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Differential Equations Arising from the 3-Variable Hermite Polynomials and Computation of Their Zeros

Written By

Cheon Seoung Ryoo

Submitted: 25 October 2017 Reviewed: 24 January 2018 Published: 20 February 2018

DOI: 10.5772/intechopen.74355

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Differential Equations - Theory and Current Research

Edited by Terry E. Moschandreou

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Abstract

In this paper, we study differential equations arising from the generating functions of the 3-variable Hermite polynomials. We give explicit identities for the 3-variable Hermite polynomials. Finally, we investigate the zeros of the 3-variable Hermite polynomials by using computer.

Keywords

• differential equations
• heat equation
• Hermite polynomials
• the 3-variable Hermite polynomials
• generating functions
• complex zeros

1. Introduction

Many mathematicians have studied in the area of the Bernoulli numbers, Euler numbers, Genocchi numbers, and tangent numbers see [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15]. The special polynomials of two variables provided new means of analysis for the solution of a wide class of differential equations often encountered in physical problems. Most of the special function of mathematical physics and their generalization have been suggested by physical problems.

In [1], the Hermite polynomials are given by the exponential generating function

n=0Hnxtnn!=e2xtt2.

We can also have the generating function by using Cauchy’s integral formula to write the Hermite polynomials as

Hnx=1nex2dndxnex2=n!2πiCe2txt2tn+1dt

with the contour encircling the origin. It follows that the Hermite polynomials also satisfy the recurrence relation

Hn+1x=2xHnx2nHn1x.

Further, the two variables Hermite Kampé de Fériet polynomials Hnxy defined by the generating function (see [3])

n=0Hnxytnn!=ext+yt2E1

are the solution of heat equation

yHnxy=2x2Hnxy,Hnx0=xn.

We note that

Hn2x1=Hnx.

The 3-variable Hermite polynomials Hnxyz are introduced [4].

Hnxyz=n!k=0n3zkHn3kxyk!n3k!.

The differential equation and he generating function for Hnxyz are given by

3z3x3+2y2x2+xxnHnxyz=0

and

ext+yt2+zt3=n=0Hnxyztnn!,E2

respectively.

By (2), we get

n=0Hnx1+x2yztnn!=ex1+x2t+yt2+zt3=n=0x2ntnn!n=0Hnx1yztnn!=n=0l=0nnlHlx1yzx2nltnn!.E3

By comparing the coefficients on both sides of (3), we have the following theorem.

Theorem 1. For any positive integer n, we have

Hnx1+x2yz=l=0nnlHlx1yzx2nl.

Applying Eq. (2), we obtain

n=0Hnxyz1+z2tnn!=ext+yt2+z1+z2t3=k=0z2nt3kk!l=0Hlxyz1tll!=n=0k=0n3Hn3kxyz1z2kn!k!n3k!.tnn!.

On equating the coefficients of the like power of t in the above, we obtain the following theorem.

Theorem 2. For any positive integer n, we have

Hnxyz1+z2=n!k=0n3Hn3kxyz1z2kk!n3k!.

Also, the 3-variable Hermite polynomials Hnxyz satisfy the following relations

yHnxyz=2x2Hnxyz,

and

zHnxyz=3x3Hnxyz.

The following elementary properties of the 3-variable Hermite polynomials Hnxyz are readily derived form (2). We, therefore, choose to omit the details involved.

Theorem 3. For any positive integer n, we have

1. Hn2x10=Hnx.

2. Hnxy1+y2z=n!k=0n2Hn2kxy1zy2kk!n2k!.

3. Hnxyz=l=0nnlHlxHnlxy+1z.

Theorem 4. For any positive integer n, we have

1. Hnx1+x2y1+y2z=l=0nnlHlx1y1zHnlx2y2.

2. Hnx1+x2y1+y2z1+z2=l=0nnlHlx1y1zHnlx2y2z2.

The 3-variable Hermite polynomials can be determined explicitly. A few of them are

