1. Introduction

Many mathematicians have studied in the area of the Bernoulli numbers, Euler numbers, Genocchi numbers, and tangent numbers see [1–15]. 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

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

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

$$H\_n(\mathbf{x}) = (-1)^n e^{\mathbf{x}^2} \frac{d^n}{d\mathbf{x}^n} e^{-\mathbf{x}^2} = \frac{n!}{2\pi i} \oint\_C e^{\frac{e^{2t\mathbf{x}\cdot\mathbf{x}-t^2}}{t^{n+1}}} dt$$

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

$$H\_{n+1}(\mathbf{x}) = 2\mathbf{x}H\_n(\mathbf{x}) - 2nH\_{n-1}(\mathbf{x}) .$$

Further, the two variables Hermite Kampé de Fériet polynomials Hnð Þ x; y defined by the generating function (see [3])

$$\sum\_{n=0}^{\infty} H\_n(x, y) \frac{t^n}{n!} = e^{xt + y t^2} \tag{1}$$

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

l¼0

t n n! ¼ e

Hn�3kð Þ <sup>x</sup>; <sup>y</sup>; <sup>z</sup><sup>1</sup> zk

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

Xn 3½ �

k¼0

∂2

∂3

The following elementary properties of the 3-variable Hermite polynomials Hnð Þ x; y; z are

<sup>∂</sup>x<sup>2</sup> Hnð Þ <sup>x</sup>; <sup>y</sup>; <sup>z</sup> ,

<sup>∂</sup>x<sup>3</sup> Hnð Þ <sup>x</sup>; <sup>y</sup>; <sup>z</sup> :

<sup>k</sup>!ð Þ <sup>n</sup> � <sup>3</sup><sup>k</sup> ! :

n l

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

� �Hlð Þ <sup>x</sup>1; <sup>y</sup>; <sup>z</sup> xn�<sup>l</sup>

xtþyt2þð Þ <sup>z</sup>1þz<sup>2</sup> <sup>t</sup>

Hlð Þ x; y; z<sup>1</sup>

1 A t n n! :

<sup>2</sup>n!

Hn�3kð Þ <sup>x</sup>; <sup>y</sup>; <sup>z</sup><sup>1</sup> <sup>z</sup><sup>k</sup>

<sup>k</sup>!ð Þ <sup>n</sup> � <sup>3</sup><sup>k</sup> ! :

2

<sup>2</sup> :

http://dx.doi.org/10.5772/intechopen.74355

83

3

t l l!

Hnð Þ¼ <sup>x</sup><sup>1</sup> <sup>þ</sup> <sup>x</sup>2; <sup>y</sup>; <sup>z</sup> <sup>X</sup><sup>n</sup>

Hnð Þ x; y; z<sup>1</sup> þ z<sup>2</sup>

Xn 3½ �

0 @

Hnð Þ¼ x; y; z<sup>1</sup> þ z<sup>2</sup> n!

∂ ∂y

> ∂ ∂z

Hn�2<sup>k</sup> <sup>x</sup>;y<sup>1</sup> ð Þ;<sup>z</sup> yk

� �Hlð Þ<sup>x</sup> Hn�<sup>l</sup>ð Þ �x; <sup>y</sup> <sup>þ</sup> <sup>1</sup>; <sup>z</sup> :

Also, the 3-variable Hermite polynomials Hnð Þ x; y; z satisfy the following relations

Hnð Þ¼ x; y; z

Hnð Þ¼ x; y; z

readily derived form (2). We, therefore, choose to omit the details involved.

2 <sup>k</sup>!ð Þ <sup>n</sup>�2<sup>k</sup> ! :

k¼0

<sup>¼</sup> <sup>X</sup><sup>∞</sup> k¼0 zn 2 t 3k k! X∞ l¼0

X∞ n¼0

> <sup>¼</sup> <sup>X</sup><sup>∞</sup> n¼0

Theorem 1. For any positive integer n, we have

Theorem 2. For any positive integer n, we have

Theorem 3. For any positive integer n, we have

n l

P n 2½ � k¼0

1 Hnð Þ¼ 2x; �1; 0 Hnð Þx :

<sup>2</sup> Hn <sup>x</sup>; <sup>y</sup><sup>1</sup> <sup>þ</sup> <sup>y</sup>2; <sup>z</sup> � � <sup>¼</sup> <sup>n</sup>!

