za}}_{2N-2}\left(N-1,x,y,z\right),\dots \\ a_{2N+2}\left(N+1,x,y,z\right)={\left(3z\right)}^{N+1}.\end{array}$$E17

Note that, here the matrix $a_i{\left(j,x,y\right)}_{0\le i\le 2N+2,0\le j\le N+1}$ is given by

$$\left(\begin{array}{llllll}1& x& 2y+x^2& \cdot & \cdots & \cdot \\ & & & & & \\ 0& 2y& 4\mathit{xy}+6z& \cdot & \cdots & \cdot \\ & & & & & \\ 0& 3z& 6\mathit{xz}+4y^2& \cdot & \cdots & \cdot \\ & & & & & \\ 0& 0& 12\mathit{yz}& \cdot & \cdots & \cdot \\ & & & & & \\ 0& 0& {\left(3z\right)}^2& \cdot & \cdots & \cdot \\ & & & & & \\ 0& 0& 0& \cdot & \cdots & \cdot \\ & & & & & \\ 0& 0& 0& {\left(3z\right)}^3& \cdots & \cdot \\ & & & & & \\ ⋮& ⋮& ⋮& ⋮& \ddots & \cdot \\ & & & & & \\ 0& 0& 0& 0& \cdots & {\left(3z\right)}^{N+1}\end{array}\right)$$

Therefore, we obtain the following theorem.

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

$$F^{\left(N\right)}={\left(\frac{\partial }{\partial t}\right)}^{N}F\left(t,x,y,z\right)=\left(\sum _{i=0}^N{a}_i\left(N,x,y,z\right)t^i\right)F\left(t,x,y,z\right)$$

has a solution

$$F=F\left(t,x,y,z\right)=e^{\mathit{xt}+{\mathit{yt}}^2+{\mathit{zt}}^3},$$

where

$$\begin{array}{l}{a}_0\left(N+1,x,y,z\right)=\sum _{i=0}^N{x}^i{a}_1\left(N-i,x,y,z\right)+x^{N+1},\\ a_1\left(N+1,x,y,z\right)=2a_2\left(N,x,y,z\right)+{\mathit{xa}}_1\left(N,x,y,z\right)+2{\mathit{ya}}_0\left(N,x,y,z\right),\\ a_{2N}\left(N+1,x,y,z\right)={\mathit{xa}}_{2N}\left(N,x,y,z\right)+2{\mathit{ya}}_{2N-1}\left(N,x,y,z\right)+3{\mathit{za}}_{2N-2}\left(N,x,y,z\right),\\ a_{2N+1}\left(N+1,x,y,z\right)=2{\mathit{ya}}_{2N}\left(N,x,y,z\right)+3{\mathit{za}}_{2N-1}\left(N,x,y,z\right),\\ a_{2N+2}\left(N+1,x,y,z\right)={\left(3z\right)}^{N+1},\end{array}$$

and

$$\begin{array}{l}{a}_i\left(N+1,x,y,z\right)=\left(i+1\right)a_{i+1}\left(N,x,y,z\right)+{\mathit{xa}}_i\left(N,x,y,z\right)\\ +2{\mathit{ya}}_{i-1}\left(N,x,y,z\right)+3{\mathit{za}}_{i-2}\left(N,x,y,z\right),\left(2\le i\le 2N-1\right).\end{array}$$

From (4), we note that

$$F^{\left(N\right)}={\left(\frac{\partial }{\partial t}\right)}^{N}F\left(t,x,y,z\right)=\sum _{k=0}^{\infty }{H}_{k+N}\left(x,y,z\right)\frac{t^k}{k!}.$$E18

By (4) and (18), we get

$$\begin{array}{c}{e}^{-\mathit{nt}}{\left(\frac{\partial }{\partial t}\right)}^{N}F\left(t,x,y,z\right)=\left(\sum _{m=0}^{\infty }{\left(-n\right)}^m\frac{t^m}{m!}\right)\left(\sum _{m=0}^{\infty }{H}_{m+N}\left(x,y,z\right)\frac{t^m}{m!}\right)\\ =\sum _{m=0}^{\infty }\left(\sum _{k=0}^m\left(\begin{array}{l}m\\ k\end{array}\right){\left(-n\right)}^{m-k}{H}_{N+k}(x,y,z)\right)\frac{t^m}{m!}.\end{array}$$E19

By the Leibniz rule and the inverse relation, we have

$$\begin{array}{c}{e}^{-\mathit{nt}}{\left(\frac{\partial }{\partial t}\right)}^{N}F\left(t,x,y,z\right)=\sum _{k=0}^N\left(\begin{array}{l}N\\ k\end{array}\right)n^{N-k}{\left(\frac{\partial }{\partial t}\right)}^k\left(e^{-\mathit{nt}}F\left(t,x,y,z\right)\right)\\ =\sum _{m=0}^{\infty }\left(\sum _{k=0}^N\left(\begin{array}{l}N\\ k\end{array}\right)n^{N-k}{H}_{m+k}\left(x-n,y,z\right)\right)\frac{t^m}{m!}.\end{array}$$E20

Hence, by (19) and (20), and comparing the coefficients of $\frac{t^m}{m!}$ gives the following theorem.