H0xyz=1,H1xyz=x,H2xyz=x2+2y,H3xyz=x3+6xy+6z,H4xyz=x4+12x2y+12y2+24xz,H5xyz=x5+20x3y+60xy2+60x2z+120yz,H6xyz=x6+30x4y+180x2y2+120y3+120x3z+720xyz+360z2,H7xyz=x7+42x5y+420x3y2+840xy3+210x4z+2520x2yz+2520y2z+2520xz2,H8xyz=x8+56x6y+840x4y2+3360x2y3+1680y4+336x5z+6720x3yz+20160xy2z+10080x2z2+20160yz2.H9xyz=x9+72x7y+1512x5y2+10080x3y3+15120xy4+504x6z+15120x4yz+90720x2y2z+60480y3z+30240x3z2+181440xyz2+60480z3,H10xyz=x10+90x8y+2520x6y2+25200x4y3+75600x2y4+30240y5+720x7z+30240x5yz+302400x3y2z+604800xy3z+75600x4z2+907200x2yz2+907200y2z2+604800xz3.

Recently, many mathematicians have studied the differential equations arising from the generating functions of special polynomials (see [7, 8, 12, 16, 17, 18, 19]). In this paper, we study differential equations arising from the generating functions of the 3-variable Hermite polynomials. We give explicit identities for the 3-variable Hermite polynomials. In addition, we investigate the zeros of the 3-variable Hermite polynomials using numerical methods. Using computer, a realistic study for the zeros of the 3-variable Hermite polynomials is very interesting. Finally, we observe an interesting phenomenon of ‘scattering’ of the zeros of the 3-variable Hermite polynomials.

2. Differential equations associated with the 3-variable Hermite polynomials

In this section, we study differential equations arising from the generating functions of the 3-variable Hermite polynomials.

Let

F=Ftxyz=ext+yt2+zt3=n=0Hnxyztnn!,x,y,z,t.E4

Then, by (4), we have

F1=tFtxyz=text+yt2+zt3=ext+yt2+zt3x+2yt+3zt2=x+2yt+3zt2Ftxyz,E5
F2=tF1txyz=2y+6ztFtxyz+x+2yt+3zt2F1txyz=x2+2y+6z+4xyt+4y2+6xzt2+12yzt3+9z2t4Ftxyz.E6

Continuing this process, we can guess that

FN=tNFtxyz=i=02NaiNxyztiFtxyz,N=0,1,2.E7

Differentiating (7) with respect to t, we have

FN+1=FNt=i=02NaiNxyziti1Ftxyz+i=02NaiNxyztiF1txyz=i=02NaiNxyziti1Ftxyz+i=02NaiNxyztix+2yt+3zt2Ftxyz=i=02NiaiNxyzti1Ftxyz+i=02NxaiNxyztiFtxyz+i=02N2yaiNxyzti+1Ftxyz+i=02N3zaiNxyzti+2Ftxyz=i=02N1i+1ai+1NxyztiFtxyz+i=02NxaiNxyztiFtxyz+i=12N+12yai1NxyztiFtxyz+i=22N+23zai2NxyztiFtxyz

Hence we have

FN+1=i=02N1i+1ai+1NxyztiFtxyz+i=02NxaiNxyztiFtxyz+i=12N+12yai1NxyztiFtxyz+i=22N+23zai2NxyztiFtxyz.E8

Now replacing N by N+1 in (7), we find

FN+1=i=02N+2aiN+1xyztiFtxyz.E9

Comparing the coefficients on both sides of (8) and (9), we obtain

a0N+1xyz=a1Nxyz+xa0Nxyz,a1N+1xyz=2a2Nxyz+xa1Nxyz+2ya0Nxyz,a2NN+1xyz=xa2NNxyz+2ya2N1Nxyz+3za2N2Nxyz,a2N+1N+1xyz=2ya2NNxyz+3za2N1Nxyz,a2N+2N+1xyz=3za2NNxyz,E10

and

aiN+1xyz=i+1ai+1Nxyz+xaiNxyz+2yai1Nxyz+3zai2Nxyz,2i2N1.E11

In addition, by (7), we have

Ftxyz=F0txyz=a00xyzFtxyz,E12

which gives

a00xyz=1.E13

It is not difficult to show that

xFtxy+2ytFtxyz+3zt2Ftxyz=F1txyz=i=02ai1xyzFtxyz=a01xyz+a1(1xyz)t+a2(1xyz)t2Ftxyz.E14

