Open access peer-reviewed chapter

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

By Cheon Seoung Ryoo

Submitted: March 19th 2020Reviewed: April 28th 2020Published: May 30th 2020

DOI: 10.5772/intechopen.92687

Downloaded: 74

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

1. Introduction

The Hermite equation is defined as

ux2xux+ε1ux=0,x,E1

where εis unrestricted. Hermite equation is encountered in the study of quantum mechanical harmonic oscillator, where εrepresent the energy of the oscillator. The ordinary Hermite numbers Hnand Hermite polynomials Hnxare usually defined by the generating functions

et2xt=n=0Hnxtnn!E2

and

et2=n=0Hntnn!.E3

Clearly, Hn=Hn0.

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 Hnxsatisfy the Hermite differential equation

d2Hxdx22xdHxdx+2nHx=0,n=0,1,2,.E4

Hence ordinary Hermite polynomials Hnxsatisfy the second-order ordinary differential equation

u2xu+2nu=0.E5

We remind that the 2-variable Hermite polynomials Hnxydefined by the generating function (see [2])

n=0Hnxytnn!=etx+ytE6

are the solution of heat equation

yHnxy=2x2Hnxy,Hnx0=xn.E7

Observe that

Hn2x1=Hnx.E8

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 Hnxyλby means of the generating function

n=0Hnxyλtnn!=1+λtx+ytλ.E9

Since 1+μtμetas μ0, it is evident that (9) reduces to (6). The 2-variable degenerate Hermite polynomials Hnxyλin generating function (9) are the solution of equation

yHnxyλ=λlog1+λ2x2Hnxyλ,Hnx0λ=log1+λλnxn.E10

Since log1+λλ1as λapproaches 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 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 Hnxyμby means of the generating function

n=0Hnxyμtnn!=1+μxtμeyt2.E11

Since 1+μxtμextas μ0, it is clear that (11) reduces to (6). Observe that degenerate Hermite polynomials Hnxyμand 2-variable modified degenerate Hermite polynomials Hnxyμare totally different.

Now, we recall that the μ-analogue of the falling factorial sequences as follows:

xμ0=1,xμn=xxμx2μxn1μ,n1.E12

Note that limμ1xμn=xx1x2xn1=xn,n1.We also need the binomial theorem: for a variable y,

1+μyt/μ=m=0tyμmμmm!=m=0l=0mS1mltyμlμmm!=l=0m=lS1mlylμmll!m!tll!.E13

We remember that the classical Stirling numbers of the first kind S1nkand the second kind S2nkare defined by the relations (see [6, 7, 8, 9, 10, 11, 12, 13])

xn=k=0nS1nkxkandxn=k=0nS2nkxk,E14

respectively. We also have

n=mS2nmtnn!=et1mm!andn=mS1nmtnn!=log1+tmm!.E15

As another application of the differential equation for Hnxyμis as follows: Note that

Gtxyμ=1+μxtμeyt2E16

satisfies

Gtxyμylog1+μμ22Gtxyμx2=0.E17

Substitute the series in (11) for Gtxyμto get

yHnxyμ=μlog1+μ22x2Hnxyμ.

Thus the 2-variable modified degenerate Hermite polynomials Hnxyμin generating function (11) are the solution of equation

log1+μμ2yHnxyμ2x2Hnxyμ=0,Hnx0μ=log1+μμnxn.E18

The generating function (11) is useful for deriving several properties of the 2-variable modified degenerate Hermite polynomials Hnxyμ. For example, we have the following expression for these polynomials:

Theorem 1. For any positive integer n, we have

Hnxyμ=k=0n2log1+μμn2kxn2kykn!k!n2k!,E19

where denotes taking the integer part.

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

n=0Hnxyμtnn!=1+μxtμeyt2=k=0ykt2kk!l=0log1+μμlxltll!=n=0k=0n2yklog1+μμn2kxn2kn!k!n2k!tnn!.E20

By comparing the coefficients of tnn!, the expected result of Theorem 1 is achieved.

