On Generalized Bell Polynomials

It is shown that the sequence of the generalized Bell polynomials Sn(x) is convex under some restrictions of the parameters involved. A kind of recurrence relation for Sn(x) is established, and some numbers related to the generalized Bell numbers and their properties are investigated.


Introduction
Hsu and Shiue 1 defined a kind of generalized Stirling number pair with three free parameters which is introduced via a pair of linear transformations between generalized factorials, viz, where n ∈ N set of nonnegative integers , α, β, and γ may be real or complex numbers with α, β, γ / 0, 0, 0 , and t | α n denotes the generalized factorial of the form In particular, t | 1 n t n with t 0 1.Various well-known generalizations were obtained by special choices of the parameters α, β, and γ cf. 1 , and the generalization of some properties of the classical Stirling numbers such as the recurrence relations the congruence relation, and a kind of asymptotic expansion was established.As a followup study of these numbers, more properties were obtained in 2 .Furthermore, some combinatorial interpretations of S n, k; α, β, γ were given in 3 in terms of occupancy distribution and drawing of balls from an urn.Hsu and Shiue 1 also defined a kind of generalized exponential polynomials S n x ≡ S n x; α, β, γ in terms of generalized Stirling numbers S n, k; α, β, γ with α, β, and γ real or complex numbers as follows: 1.6 We may call these polynomials generalized Bell polynomials.Note that when x 1, we get 1.9 Note that, when α → 0, 1 αt γ/α → exp γt .Hence, Also obtained by Hsu and Shiue is an explicit formula for S n x of the form Consequently, with x 1, we have Note that, by taking α 0, 1.16 gives the explicit formula for r, β -Bell polynomial.When x 1, this gives a kind of the Dobinski formula for r, β -Bell numbers.This reduces further to the Dobinski formula for r-Bell numbers 4 when β 1.Moreover, with γ 0, we get which is the Dobinski formula for the ordinary Bell numbers 5 .
In this paper, a recurrence relation and convexity of the generalized Bell numbers will be established and some numbers related to W n will be investigated.Some theorems on r, β -Bell polynomials will be established including the asymptotic approximation of the r, β -Bell numbers.

More Properties of S n x
Recurrence relation is one of the useful tools in constructing tables of values.The recurrence relation for the ordinary Bell numbers 6 is given by with initial condition B 0 1.Carlitz's Bell numbers 7 also satisfy the recurrence relation: with A 0 λ 1.Note that for λ 1, A n 1 B n and 2.2 will reduce to 2.1 .Moreover, Mező 4 obtained certain recurrence relations for the r-Bell polynomials, respectively, as The following theorem will generalize all of these recurrence relations.
Theorem 2.1.The generalized exponential polynomials satisfy the following recurrence relation: with S 0 x 1.Moreover, the generalized Bell numbers W n S n 1 satisfy Proof.Differentiating both sides of 1.8 with respect to t will give Applying binomial theorem and Cauchy's rule for product of two power series will yield

2.7
Comparing the coefficients of t n /n!, we obtain which is precisely equivalent to 1.10 .
Mathematicians have been aware for quite a while that the global behaviour of combinatorial sequences can be used in asymptotic estimates.One of these interesting behaviours is convexity 5 .A real sequence v k , k 0, 1, 2, . . . is called convex on an interval a, b containing at least 3 consecutive integers when

2.13
For instance, the sequence of binomial coefficients n k satisfies the convexity property since

2.14
This implies that The next theorem asserts that the sequence of generalized exponential polynomials as well as the generalized Bell numbers is convex under some restrictions.

3.1
The numbers A n, k; α, β, γ were given combinatorial interpretation in 2 , for nonnegative integers α, β, and γ, as the number of ways to distribute n distinct balls, one ball at a time, into k 1 distinct cells, first k of which has β distinct compartments and the last cell with γ distinct compartments such that i the compartments in each cell are given cyclic ordered numbering, ii the capacity of each compartment is limited to one ball, iii each successive α available compartments in a cell can only have the leading compartment getting the ball, iv the first k cells are nonempty.
If k 1 cells will be changed to any number of cells with the last cell containing γ distinct compartments and the rest of the cells each has β distinct compartments such that only the last cell could be empty, then this gives the combinatorial interpretation of B n α, β, γ .
The following theorem contains a kind of exponential generating function for B n α, β, γ .
Theorem 3.1.The numbers B n α, β, γ have the following exponential generating function: Proof.Using the exponential generating function in 1.4 , we get

