Differential equations associated with generalized Bell polynomials ...

Open Math. 2016; 14: 807–815

Open Mathematics

Open Access

Research Article Cheon Seoung Ryoo*

Differential equations associated with generalized Bell polynomials and their zeros DOI 10.1515/math-2016-0075 Received August 2, 2016; accepted September 24, 2016.

Abstract: In this paper, we study differential equations arising from the generating functions of the generalized Bell

polynomials. We give explicit identities for the generalized Bell polynomials. Finally, we investigate the zeros of the generalized Bell polynomials by using numerical simulations. Keywords: Differential equations, Bell polynomials, Generalized Bell polynomials, Zeros MSC: 05A19, 11B83, 34A30, 65L99

1 Introduction Recently, many mathematicians have worked in the are of the Bernoulli numbers, Euler numbers, Genocchi numbers, and tangent numbers (see [1–9]). The moments of the Poisson distribution are well-known to be connected to the combinatorics of the Bell and Stirling numbers. As is well known, the Bell numbers Bn are given by the generating function 1 X tn t e .e 1/ D Bn : (1) nŠ nD0

The Bell polynomials Bn ./ are given by the generating function e .e

t

1/

1 X

D

Bn ./

nD0

tn : nŠ

(2)

The generalized Bell polynomials Bn .x; / are defined by the generating function 1 X

F D F .t; x; / D

Bn .x; /

nD0

tn D e xt nŠ

.e t

t

1/

(see [10]):

(3)

In particular the generalized Bell polynomials Bn .x; / D E Œ.Z C x /n ; ; x 2 R; n 2 N; where Z is a Poission random variable with parameter > 0 (see [10]). The first few examples of generalized Bell polynomials are B0 .x; / D 1;

B1 .x; / D x;

B2 .x; / D x 2

;

B3 .x; / D x

3

3x;

B4 .x; / D x

4

4x

6x 2 C 32 ;

B5 .x; / D x 5

5x

10x 2

10x 3 C 102 C 15x2 ;

B6 .x; / D x 6

6x

15x 2

20x 3

15x 4 C 252 C 60x2 C 45x 2 2

153 ;

*Corresponding Author: Cheon Seoung Ryoo: Department of Mathematics, Hannam University, Daejeon 306-791, Korea, E-mail: [email protected] © 2016 Ryoo, published by De Gruyter Open. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivs 3.0 License.

Unauthenticated Download Date | 11/3/16 1:22 AM

808

C.S. Ryoo

B7 .x; / D x 7

7x

21x 2

35x 3

C 210x 2 2 C 105x 3 2

35x 4

1053

21x 5 C 562 C 175x2

105x3 :

From (2) and (3), we see that 1 X nD0

tn Bn .x; / D e .xC/t e . nŠ D

1 X

n X

nD0

kD0

1 X

! tk D Bk . / kŠ kD0 ! ! n tn Bk . /.x C /n k : k nŠ

/.e t

1/

Comparing the coefficients on both sides of (4), we obtain ! n X n Bn .x; / D Bk . /.x C /n k

k

1 X

.x C /

mD0

mt

m

!

mŠ (4)

.n 0/:

(5)

kD0

Recently, many mathematicians have studied the differential equations arising from the generating functions of special polynomials (see [11–13]). In this paper, we study differential equations arising from the generating functions of generalized Bell polynomials. We give explicit identities for the generalized Bell polynomials. In addition, we investigate the zeros of the generalized Bell polynomials with numerical methods. Finally, we observe an interesting phenomenon of ‘scattering’ of the zeros of generalized Bell polynomials.

