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

and

From (4), we note that

By (4) and (18), we get

e �nt <sup>∂</sup> ∂t � �<sup>N</sup>

88 Differential Equations - Theory and Current Research

e �nt <sup>∂</sup> ∂t � �<sup>N</sup>

Xm k¼0

m k � �

Corollary 7. For N ¼ 0; 1; 2, …, we have

For N ¼ 0; 1; 2, …, the differential equation

has a solution

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

∂t � �<sup>N</sup>

aið Þ¼ N þ 1; x; y; z ð Þ i þ 1 aiþ<sup>1</sup>ð Þþ N; x; y; z xaið Þ N; x; y; z

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

ð Þ �<sup>n</sup> <sup>m</sup> <sup>t</sup> m m! ! <sup>X</sup><sup>∞</sup>

m

!

k

N

!

k

N

!

k

N

N k � �

k¼0

n<sup>N</sup>�<sup>k</sup>

N

i¼0

m¼0

N

k¼0

X N

k¼0

HNþ<sup>k</sup>ð Þ¼ <sup>x</sup>; <sup>y</sup>; <sup>z</sup> <sup>X</sup>

N

N k � �

k¼0

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

Xm k¼0

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

By the Leibniz rule and the inverse relation, we have

Hence, by (19) and (20), and comparing the coefficients of <sup>t</sup>

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

ð Þ �<sup>n</sup> <sup>m</sup>�<sup>k</sup>

If we take m ¼ 0 in (21), then we have the following corollary.

HNð Þ¼ <sup>x</sup>; <sup>y</sup>; <sup>z</sup> <sup>X</sup>

∂t � �<sup>N</sup>

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

<sup>¼</sup> <sup>X</sup><sup>∞</sup> m¼0

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

<sup>¼</sup> <sup>X</sup><sup>∞</sup> m¼0

<sup>þ</sup> <sup>2</sup>yai�<sup>1</sup>ð Þþ <sup>N</sup>; <sup>x</sup>; <sup>y</sup>; <sup>z</sup> <sup>3</sup>zai�<sup>2</sup>ð Þ <sup>N</sup>; <sup>x</sup>; <sup>y</sup>; <sup>z</sup> ,ð Þ <sup>2</sup> <sup>≤</sup> <sup>i</sup> <sup>≤</sup> <sup>2</sup><sup>N</sup> � <sup>1</sup> :

k¼0

Hkþ<sup>N</sup>ð Þ x; y; z

m¼0

ð Þ �<sup>n</sup> <sup>m</sup>�<sup>k</sup>

nN�<sup>k</sup> <sup>∂</sup> ∂t � �<sup>k</sup>

!

n<sup>N</sup>�<sup>k</sup>

!

t k k!

Hmþ<sup>N</sup>ð Þ x; y; z

HNþ<sup>k</sup>ðx; y; zÞ

�ntF tð Þ ; <sup>x</sup>; <sup>y</sup>; <sup>z</sup> � �

e

Hmþ<sup>k</sup>ð Þ x � n; y; z

m

nN�<sup>k</sup>

Hkð Þ x � n; y; z :

i

F tð Þ ; x; y; z

aið Þ N; x; y; z t

!

!

t m m!

> t m m! :

t m m! :

<sup>m</sup>! gives the following theorem.

Hmþ<sup>k</sup>ð Þ x � n; y; z : (21)

: (18)

(19)

(20)

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 Hnð Þ x; y; z . By using computer, the 3-variable Hermite polynomials Hnð Þ x; y; z can be determined explicitly. We display the shapes of the 3-variable Hermite polynomials Hnð Þ x; y; z and investigate the zeros of the 3-variable Hermite polynomials Hnð Þ x; y; z . We investigate the beautiful zeros of the 3-variable Hermite polynomials Hnð Þ x; y; z by using a computer. We plot the zeros of the Hnð Þ x; y; z for n ¼ 20, y ¼ 1, � 1, 1 þ i, � 1 � i, z ¼ 3, � 3, 3 þ i, � 3 � i and x∈ C (Figure 2). In Figure 2(top-left), we choose n ¼ 20, y ¼ 1, and z ¼ 3. In Figure 2(top-right), we choose n ¼ 20, y ¼ �1, and z ¼ �3. In Figure 2(bottomleft), we choose n ¼ 20, y ¼ 1 þ i, and z ¼ 3 þ i. In Figure 2(bottom-right), we choose n ¼ 20, y ¼ �1 � i, and z ¼ �3 � i.

In Figure 3(top-left), we choose n ¼ 20, x ¼ 1, and y ¼ 1. In Figure 3(top-right), we choose n ¼ 20, x ¼ �1, and y ¼ �1. In Figure 3(bottom-left), we choose n ¼ 20, x ¼ 1 þ i, and y ¼ 1 þ i. In Figure 3(bottom-right), we choose n ¼ 20, x ¼ �1 � i, and y ¼ �1 � i.

