Some properties for 2-variable modified partially degenerate Hermite (MPDH) polynomials derived from di ff erential equations and their zeros distributions

: The 2-variable modified partially degenerate Hermite (MPDH) polynomials are the subject of our study in this paper. We find basic properties of these polynomials and obtain several types of di ff erential equations related to MPDH polynomials. Based on the MPDH polynomials, we look at the structures of the approximation roots for a particular polynomial and check the values of the approximate roots. Further, we present some conjectures for MPDH polynomials


Introduction
A solution of the following second-order differential equation is the ordinary Hermite polynomials H n (γ).Actually, for n ∈ {0, 1, 2, • • • }, the following form has a solution which is the generating function of the ordinary Hermite polynomials H n (γ).
In other words, this polynomial provides with the solution of the following heat equation: Obviously that H n (γ, 0) = γ n .It is also clear that H n (2γ, −1) = H n (γ).
Mathematicians who study polynomials have introduced and continuously developed new special polynomials that can be applied in fields such as combinatorics, numerical analysis, and physics, etc (see [1]- [6], [14], [15]).Carlitz pioneered the concept of degenerate polynomials, and since then, several mathematicians have worked to extend well-known special polynomials such as Bernoulli ( [7], [8]), Euler [7], and tangent [9] to encompass their degenerate counterparts.With the discovery of new polynomials, they studied various properties and identities of polynomials, the structure of approximate roots of polynomials, differential equations, etc (see [12], [13]).Young [8] showed the symmetric properties, congruence and identities, the relationship with the Stirling number for degenerate Bernoulli polynomials.Ryoo [9] confirmed the structures of approximate roots, properties of approximated real and imaginary roots, and identity for degenerate tangent polynomials.Carlitz [7] has defined the degenerate Stirling, Bernoulli and Eulerian numbers and he studied some properties of degenerate these numbers.In [10], Hwang and Ryoo introduced the new generating function of the 2-variable degenerate Hermite polynomials H n (γ, ν, ζ) as follows: We can see that (3) → 1 as ζ approaches to 0, it is clear that (4) becomes (2).
The purpose of this paper is to construct new MPDH polynomials based on the results mentioned above and find some conjectures by looking at the properties of differential equations and approximate roots related to these polynomials.
The structure of this paper is as follows.We find some properties of the newly defined MPDH polynomials in Section 2. Here, we also obtain the relationship between the Stirling numbers of first kind and the MPDH polynomials.In section 3, we derive the symmetric properties, which are important properties of MPDH polynomials, and Section 4 presents several differential equations with MPDH polynomials as solutions.Section 5 introduces the structures and data of approximation roots of MPDH polynomials.We try to understand higher-order MPDH polynomials by looking at various experimental results by substituting various values of variables into the MPDH polynomials.In Section 6, we present some conjectures as a result of the paper and directions for further research.

Basic properties for the MPDH polynomials
In this section, a new class of the MPDH polynomials are considered.Further, some properties of these polynomials are also obtained.
We define the MPDH polynomials H n (γ, ν, ζ) by means of the generating function Since (1 ζ → e νt 2 as ζ → 0, it is evident that (5) reduces to (1).Observe that degenerate Hermite polynomials H n (γ, ν, ζ) and MPDH polynomials H n (γ, ν, ζ) are completely different.Now, we recall that the ζ-analogue of the falling factorial sequences as follows: Note that lim ζ→1 (γ|ζ . We remember that the classical Stirling numbers of the first kind S 1 (n, k) and the second kind S 2 (n, k) are defined by the relations(see [6][7][8][9][10][11][12][13]) respectively.We also have We also need the binomial theorem: for a variable v, Note that Substitute the series in (5) for G(τ, γ, ν, ζ) to get This is the recurrence relation for MPDH polynomials.Another recurrence relation comes from This implies Eliminate H n−1 (γ, ν, ζ) from ( 8) and ( 9) to obtain Differentiate this equation and use (9) again to get Thus the MPDH polynomials H n (γ, ν, ζ) in generating function (5) are the solution of differential equation 2ν log(1 As another application of the differential equation for H n (γ, ν, ζ), we derive The generating function ( 5) is useful for deriving several properties of the MPDH polynomials The following basic properties of the MPDH polynomials H n (γ, ν, ζ) are derived form (5). We, therefore, choose to omit the details involved.
Since (8), we have where Using the well-known identity On comparing the coefficients of τ n n! , we have the following theorem.
Theorem 1.For any positive integer n, we have By (5) and Theorem 1, we have the following corollary, Corollary 2. For any positive integer n, we have The following basic properties of the MPDH polynomials H n (γ, ν, ζ) are derived form (5). We, therefore, choose to omit the details involved.
Theorem 2. For any positive integer n, we have (