2 Differential equations associated with generalized Bell polynomials Differential equations arising from the generating functions of special polynomials are studied by many authors in order to give explicit identities for special polynomials (see [11–13]). In this section, we study differential equations arising from the generating functions of generalized Bell polynomials. Let 1 X tn t F D F .t; x; / D e xt .e t 1/ D Bn .x; / ; ; x; t 2 C: (6) nŠ nD0

Then, by (6), we have F .1/ D

F .2/ D

d d xt .et t 1/ t F .t; x; / D e D e xt .e t dt dt D .x C /F .t; x; / F .t; x C 1; /;

1/

.x

.e t

1//

d .1/ F D .x C /F .1/ .t; x; / F .1/ .t; x C 1; / dt D .x C /2 F .t; x; / .2x C 2 C 1/F .t; x C 1; / C 2 F .t; x C 2; /;

and F .3/ D

(7)

(8)

d .2/ F D .x C /2 F .t; x; / dt C . 1/ .x C /2 C .2x C 2 C 1/.x C 1 C / F .t; x C 1; / C . 1/2 2 .3x C 3 C 3/ F .t; x C 2; / C . 1/3 3 F .t; x C 3; /:

Continuing this process, we can guess that F

.N /

D

d dt

N F .t; x; / D

N X

. 1/i ai .N; x; /F .t; x C i; /; .N D 0; 1; 2; : : :/:

iD0


(9)

Differential equations associated with generalized Bell polynomials and their zeros

809

Taking the derivative with respect to t in (9), we have F .N C1/ D

N X dF .N / D . 1/i ai .N; x; /F .1/ .t; x C i; / dt i D0

D

N X

. 1/i ai .N; x; / f.x C i C /F .t; x C i; /

F .t; x C i C 1; /g

i D0

D

D

N X

i

. 1/ ai .N; x; /.x C i C /F .t; x C i; / C

(10)

N X

i C1

. 1/

i D0

iD0

N X

N C1 X

. 1/i ai .N; x; /.x C i C /F .t; x C i; / C

i D0

ai .N; x; /F .t; x C i C 1; /

. 1/i ai

1 .N; x; /F .t; x

C i; /:

iD1

On the other hand, by replacing N by N C 1 in (9), we get F .N C1/ D

N C1 X

. 1/i ai .N C 1; x/F .t; x C i; /:

(11)

i D0

Comparing the coefficients on both sides of (10) and (11), we obtain a0 .N C 1; x; / D .x C /a0 .N; x; /;

aN C1 .N C 1; x; / D aN .N; x; /;

(12)

and ai .N C 1; x; / D ai

1 .N; x; /

C .x C i C /ai .N; x; /; .1 i N /:

(13)

In addition, by (9), we get F .t; x; / D F .0/ .t; x; / D a0 .0; x; /F .t; x; /:

(14)

a0 .0; x; / D 1:

(15)

By (14), we get It is not difficult to show that .x C /F .t; x; /

F .t; x C 1; / D F .1/ .t; x; / D

1 X

. 1/i ai .1; x; /F .t; x C i; /

iD0

D a0 .1; x; /F .t; x; /

(16)

a1 .1; x; /F .t; x C 1; /:

Thus, by (16), we also get a0 .1; x; / D x C ;

a1 .1; x; / D :

(17)

From (12), we note that a0 .N C 1; x; / D .x C /a0 .N; x; / D D .x C /N a0 .1; x; / D .x C /N C1 ;

(18)

aN C1 .N C 1; x; / D aN .N; x; / D D N a1 .1; x; / D N C1 :

(19)

and For i D 1; 2; 3 in (13), we have a1 .N C 1; x; / D

N X

.x C 1 C /k a0 .N

k; x; /;

(20)

k; x; /;

(21)

kD0

a2 .N C 1; x; / D

N X1

.x C 2 C /k a1 .N

kD0


810

C.S. Ryoo

and a3 .N C 1; x; / D

N X2

.x C 3 C /k a2 .N

k; x; /:

(22)

kD0

Continuing this process, we can deduce that, for 1 i N; ai .N C 1; x; / D

NX i C1

.x C i C /k ai

1 .N

k; x; /:

(23)

kD0

Here, we note that the matrix ai .j; x; /0i;j N C1 is given by 0 1 x C .x C /2 .x C /3 B0 B B0 0 2 B B B0 0 0 3 B :: :: :: B :: @: : : : 0 0 0 0