Stacks of zeros of the 3-variable Hermite polynomials Hnð Þ x; y; z for 1 ≤ n ≤ 20 from a 3-D structure are presented (Figure 3). In Figure 4(top-left), we choose n ¼ 20, y ¼ 1, and z ¼ 3. In Figure 4 (top-right), we choose n ¼ 20, y ¼ �1, and z ¼ �3. In Figure 4(bottom-left), we choose n ¼ 20, y ¼ 1 þ i, and z ¼ 3 þ i. In Figure 4(bottom-right), we choose n ¼ 20, y ¼ �1 � i, and z ¼ �3 � i.

Figure 1. The surface for the solution F tð Þ ; x; y; z .

Figure 2. Zeros of Hnð Þ x; y; z .

Our numerical results for approximate solutions of real zeros of the 3-variable Hermite polynomials Hnð Þ x; y; z are displayed (Tables 1–3).

and z axes but no x axis in three dimensions. In Figure 6(bottom-right), we draw x and z axes

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

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It is expected that Hnð Þ x; y; z , x ∈ C, y, z∈ R, has Imð Þ¼ x 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 Hnð Þ x; y; z for y, z ∈ R. We also hope to verify a remarkable regular structure of the complex roots of the 3-variable Hermite polynomials Hnð Þ x; y; z for y, z∈ R (Tables 1 and 2). Next, we calculated an approximate solution satisfying

but no y axis in three dimensions.

Figure 3. Zeros of Hnð Þ x; y; z .

Hnð Þ¼ x; y; z 0, x∈ C. The results are given in Tables 3 and 4.

The plot of real zeros of the 3-variable Hermite polynomials Hnð Þ x; y; z for 1 ≤ n ≤ 20 structure are presented (Figure 5).

In Figure 5(left), we choose y ¼ 1 and z ¼ 3. In Figure 5(right), we choose y ¼ �1 and z ¼ �3.

Stacks of zeros of Hnð Þ x; �2; 4 for 1 ≤ n ≤ 40, forming a 3D structure are presented (Figure 6). In Figure 6(top-left), we plot stacks of zeros of Hnð Þ x; �2; 4 for 1 ≤ n ≤ 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 Differential Equations Arising from the 3-Variable Hermite Polynomials and Computation of Their Zeros http://dx.doi.org/10.5772/intechopen.74355 91

Figure 3. Zeros of Hnð Þ x; y; z .

Our numerical results for approximate solutions of real zeros of the 3-variable Hermite poly-

The plot of real zeros of the 3-variable Hermite polynomials Hnð Þ x; y; z for 1 ≤ n ≤ 20 structure

In Figure 5(left), we choose y ¼ 1 and z ¼ 3. In Figure 5(right), we choose y ¼ �1 and z ¼ �3. Stacks of zeros of Hnð Þ x; �2; 4 for 1 ≤ n ≤ 40, forming a 3D structure are presented (Figure 6). In Figure 6(top-left), we plot stacks of zeros of Hnð Þ x; �2; 4 for 1 ≤ n ≤ 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

nomials Hnð Þ x; y; z are displayed (Tables 1–3).

90 Differential Equations - Theory and Current Research

are presented (Figure 5).

Figure 2. Zeros of Hnð Þ x; y; z .

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.

It is expected that Hnð Þ x; y; z , x ∈ C, y, z∈ R, has Imð Þ¼ x 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 Hnð Þ x; y; z for y, z ∈ R. We also hope to verify a remarkable regular structure of the complex roots of the 3-variable Hermite polynomials Hnð Þ x; y; z for y, z∈ R (Tables 1 and 2). Next, we calculated an approximate solution satisfying Hnð Þ¼ x; y; z 0, x∈ C. The results are given in Tables 3 and 4.

Degree n Real zeros Complex zeros

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Degree n Real zeros Complex zeros

11 0 22 0 31 2 42 2 53 2 62 4 73 4 84 4 93 6 10 4 6 11 5 6 12 6 6 13 5 8 14 6 8

11 0 20 2 31 2 42 2 51 4 62 4 73 4 82 6 93 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 Hnð Þ x; 1; 3 .

Table 2. Numbers of real and complex zeros of Hnð Þ x; �1; �3 .

Figure 4. Stacks of zeros of Hnð Þ x; y; z , 1 ≤ n ≤ 20.

The plot of real zeros of the 3-variable Hermite polynomials Hnð Þ x; y; z for 1 ≤ n ≤ 20 structure are presented (Figure 7).

In Figure 7(left), we choose x ¼ 1 and y ¼ 2. In Figure 7(right), we choose x ¼ �1 and y ¼ �2.