Symmetric identities for the MPDH polynomials
In this section, we give some new symmetric identities for the MPDH polynomials.We also get some explicit formulas and properties for the MPDH polynomials.
Theorem 3. Let w 1 , w 2 > 0 and w 1 w 2 .The following identity holds true: Proof.Let w 1 , w 2 > 0 and w 1 w 2 .We start with Then the expression for G(τ, µ) is symmetric in w 1 and w 2 On the similar lines we can obtain that Comparing the coefficients of τ n n! in last two equations, the expected result of Theorem 1 is achieved.
k is called sums of powers of consecutive integers.A generalized falling factorial sum σ k (n, µ) can be defined by the generating function(see [6,7,9]) For µ ∈ C, we defined the degenerate Bernoulli polynomials given by the generating function When γ = 0 and where B n are called the Bernoulli numbers(see [6,7]).
The first few of them are Again, we now use .
From F(τ, µ), we get the following result: In a similar fashion we have By comparing the coefficients of τ n n! on the right hand sides of the last two equations we have the below theorem.
Theorem 4. Let w 1 , w 2 > 0 and w 1 w 2 .Then the following identity holds true: By taking the limit as µ → 0, we have the following corollary.

Differential equations associated with MPDH polynomials
In this section, we construct the differential equations with coefficients a i (N, γ, ν, ζ) arising from the generating functions of the MPDH polynomials: By using the coefficients of this differential equation, we can derive explicit identities for the 2variable modified partially degenerate Hermite polynomials H n (γ, ν, ζ).Recall that Then, by (10), we have Continuing this process as shown in (12), we can guess that Differentiating ( 13) with respect to τ, we have Now replacing N by N + 1 in (13), we find Comparing the coefficients on both sides of ( 14) and ( 15), we obtain For 1 ≤ i ≤ N − 1, we obtain For i = N, we obtain For i = N + 1, we obtain In addition, by ( 13), we have By (20), we get Thus, by (10) and ( 22), we also get From ( 16), we note that For 1 ≤ i ≤ N − 1, from (17), we note that AIMS Mathematics Volume 8, Issue x, xxx-xxx From (18), we have Again, by ( 19), we have Note that, here the matrix a i ( j, γ, ν, ζ) 0≤i≤N+1,0≤ j≤N+1 is given by Therefore, by ( 24)-( 27), we obtain the following theorem.
Theorem 6.For N = 0, 1, 2, . . ., the differential equation where Here is a plot of the surface for this solution.In the left picture of Figure 1, we choose −3 ≤ γ ≤ Making N-times derivative for (8) with respect to τ, we have By Cauchy product and multiplying the exponential series e γτ = ∞ m=0 γ m τ m m! in both sides of (28), we get By the Leibniz rule and inverse relation, we have Hence, by ( 29) and (30), and comparing the coefficients of τ m m! gives the following theorem.
Theorem 7. Let m, n, N be nonnegative integers.Then If we take m = 0 in (31), then we have the following corollary.
The first few of them are

Zeros of the MPDH polynomials
This section shows the benefits of supporting theoretical prediction through numerical experiments and finding new interesting pattern of the zeros of the MPDH equations H n (γ, ν, ζ) = 0.By using computer, the MPDH polynomials H n (γ, ν, ζ) can be determined explicitly.We investigate the zeros of the MPDH equations  Our numerical results for approximate solutions of real zeros of the MPDH equations H n (γ, ν, ζ) = 0 are displayed as Tables 1 and 2.

Observations
In this article, we introduced the MPDH polynomials and got new symmetric identities for MPDH polynomials.We derived the symmetric property, one of the important properties of MPDH polynomials.We have shown several types of differential equations with H m (γ, ν, ζ) as their solution.We also observed the symmetric properties of the roots of H m (γ, ν, ζ) = 0, which appear differently as the values of the variables γ and ν change.As a result, it was found that the distribution of the roots of H m (γ, ν, ζ) = 0 had a very regular pattern, and through numerical experiments we found the following conjectures are possible.
Here, we use the notation as follows.We realize regular pattern of the complex roots of the MPDH equations H n (γ, ν, ζ) = 0 related to ν and ζ.We made these conjectures.Proving or disproving the following conjectures will be our future task.