l¼0

<sup>3</sup> Hnð Þ¼ <sup>x</sup>; <sup>y</sup>; <sup>z</sup> <sup>P</sup><sup>n</sup>

Applying Eq. (2), we obtain

theorem.

and

are the solution of heat equation

$$\frac{\partial}{\partial y}H\_n(\mathbf{x},y) = \frac{\partial^2}{\partial \mathbf{x}^2}H\_n(\mathbf{x},y), \quad H\_n(\mathbf{x},0) = \mathbf{x}^n.$$

We note that

$$H\_n(\mathfrak{A}\mathfrak{x}, -1) = H\_n(\mathfrak{x}).$$

The 3-variable Hermite polynomials Hnð Þ x; y; z are introduced [4].

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

The differential equation and he generating function for Hnð Þ x; y; z are given by

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

and

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

respectively.

By (2), we get

$$\sum\_{n=0}^{\infty} H\_n(\mathbf{x}\_1 + \mathbf{x}\_2, \mathbf{y}, z) \frac{t^n}{n!} = e^{(\mathbf{x}\_1 + \mathbf{x}\_2)t + yt^2 + zt^3}$$

$$= \sum\_{n=0}^{\infty} \mathbf{x}\_2^n \frac{t^n}{n!} \sum\_{n=0}^{\infty} H\_n(\mathbf{x}\_1, \mathbf{y}, z) \frac{t^n}{n!} \tag{3}$$

$$= \sum\_{n=0}^{\infty} \left( \sum\_{l=0}^n \binom{n}{l} H\_l(\mathbf{x}\_1, \mathbf{y}, z) \mathbf{x}\_2^{n-l} \right) \frac{t^n}{n!} .$$

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

Theorem 1. For any positive integer n, we have

$$H\_n(\mathbf{x}\_1 + \mathbf{x}\_2, y, z) = \sum\_{l=0}^n \binom{n}{l} H\_l(\mathbf{x}\_1, y, z) \mathbf{x}\_2^{n-l}.$$

Applying Eq. (2), we obtain

Hnð Þ¼ � <sup>x</sup> ð Þ<sup>1</sup> <sup>n</sup>

recurrence relation

We note that

and

respectively. By (2), we get

generating function (see [3])

82 Differential Equations - Theory and Current Research

are the solution of heat equation

∂ ∂y e <sup>x</sup><sup>2</sup> dn dxn <sup>e</sup> �x<sup>2</sup> <sup>¼</sup> <sup>n</sup>! 2πi ∮ C e<sup>2</sup>tx�<sup>t</sup> 2

X∞ n¼0

Hnð Þ¼ x; y

The 3-variable Hermite polynomials Hnð Þ x; y; z are introduced [4].

3z ∂3 <sup>∂</sup>x<sup>3</sup> <sup>þ</sup> <sup>2</sup><sup>y</sup>

> e xtþyt2þzt<sup>3</sup>

X∞ n¼0

> <sup>¼</sup> <sup>X</sup><sup>∞</sup> n¼0

Hnð Þ¼ x; y; z n!

The differential equation and he generating function for Hnð Þ x; y; z are given by

∂2 <sup>∂</sup>x<sup>2</sup> <sup>þ</sup> <sup>x</sup>

� �

Hnð Þ x<sup>1</sup> þ x2; y; z

Xn l¼0

<sup>¼</sup> <sup>X</sup><sup>∞</sup> n¼0 xn 2 t n n! X∞ n¼0

n

!

l

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

Hnþ<sup>1</sup>ð Þ¼ x 2xHnð Þ� x 2nHn�<sup>1</sup>ð Þx :

Further, the two variables Hermite Kampé de Fériet polynomials Hnð Þ x; y defined by the

t n n! ¼ e xtþyt<sup>2</sup>

Hnð Þ¼ 2x; �1 Hnð Þx :

Xn 3½ �

zk

Hn�3kð Þ x; y <sup>k</sup>!ð Þ <sup>n</sup> � <sup>3</sup><sup>k</sup> ! :

Hnð Þ¼ x; y; z 0

, (2)

t n n!