Theorem 6. Let $m,n,N$ be nonnegative integers. Then

$$\sum _{k=0}^m\left(\begin{array}{l}m\\ k\end{array}\right){\left(-n\right)}^{m-k}{H}_{N+k}\left(x,y,z\right)=\sum _{k=0}^N\left(\begin{array}{l}N\\ k\end{array}\right)n^{N-k}{H}_{m+k}\left(x-n,y,z\right).$$E21

If we take $m=0$ in (21), then we have the following corollary.

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

$$H_N\left(x,y,z\right)=\sum _{k=0}^N\left(\begin{array}{l}N\\ k\end{array}\right)n^{N-k}{H}_k\left(x-n,y,z\right).$$

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

$$F^{\left(N\right)}={\left(\frac{\partial }{\partial t}\right)}^{N}F\left(t,x,y,z\right)=\left(\sum _{i=0}^N{a}_i\left(N,x,y,z\right)t^i\right)F\left(t,x,y,z\right)$$

has a solution

$$F=F\left(t,x,y,z\right)=e^{\mathit{xt}+{\mathit{yt}}^2+{\mathit{zt}}^3}.$$

Here is a plot of the surface for this solution. In Figure 1(left), we choose $-2\le z\le 2,$ $-1\le t\le 1,$ $x=2,$ and $y=-4$. In Figure 1(right), we choose $-5\le x\le 5,-1\le t\le 1,y=-3,$ and $z=-1$.

3. Distribution of zeros of the 3-variable Hermite polynomials

Figure 1.

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

This section aims to demonstrate the benefit of using numerical investigation to support theoretical prediction and to discover new interesting pattern of the zeros of the 3-variable Hermite polynomials $H_n\left(x,y,z\right)$. By using computer, the 3-variable Hermite polynomials $H_n\left(x,y,z\right)$ can be determined explicitly. We display the shapes of the 3-variable Hermite polynomials $H_n\left(x,y,z\right)$ and investigate the zeros of the 3-variable Hermite polynomials $H_n\left(x,y,z\right)$. We investigate the beautiful zeros of the 3-variable Hermite polynomials $H_n\left(x,y,z\right)$ by using a computer. We plot the zeros of the $H_n\left(x,y,z\right)$ for $n=20,$ $y=1,-1,$ $1+i,-1-i,$ $z=3,-3,3+i,-3-i$ and $x\in \mathbb{C}$ (Figure 2). In Figure 2(top-left), we choose $n=20$, $y=1$, and $z=3$. In Figure 2(top-right), we choose $n=20$, $y=-1$, and $z=-3$. In Figure 2(bottom-left), we choose $n=20$, $y=1+i$, and $z=3+i$. In Figure 2(bottom-right), we choose $n=20$, $y=-1-i$, and $z=-3-i$.

Figure 2.

Zeros of $H_n\left(x,y,z\right)$.

In Figure 3(top-left), we choose $n=20$, $x=1$, and $y=1$. In Figure 3(top-right), we choose $n=20$, $x=-1$, and $y=-1$. In Figure 3(bottom-left), we choose $n=20$, $x=1+i$, and $y=1+i$. In Figure 3(bottom-right), we choose $n=20$, $x=-1-i$, and $y=-1-i$.

Figure 3.

Zeros of $H_n\left(x,y,z\right)$.

Stacks of zeros of the 3-variable Hermite polynomials $H_n\left(x,y,z\right)$ for $1\le n\le 20$ from a 3-D structure are presented (Figure 3). In Figure 4(top-left), we choose $n=20$, $y=1$, and $z=3$. In Figure 4(top-right), we choose $n=20$, $y=-1$, and $z=-3$. In Figure 4(bottom-left), we choose $n=20$, $y=1+i$, and $z=3+i$. In Figure 4(bottom-right), we choose $n=20$, $y=-1-i$, and $z=-3-i$.

Figure 4.

Stacks of zeros of $H_n\left(x,y,z\right),1\le n\le 20$.

Our numerical results for approximate solutions of real zeros of the 3-variable Hermite polynomials $H_n\left(x,y,z\right)$ are displayed (Tables 1–3).

Degree $n$	Real zeros	Complex zeros
1	1	0
2	0	2
3	1	2
4	2	2
5	1	4
6	2	4
7	3	4
8	2	6
9	3	6
10	4	6
11	3	8
12	4	8
13	3	10
14	4	10

Table 1.

Numbers of real and complex zeros of $H_n\left(x\mathrm{,1,3}\right)$.

Degree $n$	Real zeros	Complex zeros
1	1	0
2	2	0
3	1	2
4	2	2
5	3	2
6	2	4
7	3	4
8	4	4
9	3	6
10	4	6
11	5	6
12	6	6
13	5	8
14	6	8

Table 2.