Since limμ0log1+μμ=1,we get

Hnxy=n!k=0n2ykxn2kk!n2k!.E21

The following basic properties of the 2-variable degenerate Hermite polynomials Hnxyμare induced form (11). Therefore, it is enough to delete involved detail explanation.

Theorem 2. For any positive integer n, we have

1.Hnxyμ=k=0n2m=n2kykS1mn2kxn2kμmn2kn!m!k!.2.Hnx1+x2yμ=l=0nnllog1+μμlx2lHnlx1yμ.3.Hnx1+x2yμ=l=0nnlHnlx1yμm=lS1mlx2lμmll!m!.4.Hnxy1+y2μ=k=0n2Hn2kxy1μy2kn!k!n2k!.5.Hnx1+x2y1+y2μ=l=0nnlHlx1y1μHnlx2y2μ.E22

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 a,b>0(ab). The following identity holds true:

bmHmaxa2yμ=amHmbxb2yμ.E23

Proof. Let a,b>0(ab). We start with

Gtμ=1+μabxtμea2b2yt2.E24

Then the expression for Gtμis symmetric in aand b

Gtμ=m=0Hmaxa2yμbtmm!=m=0bmHmaxa2yμtmm!.E25

By the similar way, we get that

Gtμ=m=0Hmbxb2yμatmm!=m=0amHmbxb2yμtmm!.E26

By comparing the coefficients of tmm!in last two equations, the expected result of Theorem 3 is achieved.

Again, we now use

Ftμ=abt1+μabxtμea2b2yt21+μabtμ11+μatμ11+μbtμ1.E27

For μC, we introduce the modified degenerate Bernoulli polynomials given by the generating function

n=0βnxμtnn!=t1+μtμ11+μxtμ,see67.E28

When x=0and βnμ=βn0μare called the modified degenerate Bernoulli numbers. Note that

limμ0βnμ=Bn,E29

where Bnare called the Bernoulli numbers. The first few of them are

β0xμ=μlog1+μ,β1xμ=12+x,β2xμ=log1+μ6μxlog1+μμ+x2log1+μμ,β3xμ=xlog1+μ22μ23x2log1+μ22μ2+x3log1+μ2μ2,β4xμ=log1+μ330μ3+x2log1+μ3μ32x3log1+μ3μ3+x4log1+μ3μ3.E30

For each integer k0, Skn=0k+1k+2k++n1kis called sum of integers. A modified generalized falling factorial sum σknμcan be defined by the generating function

k=0σknμtkk!=1+μn+1tμ11+μtμ1.E31

Note that limμ0σknμ=Skn.From Ftμ, we get the following result:

Ftμ=abt1+μabxtμea2b2yt21+μabtμ11+μatμ11+μbtμ1=abt1+μatμ11+μabxtμea2b2yt21+μabtμ11+μbtμ1=bn=0βnμatnn!n=0Hnbxb2yμatnn!n=0σka1μbtnn!=n=0i=0nm=0iniimaibn+1iβmμHimbxb2yμσnia1μtnn!.E32

In a similar fashion we have

Ftμ=abt1+μbtμ11+μabxtμea2b2yt21+μabtμ11+μatμ1=an=0βnμbtnn!n=0Hnaxa2yμbtnn!n=0σkb1μatnn!=n=0i=0nm=0iniimbian+1iβmμHimaxa2yμσnib1μtnn!.E33

By comparing the coefficients of tmm!on the right hand sides of the last two equations, we have the below theorem.

Theorem 4. Let a,b>0(ab). The the following identity holds true:

i=0nm=0iniimaibn+1iβmμHimbxb2yμσnia1μ=i=0nm=0iniimbian+1iβmμHimaxa2yμσnib1μ.E34

By taking the limit as μ0, we have the following corollary.

