Differential equations associated with generalized Bell polynomials and their zeros

Seoung Cheon Ryoo

doi:10.1515/math-2016-0075

40% Rabatt

auf Fachbücher bei De Gruyter Brill *

Artikel Open Access

Differential equations associated with generalized Bell polynomials and their zeros

Seoung Cheon Ryoo

Veröffentlicht/Copyright: 29. Oktober 2016

Veröffentlicht von

Veröffentlichen auch Sie bei De Gruyter Brill

Manuskript einreichen Informationen für Autor*innen Erkunden Sie dieses Fachgebiet

$Open Mathematics$

Aus der Zeitschrift Open Mathematics Band 14 Heft 1

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 2010: 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 B_n are given by the generating function

(1)e(et−1)=∑n=0∞Bntnn!.

The Bell polynomials B_n(λ) are given by the generating function

(2)eλ(et−1)=∑n=0∞Bn(λ)tnn!.

The generalized Bell polynomials B_n(x, λ) are defined by the generating function

(3)F=F(t,x,λ)=∑n=0∞Bn(x,λ)tnn!=ext−λ(et−t−1)(see[10]).

In particular the generalized Bell polynomials B_n(x, −λ) = E_λ[(Z+x−λ)ⁿ], λ, x ∈ ℝ, n ∈ ℕ where Z is a Poission random variable with parameter λ > 0 (see 10). The first few examples of generalized Bell polynomials are

B0(x,λ)=1,B1(x,λ)=x,B2(x,λ)=x2−λ,B3(x,λ)=x3−λ−3xλ,B4(x,λ)=x4−λ−4xλ,−6x2λ+3λ2,B5(x,λ)=x5−λ−5xλ,−10x2λ−10x3λ+10λ2+15xλ2,B6(x,λ)=x6−λ−6xλ,−15x2λ−20x3λ+15x4λ+25xλ2+60λ2+45x2λ2−15λ3,B7(x,λ)=x7−λ−7xλ,−21x2λ−35x3λ+35x4λ+21x5λ+56λ2+175xλ2+210x2λ2+105x3λ2−105λ3−105xλ3.

From (2) and (3), we see that

(4)∑n=0∞Bn(x,λ)tnn!=e(x+λ)te(−λ)(et−1)=(∑k=0∞Bk(−λ)tkk!)(∑m=0∞Bk(x+λ)mtmm!)=∑n=0∞(∑k=0n(nk)Bk(−λ)(x+λ)n−k)tnn!.

Comparing the coefficients on both sides of (4), we obtain

(5)Bn(x,λ)=∑k=0n(nk)Bk(−λ)(x+λ)n−k(n≥0).

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

(6)F=F(t,x,λ) = ext−λ(et−t−1) = ∑n=0∞Bn(x,λ)tnn!, λ,x,t ∈ ℂ.

Then, by (6), we have

(7)F(1)=ddtF(t,x,λ)=ddt(ext−λ(et−t−1))=ext−λ(et−t−1)(x−λ(et−1))=(x+λ)F(t,x,λ)−λF(t,x+1,λ),

(8)F(2)=ddtF(1)=(x+λ)F(1)(t,x,λ)−λF(1)(t,x+1,λ)=(x+λ)2F(t,x,λ)−λ(2x+2λ+1)(t,x+1,λ)+λ2F(t,x+2,λ),

and

F3=ddtF(2)=(x+λ)2F(t,x,λ)+(−1)λ((x+λ)2+(2x+2λ+1)(x+1+λ))F(t,x+1,λ)+(−1)2λ2(3x+3λ+3)F(t,x+2,λ)+(−1)3λ3F(t,x+3,λ).

Continuing this process, we can guess that

(9)F(N)=(ddt)NF(t,x,λ)=∑i=0N(−1)iai(N,x,λ)F(t,x+i,λ),(N=0,1,2,...).

Taking the derivative with respect to t in (9), we have

(10)F(N+1)=dFNdt=∑!i=0N−1iaiN,x,λF1t,x+i,λ=∑i=0N−1iaiN,x,λx+i+λFt,x+i,λ−λFt,x+i+1,λ=∑i=0N−1iaiN,x,λx+i+λFt,x+i,λ+∑i=0N−1i+1λaiN,x,λFt,x+i+1,λ=∑i=0N−1iaiN,x,λx+i+λFt,x+i,λ+∑i=1N+1−1iλai−1N,x,λFt,x+i+1,λ.

On the other hand, by replacing N by N + 1 in (9), we get

(11)F(N+1)=∑i=0N+1(−1)iai(N+1,x)F(t,x+i,λ).