ð Þ <sup>x</sup>1þx<sup>2</sup> <sup>t</sup>þyt2þzt<sup>3</sup>

t n n!

t n n! :

Hnð Þ x1; y; z

2

Hlð Þ <sup>x</sup>1; <sup>y</sup>; <sup>z</sup> xn�<sup>l</sup>

k¼0

∂ <sup>∂</sup><sup>x</sup> � <sup>n</sup>

<sup>¼</sup> <sup>X</sup><sup>∞</sup> n¼0

> t n n! ¼ e

!

Hnð Þ x; y; z

<sup>∂</sup>x<sup>2</sup> Hnð Þ <sup>x</sup>; <sup>y</sup> , Hnð Þ¼ <sup>x</sup>; <sup>0</sup> xn:

Hnð Þ x; y

∂2

t <sup>n</sup>þ<sup>1</sup> dt

(1)

(3)

$$\sum\_{n=0}^{\infty} H\_n(\mathbf{x}, y, z\_1 + z\_2) \frac{t^n}{n!} = e^{\mathbf{x}t + yt^2 + (z\_1 + z\_2)t^3}$$

$$= \sum\_{k=0}^{\infty} z\_2^n \frac{t^{3k}}{k!} \sum\_{l=0}^{\infty} H\_l(\mathbf{x}, y, z\_1) \frac{t^l}{l!}$$

$$= \sum\_{n=0}^{\infty} \left( \sum\_{k=0}^{\left[\frac{n}{3}\right]} \frac{H\_{n-3k}(\mathbf{x}, y, z\_1) z\_2^k n!}{k!(n-3k)!} \cdot \right) \frac{t^n}{n!} \cdot 1$$

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

$$H\_n(\mathbf{x}, y, z\_1 + z\_2) = n! \sum\_{k=0}^{\left[\frac{n}{3}\right]} \frac{H\_{n-3k}(\mathbf{x}, y, z\_1) z\_2^k}{k!(n-3k)!}.$$

Also, the 3-variable Hermite polynomials Hnð Þ x; y; z satisfy the following relations

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

and

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

The following elementary properties of the 3-variable Hermite polynomials Hnð Þ x; y; z are readily derived form (2). We, therefore, choose to omit the details involved.

Theorem 3. For any positive integer n, we have

$$\mathbf{1} \quad H\_n(2\mathbf{x}, -1, 0) = H\_n(\mathbf{x}) .$$

$$\mathbf{2} \quad H\_n\left(\mathbf{x}, y\_1 + y\_2, z\right) = n! \sum\_{k=0}^{\left[\frac{n}{2}\right]} \frac{H\_{n-2k}\left(\mathbf{x}, y\_1, z\right) y\_2^k}{k!(n-2k)!} \dots$$

$$\mathbf{3} \quad H\_n(\mathbf{x}, y, z) = \sum\_{l=0}^n \binom{n}{l} H\_l(\mathbf{x}) H\_{n-l}(-\mathbf{x}, y+1, z) \,.$$

Theorem 4. For any positive integer n, we have

$$\mathbf{1} \quad H\_n(\mathbf{x}\_1 + \mathbf{x}\_2, y\_1 + y\_2, z) = \sum\_{l=0}^n \binom{n}{l} H\_l(\mathbf{x}\_1, y\_1, z) H\_{n-l}(\mathbf{x}\_2, y\_2).$$

$$\mathbf{2} \quad H\_n(\mathbf{x}\_1 + \mathbf{x}\_2, y\_1 + y\_2, z\_1 + z\_2) = \sum\_{l=0}^n \binom{n}{l} H\_l(\mathbf{x}\_1, y\_1, z) H\_{n-l}(\mathbf{x}\_2, y\_2, z\_2).$$

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

Then, by (4), we have

<sup>F</sup>ð Þ<sup>2</sup> <sup>¼</sup> <sup>∂</sup> ∂t

<sup>F</sup>ð Þ<sup>1</sup> <sup>¼</sup> <sup>∂</sup> ∂t

Continuing this process, we can guess that

∂t � �<sup>N</sup>

Differentiating (7) with respect to t, we have

<sup>∂</sup><sup>t</sup> <sup>¼</sup> <sup>X</sup> 2N

aið Þ <sup>N</sup>; <sup>x</sup>; <sup>y</sup>; <sup>z</sup> it<sup>i</sup>�<sup>1</sup>