Corollary 5. Let a,b>0(ab). The the following identity holds true:

i=0nm=0iniimaibn+1iSnia1BmHimbxb2y=i=0nm=0iniimbian+1iSnib1BmHimaxa2y.E35

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

In this section, we construct the differential equations with coefficients aiNxyμarising from the generating functions of the 2-variable modified degenerate Hermite polynomials:

tNGtxyμa0NxyμGtxyμa2NNxyμtNGtxyμ=0.E36

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

G=Gtxyμ=1+μxtμeyt2=n=0Hnxyμtnn!,μ,x,tC.E37

Then, by (37), we have

G1=tGtxyμ=t1+μxtμeyt2=log1+μμx+2yt1+μxtμeyt2=log1+μμx+2ytGtxyμ,E38
G2=tG1txyμ=2yGtxyμ+log1+μμx+2ytG1txyμ=2y+log1+μμ2x2Gtxyμ+log1+μμ4xytGtxyμ+2y2t2Gtxyμ.E39

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

GN=tNGtxyμ=i=0NaiNxyμtiGtxyμ,N=0,1,2.E40

By differentiating (40) with respect to t, we have

GN+1=GNt=i=0NaiNxyμiti1Gtxyμ+i=0NaiNxyμtiG1txyμ=i=0NiaiNxyμti1Gtxyμ+i=0Nlog1+μμxaiNxyμtiGtxyμ+i=0N2yaiNxyμti+1Gtxyμ=i=0N1i+1ai+1NxyμtiGtxyμ+i=0Nlog1+μμxaiNxyμtiGtxyμ+i=1N+12yai1NxyμtiGtxyμ.E41

Now we replace Nby N+1in (40). We find

GN+1=i=0N+1aiN+1xyμtiGtxyμ.E42

By comparing the coefficients on both sides of (41) and (42), we get

a0N+1xyμ=a1Nxyμ+log1+μμxa0Nxyμ,aNN+1xyμ=log1+μμxaNNxyμ+2yaN1Nxyμ,aN+1N+1xyμ=2yaNNxyμ,E43

and

aiN+1xyμ=i+1ai+1Nxyμ+log1+μμxaiNxyμ+2yai1Nxyμ,1iN1.E44

In addition, by (37), we have

Gtxyμ=G0txyμ=a00xyμGtxyμ.E45

By (45), we get

a00xyμ=1.E46

It is not difficult to show that

xlog1+μμGtxyμ+2ytGtxyμ=G1txyμ=i=01ai1xyμtiGtxyμ=a01xyμGtxyμ+a11xyμtGtxyμ.E47

Thus, by (38) and (47), we also get

a01xyμ=xlog1+μμ,a11xyμ=2y.E48

From (43) and (44), we note that

a0N+1xyμ=a1Nxyμ+xlog1+μμa0Nxyμ,a0Nxyμ=a1N1xyμ+xlog1+μμa0N1xyμ,a0N+1xyμ=i=0Nxlog1+μμia1Nixyμ+log1+μμN+1xN+1,E49
aNN+1xyμ=xlog1+μμaNNxyμ+2yaN1Nxyμ,aN1Nxyμ=xlog1+μμaN1N1xyμ+2yaN2N1xyμ,aNN+1xyμ=N+1x2yNlog1+μμ,E50