Comparing the coefficients on both sides of (10) and (11), we obtain

(12)a0(N+1,x,λ)=(x+λ)a0(N,x,λ),aN+1(N+1,x,λ)=λaN(N,x,λ),

and

(13)ai(N+1,x,λ)=λai−1(N,x,λ)+(x+i+λ)ai(N,x,λ),(1≤i≤N).

In addition, by (9), we get

(14)F(t,x,λ)=F(0)(t,x,λ)=a0(0,x,λ)F(t,x,λ).

By (14), we get

(15)a0(0,x,λ)=1.

It is not difficult to show that

(16)(x+λ)F(t,x,λ)−λF(t,x+1,λ)=F(1)(t,x,λ)=∑i=01(−1)iai(1,x,λ)F(t,x+i,λ)=a0(1,x,λ)F(t,x,λ)−a1(1,x,λ)F(t,x+1,λ).

Thus, by (16), we also get

(17)a0(1,x,λ)=x+λ,a1(1,x,λ)=λ.

From (12), we note that

(18)a0(N+1,x,λ)=(x+λ)a0(N,x,λ)=⋯=x+λNa0(1,x,λ)=x+λN+1,

and

(19)aN+1(N+1,x,λ)=λaN(N,x,λ)=⋯=λNa1(1,x,λ)=λN+1.

For i = 1, 2, 3 in (13), we have

(20)a1(N+1,x,λ)=λ∑k=0N⁡(x+1+λ)ka0(N−k,x,λ),

(21)a2(N+1,x,λ)=λ∑k=0N−1⁡(x+2+λ)ka1(N−k,x,λ),

and

(22)a3(N+1,x,λ)=λ∑k=0N−2⁡(x+3+λ)ka2(N−k,x,λ).

Continuing this process, we can deduce that, for 1 ≤ i ≤ N,

(23)ai(N+1,x,λ)=∑k=0N−i+1⁡(x+i+λ)kai−1(N−k,x,λ).

Here, we note that the matrix a_i (j,x,λ)_{0≤i,j≤N+1} is given by

(1x+λ(x+λ)2(x+λ)3⋯(x+λ)N+10λ⋅⋅⋯⋅00λ2⋅⋯⋅000λ3⋯⋅⋮⋮⋮⋮⋱⋮0000⋯λN+1)

Now, we give explicit expressions for a_i (N + 1, x, λ). By (20), (21) and (22), we get

a1(N+1,x,λ)=λ∑K1=0N(x+1+λ)k1a0(N−k1,x,λ)=λ∑k1=0N(x+1+λ)k1(x+λ)N−k1,a2(N+1,x,λ)=λ∑K2=0N−1(x+2+λ)k2a1(N−k2,x,λ)=λ2∑K2=0N−1∑k1=0N−1−k2(x+2+λ)k2(x+1+λ)k1(x+λ)N−k2−k1−1,

and

a3(N+1,x,λ)=λ∑k3=0N−2(x+2+λ)k3a2(N−k3,x,λ)=λ3∑k3=0N−2∑k2=0N−2−k3∑k1=0N−2−K3−k2(x+2+λ)k3(x+2+λ)k2(x+2+λ)k1(x+λ)N−k3−k2−k1−2.

Continuing this process, we have

(24)ai(N+1,x,λ)=λi∑ki=0N−i+1∑ki−1=0N−i+1−ki⋯∑k1=0N−i+1−ki−⋯−k2.(∏l=1i(x+l+λ)kl)(x+λ)N−i+1=l=1ikl.

Therefore, by (24), we obtain the following theorem.

Theorem 2.1

For N = 0, 1, 2, . . . , the differential equations

F(N)=∑i=0N(−1)iai(N,x,λ)F(t,x+i,λ)=(∑i=0N(−1)iai(N,x,λ)eit)F(t,x,λ)

have a solution

F=F(t,x,λ)=ext−λ(et−t−1),

where

a0(N,x,λ)=(x+λ)N,aN(N,x,λ)=(λ)Nai(N,x,λ)=λi∑ki=0N−i∑ki−1=0N−i−ki⋯∑k1=0N−i−ki−⋯−k2(∏l=1i(x+l+λ)kl)(x+λ)N−i−∑l=1ikl,(1≤i≤N).

From (6), we note that

(25)F(N)=(ddt)NF(t,x,λ)=∑k=0∞⁡Bk+N(x,λ)tkk!.