Numbers of real and complex zeros of $H_n\left(x,-1,-3\right)$.

Degree n	x
1	0
2	—
3	− 1.8845
4	3.1286, −0.17159
5	−4.5385
6	−5.8490, −1.3476
7	−7.1098, −2.1887, −0.36350
8	−8.3241, −3.4645
9	−9.4984,  − 4.6021,  − 1.1118
10	−10.637,  − 5.7212,  − 1.5785, −0.61919
11	−11.745,  − 6.8105,  − 2.8680
12	−12.824,  − 7.8743,  − 3.8894,  − 0.99513

Table 3.

Approximate solutions of $H_n\left(x,1,3\right)=0,x\in \mathrm{R}$.

The plot of real zeros of the 3-variable Hermite polynomials $H_n\left(x,y,z\right)$ for $1\le n\le 20$ structure are presented (Figure 5).

Figure 5.

Real zeros of $H_n\left(x,y,z\right),1\le n\le 20$.

In Figure 5(left), we choose $y=1$ and $z=3$. In Figure 5(right), we choose $y=-1$ and $z=-3$.

Stacks of zeros of $H_n\left(x,-2,4\right)$ for $1\le n\le 40$, forming a 3D structure are presented (Figure 6). In Figure 6(top-left), we plot stacks of zeros of $H_n\left(x,-2,4\right)$ for $1\le n\le 20$. In Figure 6(top-right), we draw $x$ and $y$ axes but no $z$ axis in three dimensions. In Figure 6(bottom-left), we draw $y$ and $z$ axes but no $x$ axis in three dimensions. In Figure 6(bottom-right), we draw $x$ and $z$ axes but no $y$ axis in three dimensions.

Figure 6.

Stacks of zeros of $H_n\left(x,-2,4\right)\mathit{\text{for}}1\le n\le 20$.

It is expected that $H_n\left(x,y,z\right),x\in \mathbb{C},y,z\in \mathbb{R}$, has $\mathit{Im}\left(x\right)=0$ reflection symmetry analytic complex functions (see Figures 2–7). We observe a remarkable regular structure of the complex roots of the 3-variable Hermite polynomials $H_n\left(x,y,z\right)$ for $y,z\in \mathbb{R}$. We also hope to verify a remarkable regular structure of the complex roots of the 3-variable Hermite polynomials $H_n\left(x,y,z\right)$ for $y,z\in \mathbb{R}$ (Tables 1 and 2). Next, we calculated an approximate solution satisfying $H_n\left(x,y,z\right)=0,x\in \mathbb{C}$. The results are given in Tables 3 and 4.

Figure 7.

Real zeros of $H_n\left(x,y,z\right),1\le n\le 20$.

degree n	x
1	0
2	−1.4142, 1.4142
3	3.3681
4	0.16229, 5.0723
5	−1.3404, 1.4745, 6.6661
6	2.9754, 8.1678
7	0.31213, 4.3783, 9.5946
8	−1.2604, 1.5304, 5.7274, 10.959
9	2.8224, 7.0271, 12.270
10	0.44594, 4.0615, 8.2834, 13.535
11	−1.1740, 1.5825, 5.2667, 9.5013, 14.760
12	−1.4659, −0.87728, 2.7469, 6.4398, 10.685, 15.949

Table 4.

Approximate solutions of $H_n\left(x,-1,-3\right)=0,x\in \mathrm{R}$.

The plot of real zeros of the 3-variable Hermite polynomials $H_n\left(x,y,z\right)$ for $1\le n\le 20$ structure are presented (Figure 7).

In Figure 7(left), we choose $x=1$ and $y=2$. In Figure 7(right), we choose $x=-1$ and $y=-2$.

Finally, we consider the more general problems. How many zeros does $H_n\left(x,y,z\right)$ have? We are not able to decide if $H_n\left(x,y,z\right)=0$ has $n$ distinct solutions. We would also like to know the number of complex zeros $C_{H_n\left(x,y,z\right)}$ of $H_n\left(x,y,z\right),\mathit{Im}\left(x\right)\ne 0.$ Since $n$ is the degree of the polynomial $H_n\left(x,y,z\right)$, the number of real zeros $R_{H_n\left(x,y,z\right)}$ lying on the real line $\mathit{Im}\left(x\right)=0$ is then $R_{H_n\left(x,y,z\right)}=n-C_{H_n\left(x,y,z\right)}$, where $C_{H_n\left(x,y,z\right)}$ denotes complex zeros. See Tables 1 and 2 for tabulated values of $R_{H_n\left(x,y,z\right)}$ and $C_{H_n\left(x,y,z\right)}$. The author has no doubt that investigations along these lines will lead to a new approach employing numerical method in the research field of the 3-variable Hermite polynomials $H_n\left(x,y,z\right)$ which appear in mathematics and physics. The reader may refer to [2, 11, 13, 20] for the details.

Acknowledgments

This work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MEST) (No. 2017R1A2B4006092).