3.3
This is exactly the desired generating function.
Differentiating both sides of 1.9 with respect to t, we yield

3.4
Since A n, k; α, β, γ vanishes when k 0 and k > n, we have

3.5
This result is embodied in the following theorem.
Theorem 3.2.The number B n α, β, γ is equal to The next theorem provides a recurrence relation for the number B n α, β, γ which can be used as a quick tool in computing its first values.
Proof.Making use of 3.6 , we have 3.8 Summing up both sides from j 0 to n − 1 and using Vandermonde's formula, we get 3.9 Hence, we have Now, by 3.6 , 12 which is precisely equivalent to 3.7 .
Note that when n 1, 3.7 gives

3.14
This implies that

3.15
The following theorem gives a kind of congruence relation for B n α, β, γ with the restriction that α → 0. We use G n,β,r to denote the following limit:

3.16
Theorem 3.4.Let r and β be integers.Then for any odd prime p and n ≥ 1, one has the following congruence relation: 3.17 Proof.Note that the explicit formula in 1.14 can be expressed in terms of a kth difference operator.That is, where Δ k denotes the kth difference operator.Hence,

3.23
This completes the proof of the theorem.

Some Theorems on r, β -Bell Polynomials
The r, β -Bell polynomials G n,β,r x have already possessed numerous properties.Some of them are obtained as special case of the properties of S n x .However, there are properties of the ordinary Bell numbers or r-Bell numbers which are difficult to establish in S n x but can be done in G n,β,r x .For instance, using the rational generating function for S n, k; β, r in 2 which is given by

4.2
It can easily be shown that

4.4
This can be expressed further as where 1 F 1 is the hypergeometric function which is defined by where a i j a i a i 1 a i 2 • • • a i j − 1 .Applying Kummer's formula 8 , we obtain the following generating function.
Theorem 4.1.The r, β -Bell polynomials satisfy the following generating function: It will be interesting if one can also obtain a generating function of this form for S n x .Now, using the integral identity in 9 , Im π 0 e je iθ sin nθ dθ π 2 and the explicit formula in 1.14 , we get e βj r e iθ sin nθ dθ ⎤ ⎦ e re iθ sin nθ dθ k! e re iθ sin nθ dθ.

4.14
It will also be compelling to establish such integral representation for S n x .
The Bell polynomials B n λ are known to be connected to the Poisson distribution.More precisely, B n λ can be expressed in terms of the moment of the Poisson random variable Z with parameter λ > 0 as

4.15
The exponential generating function for the r, β -Bell polynomials in 1.12 can be written as follows: e r/β βt e x/β e βt −1 e r/β βt E x/β e βt Z n≥0

4.16
Hence, we can also express the r, β -Bell polynomials in terms of the following moment:

4.18
Thus, we have the following theorem.S k, j λ j .

4.22
This is exactly the identity obtained by Privault in 10 .

An Asymptotic Approximation for G n,β,r
Using the exponential generating function for G n,r,β in 1.12 with x 1 and Cauchy's theorem for integrals, we obtain the integral representation

5.11
Observe that F θ is analytic at θ 0.Thus F θ has a Maclaurin series expansion about θ 0. This Maclaurin expansion can be written in the form

5.12
where we define ρ to be the operator ρ R dθ/dR .Choose R such that Re βR rR − n 0; that is, R satisfies the equation xe βR rx − n 0. This R is shown to exist in the following lemma.

Lemma 5.2.
There exists a unique positive real solution to the equation xe βR rx − n 0.
Proof.We can rewrite the given equation in the form x n − rx e −βx .

5.13
The desired solution is the x-coordinate of the intersection of the functions h x x/ n − rx and g x e −βx .
It can be seen from the preceding lemma that R → ∞ as n → ∞.With this choice of R, we have β −1 ρ k e βR rR iθ k .

5.14
We now introduce the following notations:
We have defined z as a function of R.However, for the moment we consider z to be an independent variable and expand e f z into a convergent Maclaurin series expansion of the form 5.17 where b 0 e f 0 1, b 1 e f 0 f 0 a 1 , and b 2 a 2 a 2 1 /2 .

Lemma 5.3.
There is a constant R o such that for all R > R o ,

5.18
Proof.We see that as R → ∞.From this, it follows that there is a constant R o satisfying 5.18 .Now, it will follow from Lemma 5.3 that the radius of convergence of 5.17 becomes large when θ is near zero.Thus, z e −βR/2 is within the domain of convergence.

5.33
Taking the first two terms of the asymptotic expansion of 5.32 , we have

Theorem 4 . 3 .
The r, β -Bell polynomials equal the Bell polynomials B n y, λ , defined by Privault 10 as in terms of the r, β -Bell polynomials as B n y, λ G n,1,λ y −λ .