1 .x C /N C1 C C C C C C C :: C :: A : : N C1

Now, we give explicit expressions for ai .N C 1; x; /. By (20), (21) and (22), we get N X

a1 .N C 1; x; / D

.x C 1 C /k1 a0 .N

k1 ; x; / D

.x C 1 C /k1 .x C /N

k1

;

k1 D0

k1 D0

a2 .N C 1; x; / D

N X

N X1

.x C 2 C /k2 a1 .N

k2 ; x; /

k2 D0

D 2

N X1 N

1 k2 X

k2

k1

1

.x C 3 C /k3 .x C 2 C /k2 .x C 1 C /k1 .x C /N

k3

k2

k2 D0

.x C 2 C /k2 .x C 1 C /k1 .x C /N

;

k1 D0

and a3 .N C 1; x; / D

N X2

.x C 3 C /k3 a2 .N

k3 ; x; /

k3 D0

D 3

N X2 N k3 D0

2 k3 N X

k2 D0

2X k3

k2 k1

2

:

k1 D0

Continuing this process, we have ai .N C 1; x; / Di

NX i C1 N ki D0

i Y

iC1 X ki ki

N

i C1X ki

1 D0

k2

k1 D0

! kl

.x C l C /

.x C /

N

iC1

(24) Pi

lD1 kl

:

lD1

Therefore, by (24), we obtain the following theorem. Theorem 2.1. For N D 0; 1; 2; : : : ; the differential equations F

.N /

D

N X

i

. 1/ ai .N; x; /F .t; x C i; / D

N X

! i

. 1/ ai .N; x; /e

it

F .t; x; /

i D0

iD0

have a solution F D F .t; x; / D e xt

.e t

t

1/

;



811

where a0 .N; x; / D .x C /N ; aN .N; x; / D N i

ai .N; x; / D

N Xi N

i ki X

ki D0 ki

N

i

1 D0

ki X

k2

k1 D0

From (6), we note that F .N / D

i Y

! .x C l C /

kl

.x C /N

i

Pi

lD1

kl

;

.1 i N /:

lD1

1 X d N tk F .t; x; / D BkCN .x; / : dt kŠ

(25)

kD0

From Theorem 2.1 and (25), we can derive the following equation: 1 X kD0

N X

tk D F .N / D BkCN .x; / kŠ D

N X

! i

. 1/ ai .N; x; /e

it

F

i D0 i

. 1/ ai .N; x; /

1 X

iD0

lD0

N X

1 X

tl i lŠ

1 X

!

l

mD0

k X

tm Bm .x; / mŠ

! (26)

! tk D . 1/ ai .N; x; / Bm .x; / kŠ kD0 mD0 iD0 ! ! k 1 N X X X k k m tk i . 1/i ai .N; x; /Bm .x; / D : m kŠ kD0

i

! k k i m

m

iD0 mD0

By comparing the coefficients on both sides of (26), we obtain the following theorem. Theorem 2.2. For k; N D 0; 1; 2; : : : ; we have BkCN .x; / D

N X k X i D0 mD0

! k k i m

m

. 1/i ai .N; x; /Bm .x; /;

(27)

where a0 .N; x; / D .x C /N ; aN .N; x; / D N ai .N; x; / D i

N Xi N

i ki X 1 D0

ki D0 ki

N

i

ki X

k2

k1 D0

i Y

! .x C l C /kl

.x C /N

i

Pi

lD1

kl

;

.1 i N /:

lD1

Let us take k D 0 in (27). Then, we have the following corollary. Corollary 2.3. For N D 0; 1; 2; : : : ; we have BN .x; / D

N X

. 1/i ai .N; x; /:

i D0

For N D 0; 1; 2; : : : ; the functional equations F

.N /

D

N X

i

. 1/ ai .N; x; /F .t; x C i; / D

i D0

N X

! i

. 1/ ai .N; x; /e

it

F .t; x; /

i D0

have a solution F D F .t; x; / D e xt

.e t

t

1/

:


812

C.S. Ryoo

Here is a plot of the surface for this solution. In Figure 1 (left), we choose 3 x 3; 1 t 1; and D 3 x 3; 1 t 1; and D 4.