and

aN+1N+1xyμ=2yaNNxyμ,aNNxyμ=2yaN1N1xyμ,aN+1N+1xyμ=2yN+1.E51

For i=1in (44), we have

a1N+1xyμ=2k=0Nxlog1+μμka2Nkxyμ+2yk=0Nxlog1+μμka0Nkxyμ,E52

Continuing this process, we can deduce that, for 1iN1,

aiN+1xyμ=i+1k=0Nxlog1+μμkai+1Nkxyμ+2yk=0Nxlog1+μμkai1Nkxyμ.E53

Note that, from (37)(53), here the matrix aijxyμ0i,jN+1is given by

1xlog1+μμ2y+log1+μμ2x202ylog1+μμ4xy002y20002yN+1E54

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

Theorem 5. For N=0,1,2,,the differential equation

tNGtxyμi=0NaiNxyμtiGtxyμ=0E55

has a solution

G=Gtxyμ=1+μxtμeyt2,E56

where

a0N+1xyμ=i=0Nxlog1+μμia1Nixyμ+log1+μμN+1xN+1,aNN+1xyμ=N+12yNxlog1+μμ,aN+1N+1xyμ=2yN+1,aiN+1xyμ=i+1k=0Nxlog1+μμkai+1Nkxyμ+2yk=0Nxlog1+μμkai1Nkxyμ,1iN1.E57

Here is a plot of the surface for this solution. In the left picture of Figure 1, we choose 2x2,1t1,μ=1/10,and y=0.1. In the right picture of Figure 1, we choose 2y2,1t1,μ=1/10,and x=0.1.

Figure 1.

The surface for the solution Gtxyμ.

Making N-times derivative for (10) with respect to t, we have

tNGtxyμ=m=0Hm+Nxyμtmm!.E58

By (58) and Theorem 5, we have

a0NxyμGtxyμ+a1NxyμtGtxyμ++aNNxyμtNGtxyμ=m=0Hm+Nxyμtmm!.E59

Hence we have the following theorem.

Theorem 6. For N=0,1,2,,we get

Hm+Nxyμ=i=0mHmixaiNxyμm!mi!.E60

If we take m=0in (60), then we have the below corollary.

Corollary 7. For N=0,1,2,,we have

HNxyμ=a0Nxyμ,E61

where

a00xyμ=1,a0N+1xyμ=i=0Nxlog1+μμia1Nixyμ+log1+μμN+1xN+1.E62

The first few of them are

H0xyμ=1,H1xyμ=log1+μμx,H2xyμ=2y+log1+μ2μ2x2,H3xyμ=6xylog1+μμ+log1+μ3μ3x3,H4xyμ=12y2+12x2ylog1+μ2μ2+log1+μ4μ4x4,H5xyμ=60xy2log1+μμ+20x3ylog1+μ3μ3+log1+μ5μ5x5.E63

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 Hnxyμ=0. By using computer, the 2-variable modified degenerate Hermite polynomials Hnxyμcan be determined explicitly. We investigate the zeros of the 2-variable modified degenerate Hermite equations Hnxyμ=0. The zeros of the Hnxyμ=0for n=30,y=3,3,3+i,3i,μ=1/2,and xCare displayed in Figure 2. In the top-left picture of Figure 2, we choose n=30and y=3. In the top-right picture of Figure 2, we choose n=30and y=3. In the bottom-left picture of Figure 2, we choose n=30and y=3+i. In the bottom-right picture of Figure 2, we choose n=30and y=3i.

Figure 2.

Zeros of Hnxyμ=0.

Stacks of zeros of the 2-variable modified degenerate Hermite equations Hnxyμ=0for 1n50,μ=1/2from a 3-D structure are presented in Figure 3. In the top-left picture of Figure 3, we choose y=3. In the top-right picture of Figure 3, we choose y=3. In the bottom-left picture of Figure 3, we choose y=3+i. In the bottom-right picture of Figure 3, we choose y=3i.

Figure 3.

Stacks of zeros of Hnxyμ=0,1≤n≤50.

Our numerical results for approximate solutions of real zeros of the 2-variable modified degenerate Hermite equations Hnxyμ=0are displayed (Tables 1 and 2).

y=3,μ=1/2y=3,μ=1/2
Degree nReal zerosComplex zerosReal zerosComplex zeros
11010
20220
31230
40440
51450
60660
71670
80880
91890
10010100

Table 1.

Numbers of real and complex zeros of Hnxyμ=0.

