Numbers of real and complex zeros of

## Abstract

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.

### Keywords

- differential equations
- symmetric identities
- modified degenerate Hermite polynomials
- complex zeros
- 2000 Mathematics Subject Classification: 05A19
- 11B83
- 34A30
- 65L99

## 1. Introduction

The Hermite equation is defined as

where

and

Clearly,

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

Hence ordinary Hermite polynomials

We remind that the 2-variable Hermite polynomials

are the solution of heat equation

Observe that

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

Since

Since

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 2-variable 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 2-variable modified degenerate Hermite equations. Our paper will finish with Section 6, where the conclusions and future directions of this work are showed.

## 2. 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

Since

Now, we recall that the

Note that

We remember that the classical Stirling numbers of the first kind

respectively. We also have

As another application of the differential equation for

satisfies

Substitute the series in (11) for

Thus the 2-variable modified degenerate Hermite polynomials

The generating function (11) is useful for deriving several properties of the 2-variable modified degenerate Hermite polynomials

**Theorem 1**. For any positive integer

where

**Proof.** By (11) and (13), we have

By comparing the coefficients of

Since

The following basic properties of the 2-variable degenerate Hermite polynomials

**Theorem 2.** For any positive integer

## 3. 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.

**Theorem 3.** Let

**Proof.** Let

Then the expression for

By the similar way, we get that

By comparing the coefficients of

Again, we now use

For

When

where

For each integer

Note that

In a similar fashion we have

By comparing the coefficients of

**Theorem 4.** Let

By taking the limit as

**Corollary 5.** Let

## 4. Differential equations associated with 2-variable modified degenerate Hermite polynomials

In this section, we construct the differential equations with coefficients

By using the coefficients of this differential equation, we can get explicit identities for the 2-variable modified degenerate Hermite polynomials

Then, by (37), we have

By continuing this process as shown in (39), we can get easily that

By differentiating (40) with respect to

Now we replace

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

Continuing this process, we can deduce that, for

Note that, from (37)–(53), here the matrix

Therefore, from (37)–(53), we obtain the following theorem.

**Theorem 5.** For

has a solution

where

Here is a plot of the surface for this solution. In the left picture of Figure 1, we choose

Making

By (58) and Theorem 5, we have

Hence we have the following theorem.

**Theorem 6.** For

If we take

**Corollary 7.** For

where

The first few of them are

## 5. 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 2-variable modified degenerate Hermite equations

Stacks of zeros of the 2-variable modified degenerate Hermite equations

Our numerical results for approximate solutions of real zeros of the 2-variable modified degenerate Hermite equations

Degree | Real zeros | Complex zeros | Real zeros | Complex zeros |
---|---|---|---|---|

1 | 1 | 0 | 1 | 0 |

2 | 0 | 2 | 2 | 0 |

3 | 1 | 2 | 3 | 0 |

4 | 0 | 4 | 4 | 0 |

5 | 1 | 4 | 5 | 0 |

6 | 0 | 6 | 6 | 0 |

7 | 1 | 6 | 7 | 0 |

8 | 0 | 8 | 8 | 0 |

9 | 1 | 8 | 9 | 0 |

10 | 0 | 10 | 10 | 0 |

Degree | |
---|---|

1 | |

2 | |

3 | |

4 | |

5 | |

6 | |

7 | |

8 |

We observed a remarkable regular structure of the complex roots of the 2-variable modified degenerate Hermite equations

Plot of real zeros of the 2-variable modified degenerate Hermite equations

Next, we calculated an approximate solution satisfying

## 6. Conclusions

In this chapter, we constructed the 2-variable modified degenerate Hermite polynomials and got some new symmetric identities for 2-variable modified degenerate Hermite polynomials. We constructed differential equations arising from the generating function of the 2-variable modified degenerate Hermite polynomials

Let us use the following notations.

We can see a good regular pattern of the complex roots of the 2-variable modified degenerate Hermite equations

**Conjecture 1.** Let

where

**Conjecture 2.** Let

As a result of investigating more

We observe that solutions of the 2-variable modified degenerate Hermite equations

**Conjecture 3.** Prove that the zeros of

Finally, we consider the more general problems. How many zeros does

**Conjecture 4.** For

As a result of investigating more

We expect that research in these directions will make a new approach using the numerical method related to the research of the 2-variable modified degenerate Hermite equations