Some Identities Involving Hermite Kamp é de F é riet Polynomials Arising from Differential Equations and Location of Their Zeros

In this paper, we study differential equations arising from the generating functions of Hermit Kampé de Fériet polynomials. Use this differential equation to give explicit identities for Hermite Kampé de Fériet polynomials. Finally, use the computer to view the location of the zeros of Hermite Kampé de Fériet polynomials.


Introduction
Numerous studies have been conducted on Bernoulli polynomials, Euler polynomials, tangent polynomials, Hermite polynomials and Laguerre polynomials (see [1][2][3][4][5][6][7][8][9][10][11][12][13]).The special polynomials of the two variables provided a new way to analyze solutions of various kinds of partial differential equations that are often encountered in physical problems.Most of the special function of mathematical physics and their generalization have been proposed as physical problems.For example, we recall that the two variables Hermite Kamp é de F ériet polynomials H n (x, y) defined by the generating function (see [2]) are the solution of heat equation We note that H n (2x, −1) = H n (x), where H n (x) are the classical Hermite polynomials (see [1]).The differential equation and relation are given by 2y respectively.
By (1) and Cauchy product, we get ( By comparing the coefficients on both sides of (2), we have the following theorem: Theorem 1.For any positive integer n, we have The following elementary properties of the two variables Hermite Kamp é de F ériet polynomials H n (x, y) are readily derived from (1).Theorem 2. For any positive integer n, we have Recently, many mathematicians have studied differential equations that occur in the generating functions of special polynomials (see [8,9,[14][15][16]).The paper is organized as follows.We derive the differential equations generated from the generating function of Hermite Kamp é de F ériet polynomials: By obtaining the coefficients of this differential equation, we obtain explicit identities for the Hermite Kamp é de F ériet polynomials in Section 2. In Section 3, we investigate the zeros of the Hermite Kamp é de F ériet polynomials using numerical methods.Finally, we observe the scattering phenomenon of the zeros of Hermite Kamp é de F ériet polynomials.

Differential Equations Associated with Hermite Kamp é de F ériet Polynomials
In order to obtain explicit identities for special polynomials, differential equations arising from the generating functions of special polynomials are studied by many authors (see [8,9,[14][15][16]).In this section, we introduce differential equations arising from the generating functions of Hermite Kamp é de F ériet polynomials and use these differential equations to obtain the explicit identities for the Hermite Kamp é de F ériet polynomials. Let Then, by (3), we have + (4xy 2 + 8xy 2 )t 2 F(t, x, y).
If we continue this process, we can guess as follows: Differentiating ( 4) with respect to t, we have Now, replacing N by N + 1 in (4), we find Comparing the coefficients on both sides of ( 5) and ( 6), we obtain and In addition, by (4), we have which gives a 0 (0, x, y) = 1. (10) Thus, by (11), we also find From ( 7), we note that For i = 1 in (8), we have Continuing this process, we can deduce that, for 1 ≤ i ≤ N − 1, Note that here the matrix a i (j, x, y) 0≤i,j≤N+1 is given by .
Therefore, we obtain the following theorem.
Theorem 3.For N = 0, 1, 2, . . ., the differential equation where Making N-times derivative for (3) with respect to t, we have By Cauchy product and multiplying the exponential series e xt = ∑ ∞ m=0 x m t m m! in both sides of (18), we get For non-negative integer m, assume that {a(m)}, {b(m)}, {c(m)}, { c(m)} are four sequences given by t m m! = 1, we have the following inverse relation: By (20) and the Leibniz rule, we have Hence, by ( 19) and ( 21), and comparing the coefficients of t m m! gives the following theorem.
Theorem 4. Let m, n, N be nonnegative integers.Then, If we take m = 0 in ( 22), then we have the following: Corollary 1.For N = 0, 1, 2, . . ., we have For N = 0, 1, 2, . . ., the differential equation Here is a plot of the surface for this solution.

Zeros of the Hermite Kamp é de F ériet Polynomials
By using software programs, many mathematicians can explore concepts more easily than in the past.These experiments allow mathematicians to quickly create and visualize new ideas, review properties of figures, create many problems, and find and guess patterns.This numerical survey is particularly interesting because it helps many mathematicians understand basic concepts and solve problems.In this section, we examine the distribution and pattern of zeros of Hermite Kamp é de F ériet polynomials H n (x, y) according to the change of degree n.Based on these results, we present a problem that needs to be approached theoretically.
Using a computer, we investigate the distribution of zeros of the Hermite Kamp é de F ériet polynomials H n (x, y).
Stacks of zeros of the Hermite Kamp é de F ériet polynomials H n (x, y) for 1 ≤ n ≤ 20 from a 3D structure are presented (Figure 3).In Figure 3 (top-left), we choose y = 2.In Figure 3 (top-right), we choose y = −2.In Figure 3 (bottom-left), we choose y = 2 + i.In Figure 3 (bottom-right), we choose y = −2 − i.Our numerical results for approximate solutions of real zeros of the Hermite Kamp é de F ériet polynomials H n (x, y) are displayed (Tables 1-3).
The plot of real zeros of the Hermite Kamp é de F ériet polynomials H n (x, y) for 1 ≤ n ≤ 20 structure are presented (Figure 4).It is expected that H n (x, y), x ∈ C, y > 0, has Im(x) = 0 reflection symmetry analytic complex functions (see Figures 2 and 3).We also expect that H n (x, y), x ∈ C, y < 0, has Re(x) = 0 reflection symmetry analytic complex functions (see Figures 2-4).We observe a remarkable regular structure of the complex roots of the Hermite Kamp é de F ériet polynomials H n (x, y) for y < 0. We also hope to verify a remarkable regular structure of the complex roots of the Hermite Kamp é de F ériet polynomials H n (x, y) for y < 0 (Table 1).Next, we calculated an approximate solution that satisfies H n (x, y) = 0, x ∈ C. The results are shown in Table 3.

Conclusions and Future Developments
This study obtained the explicit identities for Hermite Kamp é de F ériet polynomials H n (x, y).The location and symmetry of the roots of the Hermite Kamp é de F ériet polynomials were investigated.We examined the symmetry of the zeros of the Hermite Kamp é de F ériet polynomials for various variables x and y, but, unfortunately, we could not find a regular pattern.However, the following special cases showed regularity.Through numerical experiments, we will make the following series of conjectures.
If y > 0, we can see that H n (x, y) has Re(x) = 0 reflection symmetry.Therefore, the following conjecture is possible.Conjecture 1. Prove or disprove that H(x, y), x ∈ C and y > 0, has Im(x) = 0 reflection symmetry analytic complex functions.Furthermore, H n (x, y) has Re(x) = 0 reflection symmetry for y < 0.
As a result of investigating more n variables, it is still unknown whether the conjecture is true or false for all variables n (see Figure 1).Conjecture 2. Prove or disprove that H n (x, y) = 0 has n distinct solutions.
Let's use the following notations.R H n (x,y) denotes the number of real zeros of H n (x, y) lying on the real plane Im(x) = 0 and C H n (x,y) denotes the number of complex zeros of H n (x, y).Since n is the degree of the polynomial H n (x, y), we have R H n (x,y) = n − C H n (x,y) (see Tables 1 and 2).Conjecture 3. Prove or disprove that R H n (x,y) = n, if y < 0, 0, if y > 0, C H n (x,y) = 0, if y < 0, n, if y > 0.

Figure 1 .
Figure 1.The surface for the solution F(t, x, y).

Table 1 .
Numbers of real and complex zeros of H n (x, −2).

Table 2 .
Numbers of real and complex zeros of H n (x, 2).