Differential equations associated with generalized Bell polynomials and their zeros

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.


Introduction
Recently, many mathematicians have worked in the are of the Bernoulli numbers, Euler numbers, Genocchi numbers, and tangent numbers (see [1][2][3][4][5][6][7][8][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 e .e t 1/ D 1 X nD0 B n t n nŠ : The Bell polynomials B n . / are given by the generating function e .e t 1/ D 1 X nD0 B n . / t n nŠ : The generalized Bell polynomials B n .x; / are defined by the generating function F D F .t; x; / D 1 X nD0 B n .x; / t n nŠ D e xt .e t t 1/ (see [10]): In particular the generalized Bell polynomials B n .x; / D E OE.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 B 0 .x; / D 1; B 1 .x; / D x; B 2 .x; / D x 2 ; From (2) and (3), we see that Comparing the coefficients on both sides of (4), we obtain Recently, many mathematicians have studied the differential equations arising from the generating functions of special polynomials (see [11][12][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.

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][12][13]). In this section, we study differential equations arising from the generating functions of generalized Bell polynomials. Let Then, by (6) and x C 3; /: Continuing this process, we can guess that .N D 0; 1; 2; : : :/: Taking the derivative with respect to t in (9), we have On the other hand, by replacing N by N C 1 in (9), we get Comparing the coefficients on both sides of (10) and (11), we obtain and In addition, by (9), we get By (14), we get a 0 .0; x; / D 1: It is not difficult to show that Thus, by (16), we also get a 0 .1; x; / D x C ; a 1 .1; x; / D : From (12), we note that and For i D 1; 2; 3 in (13), we have and Continuing this process, we can deduce that, for 1 Ä i Ä N; Here, we note that the matrix a i .j; x; / 0Äi;j ÄN C1 is given by : : : : : : : : : : : : : : : : : : Now, we give explicit expressions for a i .N C 1; x; /. By (20), (21) and (22), we get Continuing this process, we have Therefore, by (24), we obtain the following theorem.
From (6), we note that From Theorem 2.1 and (25), we can derive the following equation: By comparing the coefficients on both sides of (26), we obtain the following theorem.
Let us take k D 0 in (27). Then, we have the following corollary. Here is a plot of the surface for this solution.

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; /. By using computer, the generalized Bell polynomials B n .x; / can be determined explicitly. We display the shapes of the generalized Bell polynomials B n .x; / and investigate the zeros of the generalized Bell polynomials B n .x; /. For n D 1; ; 10, we can draw a plot of the generalized Bell polynomials B n .x; /, respectively. This shows the ten plots combined into one. We display the shape of B n .x; /, 10 Ä x Ä 10; D 4 ( Figure 2).  Our numerical results for approximate solutions of real zeros of the generalized Bell polynomials B n .x; / are displayed (Tables 1, 2).
Plot of real zeros of B n .x; / for 1 Ä n Ä 20 structure are presented ( Figure 5).  We observe a remarkably regular structure of the complex roots of the generalized Bell polynomials B n .x; /. We hope to verify a remarkably regular structure of the complex roots of the generalized Bell polynomials B n .x; / (Table 1). Next, we calculated an approximate solution satisfying B n .x; / D 0; x 2 C. The results are given in Table 2. Finally, we shall consider the more general problems. How many zeros does B n .x; / have? Prove or disprove: B n .x; / D 0 has n distinct solutions (see Table 2). Find the numbers of complex zeros C B n .x; / of B n .x; /; I m.x/ ¤ 0: Since n is the degree of the polynomial B n .x; /, the number of real zeros R B n .x; / lying on the real line I m.
x/ D 0 is then R B n .x; / D n C B n .x; / , where C B n .x; / denotes complex zeros. See Table 1 for tabulated values of R B n .x; / and C B n .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 B n .x; / to appear in mathematics and physics. The reader may refer to [14,15] for the details.