Thus, by (14), we also find

a01xyz=x,a11xyz=2y,a21xyz=3z.E15

From (10), we note that

a0N+1xyz=a1Nxyz+xa0Nxyz,a0Nxyz=a1N1xyz+xa0N1xyz,a0N+1xyz=i=0Nxia1Nixyz+xN+1,E16

and

a2N+2N+1xyz=3za2NNxyz,a2NNxyz=3za2N2N1xyz,a2N+2N+1xyz=3zN+1.E17

Note that, here the matrix aijxy0i2N+2,0jN+1 is given by

1x2y+x202y4xy+6z03z6xz+4y20012yz003z20000003z300003zN+1

Therefore, we obtain the following theorem.

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

FN=tNFtxyz=i=0NaiNxyztiFtxyz
has a solution
F=Ftxyz=ext+yt2+zt3,

where

a0N+1xyz=i=0Nxia1Nixyz+xN+1,a1N+1xyz=2a2Nxyz+xa1Nxyz+2ya0Nxyz,a2NN+1xyz=xa2NNxyz+2ya2N1Nxyz+3za2N2Nxyz,a2N+1N+1xyz=2ya2NNxyz+3za2N1Nxyz,a2N+2N+1xyz=3zN+1,
and
aiN+1xyz=i+1ai+1Nxyz+xaiNxyz+2yai1Nxyz+3zai2Nxyz,2i2N1.

From (4), we note that

FN=tNFtxyz=k=0Hk+Nxyztkk!.E18

By (4) and (18), we get

enttNFtxyz=m=0nmtmm!m=0Hm+Nxyztmm!=m=0k=0mmknmkHN+k(xyz)tmm!.E19

By the Leibniz rule and the inverse relation, we have

enttNFtxyz=k=0NNknNktkentFtxyz=m=0k=0NNknNkHm+kxnyztmm!.E20

Hence, by (19) and (20), and comparing the coefficients of tmm! gives the following theorem.

Theorem 6. Let m,n,N be nonnegative integers. Then

k=0mmknmkHN+kxyz=k=0NNknNkHm+kxnyz.E21

If we take m=0 in (21), then we have the following corollary.

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

HNxyz=k=0NNknNkHkxnyz.

For N=0,1,2,, the differential equation

FN=tNFtxyz=i=0NaiNxyztiFtxyz

has a solution

F=Ftxyz=ext+yt2+zt3.

Here is a plot of the surface for this solution. In Figure 1(left), we choose 2z2, 1t1, x=2, and y=4. In Figure 1(right), we choose 5x5,1t1,y=3, and z=1.

3. Distribution of zeros of the 3-variable Hermite polynomials

This section aims to demonstrate the benefit of using numerical investigation to support theoretical prediction and to discover new interesting pattern of the zeros of the 3-variable Hermite polynomials Hnxyz. By using computer, the 3-variable Hermite polynomials Hnxyz can be determined explicitly. We display the shapes of the 3-variable Hermite polynomials Hnxyz and investigate the zeros of the 3-variable Hermite polynomials Hnxyz. We investigate the beautiful zeros of the 3-variable Hermite polynomials Hnxyz by using a computer. We plot the zeros of the Hnxyz for n=20, y=1,1, 1+i,1i, z=3,3,3+i,3i and xC (Figure 2). In Figure 2(top-left), we choose n=20, y=1, and z=3. In Figure 2(top-right), we choose n=20, y=1, and z=3. In Figure 2(bottom-left), we choose n=20, y=1+i, and z=3+i. In Figure 2(bottom-right), we choose n=20, y=1i, and z=3i.

In Figure 3(top-left), we choose n=20, x=1, and y=1. In Figure 3(top-right), we choose n=20, x=1, and y=1. In Figure 3(bottom-left), we choose n=20, x=1+i, and y=1+i. In Figure 3(bottom-right), we choose n=20, x=1i, and y=1i.

Stacks of zeros of the 3-variable Hermite polynomials Hnxyz for 1n20 from a 3-D structure are presented (Figure 3). In Figure 4(top-left), we choose n=20, y=1, and z=3. In Figure 4(top-right), we choose n=20, y=1, and z=3. In Figure 4(bottom-left), we choose n=20, y=1+i, and z=3+i. In Figure 4(bottom-right), we choose n=20, y=1i, and z=3i.