4. In Figure 1 (right), we choose

Fig. 1. The surface for the solution F .t; x; /

3 Zeros of the generalized Bell 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 generalized Bell polynomials Bn .x; /. By using computer, the generalized Bell polynomials Bn .x; / can be determined explicitly. We display the shapes of the generalized Bell polynomials Bn .x; / and investigate the zeros of the generalized Bell polynomials Bn .x; /. For n D 1; ; 10, we can draw a plot of the generalized Bell polynomials Bn .x; /, respectively. This shows the ten plots combined into one. We display the shape of Bn .x; /, 10 x 10; D 4 (Figure 2). Fig. 2. Zeros of Bn .x; /

We investigate the beautiful zeros of the generalized Bell polynomials Bn .x; / by using a computer. We plot the zeros of the Bn .x; / for n D 5; 10; 15; 20; D 4; and x 2 C (Figure 3). In Figure 3 (top-left), we choose n D 5 and D 4. In Figure 3 (top-right), we choose n D 10 and D 4. In Figure 3 (bottom-left), we choose n D 15 and D 4 . In Figure 3 (bottom-right), we choose n D 20 and D 4. Prove that Bn .x; /; x 2 C, has I m.x/ D 0 reflection symmetry analytic complex functions (see Figure 3). Stacks of zeros of the generalized Bell polynomials Bn .x; / for 1 n 20; D 4 from a 3-D structure are presented (Figure 4).



813

Fig. 3. Zeros of Bn .x; /

Fig. 4. Stacks of zeros of Bn .x; /; 1 n 20

Our numerical results for approximate solutions of real zeros of the generalized Bell polynomials Bn .x; / are displayed (Tables 1, 2). Plot of real zeros of Bn .x; / for 1 n 20 structure are presented (Figure 5).


814

C.S. Ryoo

Table 1. Numbers of real and complex zeros of Bn .x; 4/ degree n

real zeros

complex zeros

1

1

0

2

2

0

3

3

0

4

4

0

5

5

0

6

6

0

7

7

0

8

6

2

9

7

2

10

8

2

11

9

2

12

10

2

13

9

4

14

10

4

Fig. 5. Real zeros of Bn .x; / for 1 n 20

We observe a remarkably regular structure of the complex roots of the generalized Bell polynomials Bn .x; /. We hope to verify a remarkably regular structure of the complex roots of the generalized Bell polynomials Bn .x; / (Table 1). Next, we calculated an approximate solution satisfying Bn .x; / D 0; x 2 C. The results are given in Table 2. Table 2. Approximate solutions of Bn .x; 4/ D 0; x 2 R degree n

x

1 2 3

0 -2.0000,

2.0000

3.62008, -3.28357, -0.336509

4

5.04407, -4.20888, -1.91657, 1.08138

5

6.34241, -4.89805, -3.12253, 2.3597, -0.681527

6

7.55109, -5.3997, -4.10205, 3.54357, -2.0558, 0.462889

7

8.69145, -5.70673, -4.95736, 4.65759, -3.19025, 1.54067, -1.03537

8

5.71699, -4.16654, 2.56659, -2.28486, -0.0564486

Finally, we shall consider the more general problems. How many zeros does Bn .x; / have? Prove or disprove: Bn .x; / D 0 has n distinct solutions (see Table 2). Find the numbers of complex zeros CBn .x;/ of



815

Bn .x; /; I m.x/ ¤ 0: Since n is the degree of the polynomial Bn .x; /, the number of real zeros RBn .x;/ lying on the real line I m.x/ D 0 is then RBn .x;/ D n CBn .x;/ , where CBn .x;/ denotes complex zeros. See Table 1 for tabulated values of RBn .x;/ and CBn .x;/ . The author has no doubt that investigations along this line will lead to a new approach employing numerical method in the research field of the generalized Bell polynomials Bn .x; / to appear in mathematics and physics. The reader may refer to [14, 15] for the details.

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