þ X 2N

¼ 2 X N�1

> þ 2 X Nþ1

Hence we have

i¼0 2yai

i¼0

i¼1

<sup>F</sup>ð Þ <sup>N</sup>þ<sup>1</sup> <sup>¼</sup>

i¼0

<sup>¼</sup> <sup>X</sup> 2N

i¼0

<sup>F</sup>ð Þ <sup>N</sup> <sup>¼</sup> <sup>∂</sup>

<sup>F</sup>ð Þ <sup>N</sup>þ<sup>1</sup> <sup>¼</sup> <sup>∂</sup>Fð Þ <sup>N</sup>

<sup>¼</sup> <sup>X</sup> 2N

i¼0

F tð Þ¼ ; x; y; z

<sup>¼</sup> <sup>x</sup><sup>2</sup> <sup>þ</sup> <sup>2</sup><sup>y</sup> � � <sup>þ</sup> ð Þ <sup>6</sup><sup>z</sup> <sup>þ</sup> <sup>4</sup>xy <sup>t</sup> <sup>þ</sup> <sup>4</sup>y<sup>2</sup> <sup>þ</sup> <sup>6</sup>xz � �<sup>t</sup>

F tð Þ¼ ; <sup>x</sup>; <sup>y</sup>; <sup>z</sup> <sup>X</sup>

aið Þ <sup>N</sup>; <sup>x</sup>; <sup>y</sup>; <sup>z</sup> it<sup>i</sup>�<sup>1</sup>

iaið Þ N; x; y; z t

ð Þ N; x; y; z t

ð Þ i þ 1 aiþ<sup>1</sup>ð Þ N; x; y; z t

<sup>2</sup>yai�<sup>1</sup>ð Þ <sup>N</sup>; <sup>x</sup>; <sup>y</sup>; <sup>z</sup> <sup>t</sup>

2 X N�1

i¼0

þ X 2N

> þ 2 X Nþ1

þ 2 X Nþ2

i¼0

i¼1

i¼2

F tð Þþ ; <sup>x</sup>; <sup>y</sup>; <sup>z</sup> <sup>X</sup>

2N

i¼0

i�1

i

F tð Þþ ; x; y; z

ð Þ i þ 1 aiþ<sup>1</sup>ð Þ N; x; y; z t

xaið Þ N; x; y; z t

<sup>2</sup>yai�<sup>1</sup>ð Þ <sup>N</sup>; <sup>x</sup>; <sup>y</sup>; <sup>z</sup> <sup>t</sup>

3zai�<sup>2</sup>ð Þ N; x; y; z t

iþ1

i

2N

i¼0

∂ ∂t e

xtþyt2þzt<sup>3</sup> � � <sup>¼</sup> <sup>e</sup>

<sup>F</sup>ð Þ<sup>1</sup> ð Þ¼ <sup>t</sup>; <sup>x</sup>; <sup>y</sup>; <sup>z</sup> ð Þ <sup>2</sup><sup>y</sup> <sup>þ</sup> <sup>6</sup>zt F tð Þþ ; <sup>x</sup>; <sup>y</sup>; <sup>z</sup> <sup>x</sup> <sup>þ</sup> <sup>2</sup>yt <sup>þ</sup> <sup>3</sup>zt<sup>2</sup> � �Fð Þ<sup>1</sup> ð Þ <sup>t</sup>; <sup>x</sup>; <sup>y</sup>; <sup>z</sup>

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

<sup>4</sup> � �F tð Þ ; <sup>x</sup>; <sup>y</sup>; <sup>z</sup> :

aið Þ N; x; y; z t

xtþyt2þzt<sup>3</sup>

<sup>2</sup> <sup>þ</sup> ð Þ <sup>12</sup>yz <sup>t</sup>

i

2N

i¼0

2N

i¼0

2N

i¼0

2N

i¼0

F tð Þþ ; <sup>x</sup>; <sup>y</sup>; <sup>z</sup> <sup>X</sup>

aið Þ N; x; y; z t

F tð Þþ ; <sup>x</sup>; <sup>y</sup>; <sup>z</sup> <sup>X</sup>

F tð Þþ ; <sup>x</sup>; <sup>y</sup>; <sup>z</sup> <sup>X</sup>

2 X Nþ2

i¼2

i

i

F tð Þ ; x; y; z

F tð Þ ; x; y; z

i

i

F tð Þ ; x; y; z

F tð Þ ; x; y; z :