Our numerical results for approximate solutions of real zeros of the 3-variable Hermite polynomials Hnxyz are displayed (Tables 13).

Degree nReal zerosComplex zeros
110
202
312
422
514
624
734
826
936
1046
1138
1248
13310
14410

Table 1.

Numbers of real and complex zeros of Hnx,1,3.

Degree nReal zerosComplex zeros
110
220
312
422
532
624
734
844
936
1046
1156
1266
1358
1468

Table 2.

Numbers of real and complex zeros of Hnx13.

Degree nx
10
2
3− 1.8845
43.1286, −0.17159
5−4.5385
6−5.8490, −1.3476
7−7.1098, −2.1887, −0.36350
8−8.3241, −3.4645
9−9.4984,  − 4.6021,  − 1.1118
10−10.637,  − 5.7212,  − 1.5785, −0.61919
11−11.745,  − 6.8105,  − 2.8680
12−12.824,  − 7.8743,  − 3.8894,  − 0.99513

Table 3.

Approximate solutions of Hnx13=0,xR.

The plot of real zeros of the 3-variable Hermite polynomials Hnxyz for 1n20 structure are presented (Figure 5).

In Figure 5(left), we choose y=1 and z=3. In Figure 5(right), we choose y=1 and z=3.

Stacks of zeros of Hnx24 for 1n40, forming a 3D structure are presented (Figure 6). In Figure 6(top-left), we plot stacks of zeros of Hnx24 for 1n20. In Figure 6(top-right), we draw x and y axes but no z axis in three dimensions. In Figure 6(bottom-left), we draw y and z axes but no x axis in three dimensions. In Figure 6(bottom-right), we draw x and z axes but no y axis in three dimensions.

It is expected that Hnxyz,xC,y,zR, has Imx=0 reflection symmetry analytic complex functions (see Figures 27). We observe a remarkable regular structure of the complex roots of the 3-variable Hermite polynomials Hnxyz for y,zR. We also hope to verify a remarkable regular structure of the complex roots of the 3-variable Hermite polynomials Hnxyz for y,zR (Tables 1 and 2). Next, we calculated an approximate solution satisfying Hnxyz=0,xC. The results are given in Tables 3 and 4.

degree nx
10
2−1.4142, 1.4142
33.3681
40.16229, 5.0723
5−1.3404, 1.4745, 6.6661
62.9754, 8.1678
70.31213, 4.3783, 9.5946
8−1.2604, 1.5304, 5.7274, 10.959
92.8224, 7.0271, 12.270
100.44594, 4.0615, 8.2834, 13.535
11−1.1740, 1.5825, 5.2667, 9.5013, 14.760
12−1.4659, −0.87728, 2.7469, 6.4398, 10.685, 15.949

Table 4.

Approximate solutions of Hnx13=0,xR.

The plot of real zeros of the 3-variable Hermite polynomials Hnxyz for 1n20 structure are presented (Figure 7).

In Figure 7(left), we choose x=1 and y=2. In Figure 7(right), we choose x=1 and y=2.

Finally, we consider the more general problems. How many zeros does Hnxyz have? We are not able to decide if Hnxyz=0 has n distinct solutions. We would also like to know the number of complex zeros CHnxyz of Hnxyz,Imx0. Since n is the degree of the polynomial Hnxyz, the number of real zeros RHnxyz lying on the real line Imx=0 is then RHnxyz=nCHnxyz, where CHnxyz denotes complex zeros. See Tables 1 and 2 for tabulated values of RHnxyz and CHnxyz. The author has no doubt that investigations along these lines will lead to a new approach employing numerical method in the research field of the 3-variable Hermite polynomials Hnxyz which appear in mathematics and physics. The reader may refer to [2, 11, 13, 20] for the details.

Acknowledgments

This work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MEST) (No. 2017R1A2B4006092).

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Written By

Cheon Seoung Ryoo

Submitted: 25 October 2017 Reviewed: 24 January 2018 Published: 20 February 2018