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

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.


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 We can also have the generating function by using Cauchy's integral formula to write the Hermite polynomials as H n x ð Þ ¼ À1 ð Þ n e x 2 d n dx n e Àx 2 ¼ n! 2πi ∮ C e 2txÀt 2 t nþ1 dt with the contour encircling the origin. It follows that the Hermite polynomials also satisfy the recurrence relation H nþ1 x ð Þ ¼ 2xH n x ð Þ À 2nH nÀ1 x ð Þ: Further, the two variables Hermite Kampé de Fériet polynomials H n x; y ð Þ defined by the generating function (see [3]) H n x; y ð Þ t n n! ¼ e xtþyt 2 (1) are the solution of heat equation ∂ ∂y H n x; y ð Þ ¼ ∂ 2 ∂x 2 H n x; y ð Þ, H n x; 0 ð Þ ¼ x n : We note that H n 2x; À1 ð Þ¼H n x ð Þ: The 3-variable Hermite polynomials H n x; y; z ð Þare introduced [4].
H n x; y; z ð Þ¼n! X n 3 ½ k¼0 z k H nÀ3k x; y ð Þ k! n À 3k ð Þ ! : The differential equation and he generating function for H n x; y; z ð Þare given by By comparing the coefficients on both sides of (3), we have the following theorem.
Theorem 1. For any positive integer n, we have Applying Eq. (2), we obtain On equating the coefficients of the like power of t in the above, we obtain the following theorem.

Differential equations associated with the 3-variable Hermite polynomials
In this section, we study differential equations arising from the generating functions of the 3variable Hermite polynomials. Let H n x; y; z ð Þ t n n! , x, y, z, t ∈ ℂ: Then, by (4), we have Continuing this process, we can guess that Differentiating (7) with respect to t, we have Now replacing N by N þ 1 in (7), we find Comparing the coefficients on both sides of (8) and (9), we obtain and In addition, by (7), we have which gives a 0 0; x; y; z ð Þ¼1: Thus, by (14), we also find a 0 1; x; y; z ð Þ¼x, a 1 1; x; y; z ð Þ¼2y, a 2 1; x; y; z ð Þ¼3z: From (10), we note that Note that, here the matrix a i j; x; y ð Þ 0 ≤ i ≤ 2Nþ2, 0 ≤ j ≤ Nþ1 is given by Therefore, we obtain the following theorem. From (4), we note that By (4) and (18), we get e Ànt ∂ ∂t By the Leibniz rule and the inverse relation, we have Hence, by (19) and (20), and comparing the coefficients of t m m! gives the following theorem. Here is a plot of the surface for this solution. In Figure 1(left), we choose À2 ≤ z ≤ 2, À1 ≤ t ≤ 1, x ¼ 2, and y ¼ À4. In Figure 1(right), we choose À5 ≤ x ≤ 5, À 1 ≤ t ≤ 1, y ¼ À3, and z ¼ À1.

Distribution of zeros of the 3-variable Hermite polynomials
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 x; y; z ð Þ. By using computer, the 3-variable Hermite polynomials H n x; y; z ð Þcan be determined explicitly. We display the shapes of the 3-variable Hermite polynomials H n x; y; z ð Þand investigate the zeros of the 3-variable Hermite polynomials H n x; y; z ð Þ. We investigate the beautiful zeros of the 3-variable Hermite polynomials H n x; y; z ð Þ by using a computer. We plot the zeros of the H n x; y; z ð Þ for n ¼ 20, y ¼ 1, À 1, 1 þ i, À 1 À i, z ¼ 3, À 3, 3 þ i, À 3 À 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 À1, and z ¼ À3. 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 À1 À i, and z ¼ À3 À i.
The plot of real zeros of the 3-variable Hermite polynomials H n x; y; z ð Þfor 1 ≤ n ≤ 20 structure are presented ( Figure 5).
Stacks of zeros of H n x; À2; 4 ð Þfor 1 ≤ n ≤ 40, forming a 3D structure are presented ( Figure 6). In Figure 6(top-left), we plot stacks of zeros of H n x; À2; 4 ð Þfor 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 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 H n x; y; z ð Þ, x ∈ C, y, z ∈ R, has Im x ð Þ ¼ 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 x; y; z ð Þ for y, z ∈ R. We also hope to verify a remarkable regular structure of the complex roots of the 3-variable Hermite polynomials H n x; y; z ð Þfor y, z ∈ R (Tables 1 and 2). Next, we calculated an approximate solution satisfying H n x; y; z ð Þ¼0, x ∈ C. The results are given in Tables 3 and 4. The plot of real zeros of the 3-variable Hermite polynomials H n x; y; z ð Þfor 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 À1 and y ¼ À2.   tabulated values of R Hn x;y;z ð Þ and C Hn x;y;z ð Þ . 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 x; y; z ð Þwhich appear in mathematics and physics. The reader may refer to [2,11,13,20] for the details.