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

(26)∑k=0∞Bk+N(x,λ)tkk!=F(N)=(∑i=0N(−1)iai(N,x,λ)eit)F=∑i=0N(−1)iai(N,x,λ)(∑l=0∞iltll!)(∑m=0∞Bm(x,λ)tmm!)=∑i=0N(−1)iai(N,x,λ)(∑k=0∞∑m=0k(km)ik−mBm(x,λ)tkk!)=∑k=0∞(∑i=0N∑m=0k(km)ik−m(−1)iai(N,x,λ)Bm(x,λ)tkk!).

By comparing the coefficients on both sides of (26), we obtain the following theorem.

Theorem 2.2

For k, N = 0, 1, 2, . . . , we have

(27)Bk+N(x,λ)=∑i=0N⁡∑m=0k⁡(km)ik−m(−1)iai(N,x,λ)Bm(x,λ),

where

a0(N,x,λ)=(x+λ)N,aN(N,x,λ)=(λ)Nai(N,x,λ)=λi∑ki=0N−i∑ki−1=0N−i−ki⋯∑k1=0N−i−ki−⋯−k2(∏l=1i(x+l+λ)kl)(x+λ)N−i−∑l=1ikl,(1≤i≤N).

Let us take k = 0 in (27). Then, we have the following corollary.

Corollary 2.3

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

BN(x,λ)=∑i=0N(−1)iai(N,x,λ).

For N = 0, 1, 2, . . . , the functional equations

F(N)=∑i=0N(−1)iai(N,x,λ)F(t,x+i,λ)=(∑i=0N(−1)iai(N,x,λ)eit)F(t,x,λ)

have a solution

F=F(t,x,λ)=ext−λ(et−t−1).

Here is a plot of the surface for this solution.

In Figure 1 (left), we choose −3 ≤ x ≤ 3, −1 ≤ t ≤ 1, and λ = −4. In Figure 1 (right), we choose −3 ≤ x ≤ 3, −1 ≤ t ≤ 1, and λ = 4.

$Fig. 1 The surface for the solution F (t, x, λ)$

Fig. 1

The surface for the solution F (t, x, λ)

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 B_n(x, λ). By using computer, the generalized Bell polynomials B_n(x, λ) can be determined explicitly. We display the shapes of the generalized Bell polynomials B_n(x, λ) and investigate the zeros of the generalized Bell polynomials B_n(x, λ). For n = 1, ..., 10, we can draw a plot of the generalized Bell polynomials B_n(x, λ), respectively. This shows the ten plots combined into one. We display the shape of B_n(x, λ), −10 ≤ x ≤ −10, λ = 4 (Figure 2).

$Fig. 2 Zeros of Bn(x, λ)$

Fig. 2

Zeros of B_n(x, λ)

We investigate the beautiful zeros of the generalized Bell polynomials B_n(x, λ) by using a computer. We plot the zeros of the B_n(x, λ) for n = 5, 10, 15, 20, λ = 4, and x ∈ ℂ (Figure 3).

$Fig. 3 Zeros of Bn(x, λ)$

Fig. 3

Zeros of B_n(x, λ)

In Figure 3 (top-left), we choose n = 5 and λ = 4. In Figure 3 (top-right), we choose n = 10 and λ = 4. In Figure 3 (bottom-left), we choose n = 15 and λ = 4 . In Figure 3 (bottom-right), we choose n = 20 and λ = 4. Prove that B_n(x, λ), x ∈ ℂ, has I m(x) = 0 reflection symmetry analytic complex functions (see Figure 3). Stacks of zeros of the generalized Bell polynomials B_n(x, λ) for 1 ≤ n ≤ 20, λ = 4 from a 3-D structure are presented (Figure 4).

$Fig. 4 Stacks of zeros of Bn(x, λ), 1 ≤ n ≤ 20$

Fig. 4

Stacks of zeros of B_n(x, λ), 1 ≤ n ≤ 20

Our numerical results for approximate solutions of real zeros of the generalized Bell polynomials B_n(x, λ) are displayed (Tables 1, 2).

Table 1

Numbers of real and complex zeros of B_n(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

Table 2

Approximate solutions of B_n(x, 4) = 0, x ∈ ℝ

degree n	x
1	0
2	-2.0000, 2.0000
3	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

Plot of real zeros of B_n(x, λ) for 1 ≤ n ≤ 20 structure are presented (Figure 5).

$Fig. 5 Real zeros of Bn(x, λ) for 1 ≤ n ≤ 20$

Fig. 5

Real zeros of B_n(x, λ) for 1 ≤ n ≤ 20

We observe a remarkably regular structure of the complex roots of the generalized Bell polynomials B_n(x, λ). We hope to verify a remarkably regular structure of the complex roots of the generalized Bell polynomials B_n(x, λ) (Table 1). Next, we calculated an approximate solution satisfying B_n(x, λ) = 0, x ∈ ℂ. The results are given in Table 2.