Degree nx
10
23.0206,3.0206
35.2318,0,5.2318
47.0513,2.2412,2.2412,7.0513
58.6297,4.0948,0,4.0948,8.6297
610.041,5.7064,1.8628,1.8628,5.7064,10.041
711.329,7.1490,3.4870,0,3.4870,7.1490,11.329
812.519,8.4652,4.9433,1.6283,1.6283,4.9433,8.4652,12.519

Table 2.

Approximate solutions of Hnxyμ=0,xR.

We observed a remarkable regular structure of the complex roots of the 2-variable modified degenerate Hermite equations Hnxyμ=0and also hope to verify same kind of regular structure of the complex roots of the 2-variable modified degenerate Hermite equations Hnxyμ=0(Table 1).

Plot of real zeros of the 2-variable modified degenerate Hermite equations Hnxyμ=0for 1n50,μ=1/2structure are presented in Figure 4. In the top-left picture of Figure 4, we choose y=3. In the top-right picture of Figure 4, we choose y=3. In the bottom-left picture of Figure 4, we choose y=3+i. In the bottom-right picture of Figure 4, we choose y=3i.

Figure 4.

Real zeros of Hnxyμ=0for1≤n≤50, μ=12.

Next, we calculated an approximate solution satisfying Hnxyμ=0,xC. The results are given in Table 2. In Table 2, we choose y=3and μ=1/2.

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 Hnxyμ. We also investigated the symmetry of the zeros of the 2-variable modified degenerate Hermite equations Hnxyμ=0for various variables xand y. As a result, we found that the distribution of the zeros of 2-variable modified degenerate Hermite equations Hnxyμ=0is very regular pattern. So, we make the following series of conjectures with numerical experiments:

Let us use the following notations. RHnxyμdenotes the number of real zeros of Hnxyμ=0lying on the real plane Imx=0and CHnxyμdenotes the number of complex zeros of Hnxyμ=0. Since nis the degree of the polynomial Hnxyμ, we have RHnxyμ=nCHnxyμ.

We can see a good regular pattern of the complex roots of the 2-variable modified degenerate Hermite equations Hnxyμ=0for yand μ. Therefore, the following conjecture is possible.

Conjecture 1. Let nbe odd positive integer. For a>0or aC\aa<0, prove or disprove that

RHnxaμ=1,CHnxaμ=2n2,E64

where Cis the set of complex numbers.

Conjecture 2. Let nbe odd positive integer and aC. Prove or disprove that

Hn0aμ=0.E65

As a result of investigating more yand μvariables, it is still unknown whether the conjecture 1 and conjecture 2 is true or false for all variables yand μ.

We observe that solutions of the 2-variable modified degenerate Hermite equations Hnxyμ=0has not Rex=breflection symmetry for bR. It is expected that solutions of the 2-variable modified degenerate Hermite equations Hnxyμ=0, has not Rex=breflection symmetry (see Figures 24).

Conjecture 3. Prove that the zeros of Hnxaμ=0,aR, has Imx=0reflection symmetry analytic complex functions. Prove that the zeros of Hnxaμ=0,a<0,aC\R, has not Imx=0reflection symmetry analytic complex functions.

Finally, we consider the more general problems. How many zeros does Hnxyμhave? We are not able to decide if Hnxyμ=0has ndistinct solutions. We would like to know the number of complex zeros CHnxyμof Hnxyμ=0.

Conjecture 4. For aC,prove or disprove that Hnxaμ=0has ndistinct solutions.

As a result of investigating more nvariables, it is still unknown whether the conjecture is true or false for all variables n(see Tables 1 and 2).

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 Hnxyμ=0which appear in applied mathematics and mathematical physics.

Additional information

Mathematics Subject Classification: 05A19, 11B83, 34A30, 65L99

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Cheon Seoung Ryoo (May 30th 2020). Some Identities Involving 2-Variable Modified Degenerate Hermite Polynomials Arising from Differential Equations and Distribution of Their Zeros, Number Theory and Its Applications, Cheon Seoung Ryoo, IntechOpen, DOI: 10.5772/intechopen.92687. Available from:

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