Some Identities Involving 2-Variable Modified Degenerate Hermite Polynomials Arising from Differential Equations and Distribution of Their Zeros

In this chapter, we introduce the 2-variable modified degenerate Hermite polynomials and obtain some new symmetric identities for 2-variable modified degenerate Hermite polynomials. In order to give explicit identities for 2-variable modified degenerate Hermite polynomials, differential equations arising from the generating functions of 2-variable modified degenerate Hermite polynomials are studied. Finally, we investigate the structure and symmetry of the zeros of the 2-variable modified degenerate Hermite equations.


Introduction
The Hermite equation is defined as where ε is unrestricted. Hermite equation is encountered in the study of quantum mechanical harmonic oscillator, where ε represent the energy of the oscillator. The ordinary Hermite numbers H n and Hermite polynomials H n x ð Þ are usually defined by the generating functions H n x ð Þ t n n! (2) and e Àt 2 ¼ X ∞ n¼0 H n t n n! : Clearly, H n ¼ H n 0 ð Þ.
It is known that these numbers and polynomials play an important role in various fields of mathematics and physics, including number theory, combinations, special functions, and differential equations. Many interested properties about that have been studied (see [1][2][3][4][5]). The ordinary Hermite polynomials H n x ð Þ satisfy the Hermite differential equation Hence ordinary Hermite polynomials H n x ð Þ satisfy the second-order ordinary differential equation u 00 À 2xu 0 þ 2nu ¼ 0: (5) We remind that the 2-variable Hermite polynomials H n x, y ð Þ defined by the generating function (see [2]) are the solution of heat equation Observe that H n 2x, À1 ð Þ¼H n x ð Þ: Motivated by their importance and potential applications in certain problems of probability, combinatorics, number theory, differential equations, numerical analysis and other areas of mathematics and physics, several kinds of some special numbers and polynomials were recently studied by many authors (see [1][2][3][4][5][6][7][8]). Many mathematicians have studied in the area of the degenerate Stiling, degenerate Bernoulli polynomials, degenerate Euler polynomials, degenerate Genocchi polynomials, and degenerate tangent polynomials (see [6,7,9]).
Recently, Hwang and Ryoo [10] proposed the 2-variable degenerate Hermite polynomials H n x, y, λ ð Þby means of the generating function Since 1 þ μ ð Þ t μ ! e t as μ ! 0, it is evident that (9) reduces to (6). The 2-variable degenerate Hermite polynomials H n x, y, λ ð Þin generating function (9) are the solution of equation Since log 1þλ ð Þ λ ! 1 as λ approaches to 0, it is apparent that (10) descends to (7). Mathematicians have studied the differential equations arising from the generating functions of special numbers and polynomials (see [10][11][12][13][14]). Now, a new class of 2-variable modified degenerate Hermite polynomials are constructed based on the results so far. We can induce the differential equations generated from the generating function of 2-variable modified degenerate Hermite polynomials. By using the coefficients of this differential equation, we obtain explicit identities for the 2-variable modified degenerate Hermite polynomials. The rest of the paper is organized as follows. In Section 2, we construct the 2-variable modified degenerate Hermite polynomials and obtain basic properties of these polynomials. In Section 3, we give some symmetric identities for 2-variable modified degenerate Hermite polynomials. In Section 4, we derive the differential equations generated from the generating function of 2-variable modified degenerate Hermite polynomials. Using the coefficients of this differential equation, we have explicit identities for the 2variable modified degenerate Hermite polynomials. In Section 5, we investigate the zeros of the 2-variable modified degenerate Hermite equations by using computer. Further, we observe the pattern of scattering phenomenon for the zeros of 2variable modified degenerate Hermite equations. Our paper will finish with Section 6, where the conclusions and future directions of this work are showed.

Basic properties for the 2-variable modified degenerate Hermite polynomials
In this section, a new class of the 2-variable modified degenerate Hermite polynomials are considered. Furthermore, some properties of these polynomials are also obtained.
We define the 2-variable modified degenerate Hermite polynomials H n x, yjμ ð Þ by means of the generating function Since 1 þ μ ð Þ xt μ ! e xt as μ ! 0, it is clear that (11) reduces to (6). Observe that degenerate Hermite polynomials H n x, y, μ ð Þand 2-variable modified degenerate Hermite polynomials H n x, yjμ ð Þare totally different. Now, we recall that the μ-analogue of the falling factorial sequences as follows: Note that lim μ!1 xjμ We also need the binomial theorem: for a variable y, We remember that the classical Stirling numbers of the first kind S 1 n, k ð Þand the second kind S 2 n, k ð Þ are defined by the relations (see [6][7][8][9][10][11][12][13]) respectively. We also have As another application of the differential equation for H n x, yjμ ð Þis as follows: Note that Thus the 2-variable modified degenerate Hermite polynomials H n x, yjμ ð Þin generating function (11) are the solution of equation The generating function (11) is useful for deriving several properties of the 2variable modified degenerate Hermite polynomials H n x, yjμ ð Þ. For example, we have the following expression for these polynomials: Theorem 1. For any positive integer n, we have where ½ denotes taking the integer part. Proof. By (11) and (13), we have By comparing the coefficients of t n n! , the expected result of Theorem 1 is achieved. □ The following basic properties of the 2-variable degenerate Hermite polynomials H n x, yjμ ð Þare induced form (11). Therefore, it is enough to delete involved detail explanation.
Theorem 2. For any positive integer n, we have