Finally, we consider the more general problems. How many zeros does Hnð Þ x; y; z have? We are not able to decide if Hnð Þ¼ x; y; z 0 has n distinct solutions. We would also like to know the number of complex zeros CHnð Þ <sup>x</sup>;y;<sup>z</sup> of Hnð Þ x; y; z , Imð Þx 6¼ 0: Since n is the degree of the polynomial Hnð Þ x; y; z , the number of real zeros RHnð Þ <sup>x</sup>;y;<sup>z</sup> lying on the real line Imð Þ¼ x 0 is then RHnð Þ <sup>x</sup>;y;<sup>z</sup> ¼ n � CHnð Þ <sup>x</sup>;y;<sup>z</sup> , where CHnð Þ <sup>x</sup>;y;<sup>z</sup> denotes complex zeros. See Tables 1 and 2 for Differential Equations Arising from the 3-Variable Hermite Polynomials and Computation of Their Zeros http://dx.doi.org/10.5772/intechopen.74355 93


Table 1. Numbers of real and complex zeros of Hnð Þ x; 1; 3 .


Table 2. Numbers of real and complex zeros of Hnð Þ x; �1; �3 .

The plot of real zeros of the 3-variable Hermite polynomials Hnð Þ x; y; z for 1 ≤ n ≤ 20 structure

In Figure 7(left), we choose x ¼ 1 and y ¼ 2. In Figure 7(right), we choose x ¼ �1 and y ¼ �2. Finally, we consider the more general problems. How many zeros does Hnð Þ x; y; z have? We are not able to decide if Hnð Þ¼ x; y; z 0 has n distinct solutions. We would also like to know the number of complex zeros CHnð Þ <sup>x</sup>;y;<sup>z</sup> of Hnð Þ x; y; z , Imð Þx 6¼ 0: Since n is the degree of the polynomial Hnð Þ x; y; z , the number of real zeros RHnð Þ <sup>x</sup>;y;<sup>z</sup> lying on the real line Imð Þ¼ x 0 is then RHnð Þ <sup>x</sup>;y;<sup>z</sup> ¼ n � CHnð Þ <sup>x</sup>;y;<sup>z</sup> , where CHnð Þ <sup>x</sup>;y;<sup>z</sup> denotes complex zeros. See Tables 1 and 2 for

are presented (Figure 7).

Figure 4. Stacks of zeros of Hnð Þ x; y; z , 1 ≤ n ≤ 20.

92 Differential Equations - Theory and Current Research


Table 3. Approximate solutions of Hnð Þ¼ x; 1; 3 0, x∈R.

Figure 6. Stacks of zeros of Hnð Þ x; �2; 4 for 1 ≤ n ≤ 20.

degree n x 1 0

3.3681

�1.4142, 1.4142

0.16229, 5.0723

2.9754, 8.1678

�1.3404, 1.4745, 6.6661

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

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0.31213, 4.3783, 9.5946

Figure 5. Real zeros of Hnð Þ x; y; z , 1 ≤ n ≤ 20.

tabulated values of RHnð Þ <sup>x</sup>;y;<sup>z</sup> and CHnð Þ <sup>x</sup>;y;<sup>z</sup> . 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 Hnð Þ x; y; z which appear in mathematics and physics. The reader may refer to [2, 11, 13, 20] for the details.

Differential Equations Arising from the 3-Variable Hermite Polynomials and Computation of Their Zeros http://dx.doi.org/10.5772/intechopen.74355 

Figure 6. Stacks of zeros of Hnð Þ x; �2; 4 for 1 ≤ n ≤ 20.

tabulated values of RHnð Þ <sup>x</sup>;y;<sup>z</sup> and CHnð Þ <sup>x</sup>;y;<sup>z</sup> . 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 Hnð Þ x; y; z which appear in mathematics and physics. The

reader may refer to [2, 11, 13, 20] for the details.

Figure 5. Real zeros of Hnð Þ x; y; z , 1 ≤ n ≤ 20.

Degree n x 0 — � 1.8845

Differential Equations - Theory and Current Research

3.1286, �0.17159

�5.8490, �1.3476

�8.3241, �3.4645

�7.1098, �2.1887, �0.36350

�9.4984, � 4.6021, � 1.1118

�11.745, � 6.8105, � 2.8680

�10.637, � 5.7212, � 1.5785, �0.61919

�12.824, � 7.8743, � 3.8894, � 0.99513

�4.5385

Table 3. Approximate solutions of Hnð Þ¼ x; 1; 3 0, x∈R.



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Table 4. Approximate solutions of Hnð Þ¼ x; �1; �3 0, x∈R.

Figure 7. Real zeros of Hnð Þ x; y; z , 1 ≤ n ≤ 20.