F tð Þþ ; <sup>x</sup>; <sup>y</sup>; <sup>z</sup> <sup>X</sup>

<sup>¼</sup> <sup>x</sup> <sup>þ</sup> <sup>2</sup>yt <sup>þ</sup> <sup>3</sup>zt<sup>2</sup> � �F tð Þ ; <sup>x</sup>; <sup>y</sup>; <sup>z</sup> ,

<sup>3</sup> <sup>þ</sup> <sup>9</sup>z<sup>2</sup> � �<sup>t</sup>

aið Þ N; x; y; z t

xaið Þ N; x; y; z t

xaið Þ N; x; y; z t

3zai�<sup>2</sup>ð Þ N; x; y; z t

3zaið Þ N; x; y; z t

<sup>x</sup> <sup>þ</sup> <sup>2</sup>yt <sup>þ</sup> <sup>3</sup>zt<sup>2</sup> � �

http://dx.doi.org/10.5772/intechopen.74355

F tð Þ ; x; y; z , Nð Þ ¼ 0; 1; 2;… : (7)

i

<sup>i</sup> <sup>x</sup> <sup>þ</sup> <sup>2</sup>yt <sup>þ</sup> <sup>3</sup>zt<sup>2</sup> � �F tð Þ ; <sup>x</sup>; <sup>y</sup>; <sup>z</sup>

i

iþ2

i

i

<sup>F</sup>ð Þ<sup>1</sup> ð Þ <sup>t</sup>; <sup>x</sup>; <sup>y</sup>; <sup>z</sup>

F tð Þ ; x; y; z

F tð Þ ; x; y; z

F tð Þ ; x; y; z

F tð Þ ; x; y; z

(8)

(5)

85

(6)