Symmetric identities for 2-variable modified degenerate Hermite polynomials
In this section, we give some new symmetric identities for 2-variable modified degenerate Hermite polynomials. We also get some explicit formulas and properties for 2-variable modified degenerate Hermite polynomials.
The following identity holds true: Then the expression for G t, μ ð Þ is symmetric in a and b By the similar way, we get that By comparing the coefficients of t m m! in last two equations, the expected result of Theorem 3 is achieved.
□ Again, we now use For μ ∈ , we introduce the modified degenerate Bernoulli polynomials given by the generating function When x ¼ 0 and β n μ ð Þ ¼ β n 0jμ ð Þ are called the modified degenerate Bernoulli numbers. Note that where B n are called the Bernoulli numbers. The first few of them are For each integer k ≥ 0, S k n ð Þ ¼ 0 k þ 1 k þ 2 k þ ⋯ þ n À 1 ð Þ k is called sum of integers. A modified generalized falling factorial sum σ k n, μ ð Þ can be defined by the generating function Note that lim μ!0 σ k njμ ð Þ ¼ S k n ð Þ: From F t, μ ð Þ, we get the following result: In a similar fashion we have By comparing the coefficients of t m m! on the right hand sides of the last two equations, we have the below theorem.
Theorem 4. Let a, b > 0(a 6 ¼ b). The the following identity holds true: By taking the limit as μ ! 0, we have the following corollary. Corollary 5. Let a, b > 0 (a 6 ¼ b). The the following identity holds true:

Differential equations associated with 2-variable modified degenerate Hermite polynomials
In this section, we construct the differential equations with coefficients a i N, x, y, μ ð Þarising from the generating functions of the 2-variable modified degenerate Hermite polynomials: By using the coefficients of this differential equation, we can get explicit identities for the 2-variable modified degenerate Hermite polynomials H n x, y, μ ð Þ.

Recall that
Then, by (37), we have By continuing this process as shown in (39), we can get easily that By differentiating (40) with respect to t, we have Now we replace N by N þ 1 in (40). We find By comparing the coefficients on both sides of (41) and (42), we get and In addition, by (37), we have By (45), we get It is not difficult to show that Thus, by (38) and (47), we also get From (43) and (44), we note that and For i ¼ 1 in (44), we have Continuing this process, we can deduce that, for 1 ≤ i ≤ N À 1, Note that, from (37)-(53), here the matrix a i j, x, y, μ ð Þ 0 ≤ i,j ≤ Nþ1 is given by Therefore, from (37)-(53), we obtain the following theorem. Theorem 5. For N ¼ 0, 1, 2, … , the differential equation
Making N-times derivative for (10) with respect to t, we have ∂ ∂t By (58) and Theorem 5, we have Hence we have the following theorem. Theorem 6. For N ¼ 0, 1, 2, … , we get If we take m ¼ 0 in (60), then we have the below corollary. Corollary 7. For N ¼ 0, 1, 2, … , we have where The first few of them are H 0 x, y, μ ð Þ¼1, x,

Zeros of the 2-variable modified degenerate Hermite polynomials
This section shows the benefits of supporting theoretical prediction through numerical experiments and finding new interesting pattern of the zeros of the 2variable modified degenerate Hermite equations H n x, yjμ ð Þ¼0. By using computer, the 2-variable modified degenerate Hermite polynomials H n x, yjμ ð Þcan be determined explicitly. We investigate the zeros of the 2-variable modified degenerate Hermite equations H n x, yjμ ð Þ¼0. The zeros of the H n x, yjμ ð Þ¼0 for n ¼ 30, y ¼ 3, À 3, 3 þ i, À 3 À i, μ ¼ 1=2, and x ∈  are displayed in Our numerical results for approximate solutions of real zeros of the 2-variable modified degenerate Hermite equations H n x, yjμ ð Þ¼0 are displayed (Tables 1 and 2).
We observed a remarkable regular structure of the complex roots of the 2variable modified degenerate Hermite equations H n x, yjμ ð Þ¼0 and also hope to verify same kind of regular structure of the complex roots of the 2-variable modified degenerate Hermite equations H n x, yjμ ð Þ¼0 ( Table 1). Plot of real zeros of the 2-variable modified degenerate Hermite equations H n x, yjμ ð Þ¼0 for 1 ≤ n ≤ 50, μ ¼ 1=2 structure are presented in Figure 4. In the topleft picture of Figure 4, we choose y ¼ 3. In the top-right picture of Figure 4, we   choose y ¼ À3. In the bottom-left picture of Figure 4, we choose y ¼ À3 þ i. In the bottom-right picture of Figure 4, we choose y ¼ À3 À i. Next, we calculated an approximate solution satisfying H n x, yjμ ð Þ¼0, x ∈ . The results are given in Table 2. In Table 2, we choose y ¼ À3 and μ ¼ 1=2.  Real zeros of H n x, yjμ ð Þ¼0 for 1 ≤ n ≤ 50, μ ¼ 1