Finally, we shall consider the more general problems. How many zeros does B_n(x, λ) have? Prove or disprove: B_n(x, λ) = 0 has n distinct solutions (see Table 2). Find the numbers of complex zeros C_{B_n(x,λ)} of B_n(x, λ), I m(x) ≠ 0. Since n is the degree of the polynomial B_n(x, λ), the number of real zeros R_{B_n}(x,λ) lying on the real line I m(x) = 0 is then R_{B_n}(x,λ) = n − C_{B_n}(x,λ), where C_{B_n}(x,λ) denotes complex zeros. See Table 1 for tabulated values of R_{B_n}(x, λ) and C_{B_n}(x,λ). 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 B_n(x, λ) to appear in mathematics and physics. The reader may refer to [14, 15] for the details.

References

[1] Acikgöz, M, Erdal, D., Araci, S., A new approach to q-Bernoulli numbers and q-Bernoulli polynomials related to q-Bernstein polynomials, Advances in Difference Equations, 2010, Article ID 951764, 9 pages10.1186/1687-1847-2010-951764Suche in Google Scholar

[2] Bayad, A., Kim, T., Higher recurrences for Apostal-Bernoulli-Euler numbers, Russ. J. Math. Phys. 2012 19(1), 1-1010.1134/S1061920812010013Suche in Google Scholar

[3] A. Erdelyi, A., Magnus, W., Oberhettinger,F., Tricomi, F. G., Higher Transcendental Functions, 1981, Vol 3. New York: KriegerSuche in Google Scholar

[4] Kang, J.Y., Lee, H.Y., Jung, N.S., Some relations of the twisted q-Genocchi numbers and polynomials with weight ˛ and weak weight ˇ , Abstract and Applied Analysis, 2012, Article ID 860921, 9 pages10.1155/2012/860921Suche in Google Scholar

[5] Kim, M.S., Hu, S., On p-adic Hurwitz-type Euler Zeta functions, J. Number Theory, 2012, 132, 2977-301510.1016/j.jnt.2012.05.037Suche in Google Scholar

[6] Roman, S., The umbral calculus, Pure and Applied Mathematics, 111, Academic Press, Inc. [Harcourt Brace Jovanovich Publishes]. New York, 1984Suche in Google Scholar

[7] Ozden, H., Simsek, Y., A new extension of q-Euler numbers and polynomials related to their interpolation functions, Appl. Math. Letters, 2008, 21, 934-93810.1016/j.aml.2007.10.005Suche in Google Scholar

[8] Simsek, Y., Complete sum of products of .h; q/-extension of Euler polynomials and numbers, Journal of Difference Equations and Applications, 2010, 16(11), 1331-134810.1080/10236190902813967Suche in Google Scholar

[9] Robert, A.M., A Course in p-adic Analysis, Graduate Text in Mathematics, 2000, Vol. 198, Springer10.1007/978-1-4757-3254-2Suche in Google Scholar

[10] Privault, N. Genrealized Bell polynomials and the combinatorics of Poisson central moments, The Electronic Journal of Combinatorics, 2011, 18, #5410.37236/541Suche in Google Scholar

[11] Kim, T., Kim, D.S., Ryoo, C. S., Kwon, H. I., Differential equations associated with Mahler and Sheffer-Mahler polynomials, submitted for publicationSuche in Google Scholar

[12] Kim, T., Kim, D.S., Identities involving degenerate Euler numbers and polynomials arising from non-linear differential equations, J. Nonlinear Sci. Appl., 2016, 9, 2086-209810.22436/jnsa.009.05.14Suche in Google Scholar

[13] Ryoo, C.S., Differential equations associated with tangent numbers, J. Appl. Math. & Informatics, 2016, 34(5-6), 487-49410.14317/jami.2016.487Suche in Google Scholar

[14] Agarwal, R.P., Kim, Y.H., Ryoo, C.S., Calculating zeros of the twisted Euler Polynomials, Neural Parallel Sci. Comput., 2008, 16, 505-516Suche in Google Scholar

[15] Ryoo, C.S., Kim, T., Agarwal, R.P., A numerical investigation of the roots of q-polynomials, Inter. J. Comput. Math., 2006, 83(2), 223-23410.1080/00207160600654811Suche in Google Scholar

Received: 2016-8-2

Accepted: 2016-9-24

Published Online: 2016-10-29

Published in Print: 2016-1-1

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.