H0ð Þ¼ x; y; z 1, H1ð Þ¼ x; y; z x, <sup>H</sup>2ð Þ¼ <sup>x</sup>; <sup>y</sup>; <sup>z</sup> <sup>x</sup><sup>2</sup> <sup>þ</sup> <sup>2</sup>y, <sup>H</sup>3ð Þ¼ <sup>x</sup>; <sup>y</sup>; <sup>z</sup> <sup>x</sup><sup>3</sup> <sup>þ</sup> <sup>6</sup>xy <sup>þ</sup> <sup>6</sup>z, <sup>H</sup>4ð Þ¼ <sup>x</sup>; <sup>y</sup>; <sup>z</sup> <sup>x</sup><sup>4</sup> <sup>þ</sup> <sup>12</sup>x<sup>2</sup><sup>y</sup> <sup>þ</sup> <sup>12</sup>y<sup>2</sup> <sup>þ</sup> <sup>24</sup>xz, <sup>H</sup>5ð Þ¼ <sup>x</sup>; <sup>y</sup>; <sup>z</sup> <sup>x</sup><sup>5</sup> <sup>þ</sup> <sup>20</sup>x<sup>3</sup><sup>y</sup> <sup>þ</sup> <sup>60</sup>xy<sup>2</sup> <sup>þ</sup> <sup>60</sup>x<sup>2</sup><sup>z</sup> <sup>þ</sup> <sup>120</sup>yz, <sup>H</sup>6ð Þ¼ <sup>x</sup>; <sup>y</sup>; <sup>z</sup> <sup>x</sup><sup>6</sup> <sup>þ</sup> <sup>30</sup>x<sup>4</sup><sup>y</sup> <sup>þ</sup> <sup>180</sup>x<sup>2</sup>y<sup>2</sup> <sup>þ</sup> <sup>120</sup>y<sup>3</sup> <sup>þ</sup> <sup>120</sup>x<sup>3</sup><sup>z</sup> <sup>þ</sup> <sup>720</sup>xyz <sup>þ</sup> <sup>360</sup>z<sup>2</sup>, <sup>H</sup>7ð Þ¼ <sup>x</sup>; <sup>y</sup>; <sup>z</sup> <sup>x</sup><sup>7</sup> <sup>þ</sup> <sup>42</sup>x<sup>5</sup><sup>y</sup> <sup>þ</sup> <sup>420</sup>x<sup>3</sup>y<sup>2</sup> <sup>þ</sup> <sup>840</sup>xy<sup>3</sup> <sup>þ</sup> <sup>210</sup>x<sup>4</sup><sup>z</sup> <sup>þ</sup> <sup>2520</sup>x<sup>2</sup>yz <sup>þ</sup> <sup>2520</sup>y<sup>2</sup><sup>z</sup> <sup>þ</sup> <sup>2520</sup>xz<sup>2</sup>, <sup>H</sup>8ð Þ¼ <sup>x</sup>; <sup>y</sup>; <sup>z</sup> <sup>x</sup><sup>8</sup> <sup>þ</sup> <sup>56</sup>x<sup>6</sup><sup>y</sup> <sup>þ</sup> <sup>840</sup>x<sup>4</sup>y<sup>2</sup> <sup>þ</sup> <sup>3360</sup>x<sup>2</sup>y<sup>3</sup> <sup>þ</sup> <sup>1680</sup>y<sup>4</sup> <sup>þ</sup> <sup>336</sup>x<sup>5</sup><sup>z</sup> <sup>þ</sup> <sup>6720</sup>x<sup>3</sup>yz <sup>þ</sup> <sup>20160</sup>xy<sup>2</sup><sup>z</sup> <sup>þ</sup> <sup>10080</sup>x<sup>2</sup>z<sup>2</sup> <sup>þ</sup> <sup>20160</sup>yz<sup>2</sup>: <sup>H</sup>9ð Þ¼ <sup>x</sup>; <sup>y</sup>; <sup>z</sup> <sup>x</sup><sup>9</sup> <sup>þ</sup> <sup>72</sup>x<sup>7</sup><sup>y</sup> <sup>þ</sup> <sup>1512</sup>x<sup>5</sup>y<sup>2</sup> <sup>þ</sup> <sup>10080</sup>x<sup>3</sup>y<sup>3</sup> <sup>þ</sup> <sup>15120</sup>xy<sup>4</sup> <sup>þ</sup> <sup>504</sup>x<sup>6</sup><sup>z</sup> <sup>þ</sup> <sup>15120</sup>x<sup>4</sup>yz <sup>þ</sup> <sup>90720</sup>x<sup>2</sup>y<sup>2</sup><sup>z</sup> <sup>þ</sup> <sup>60480</sup>y<sup>3</sup><sup>z</sup> <sup>þ</sup> <sup>30240</sup>x<sup>3</sup>z<sup>2</sup> <sup>þ</sup> <sup>181440</sup>xyz<sup>2</sup> <sup>þ</sup> <sup>60480</sup>z<sup>3</sup>, <sup>H</sup>10ð Þ¼ <sup>x</sup>; <sup>y</sup>; <sup>z</sup> <sup>x</sup><sup>10</sup> <sup>þ</sup> <sup>90</sup>x<sup>8</sup><sup>y</sup> <sup>þ</sup> <sup>2520</sup>x<sup>6</sup>y<sup>2</sup> <sup>þ</sup> <sup>25200</sup>x<sup>4</sup>y<sup>3</sup> <sup>þ</sup> <sup>75600</sup>x<sup>2</sup>y<sup>4</sup> <sup>þ</sup> <sup>30240</sup>y<sup>5</sup> <sup>þ</sup> <sup>720</sup>x<sup>7</sup><sup>z</sup> <sup>þ</sup> <sup>30240</sup>x<sup>5</sup>yz <sup>þ</sup> <sup>302400</sup>x<sup>3</sup>y<sup>2</sup><sup>z</sup> <sup>þ</sup> <sup>604800</sup>xy<sup>3</sup><sup>z</sup> <sup>þ</sup> <sup>75600</sup>x<sup>4</sup>z<sup>2</sup> <sup>þ</sup> <sup>907200</sup>x<sup>2</sup>yz<sup>2</sup> <sup>þ</sup> <sup>907200</sup>y<sup>2</sup>z<sup>2</sup> <sup>þ</sup> <sup>604800</sup>xz<sup>3</sup>:

Recently, many mathematicians have studied the differential equations arising from the generating functions of special polynomials (see [7, 8, 12, 16–19]). 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.
