On Generalized Bell Polynomials

Abstract

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

1. 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, (𝑡∣𝛼)_𝑛=^𝑛_𝑘=0𝑆𝑛,𝑘;𝛼,𝛽,𝛾𝑡−𝛾∣𝛽_𝑘,𝑡∣𝛽_𝑛=^𝑛_𝑘=0𝑆𝑛,𝑘;𝛽,𝛼,−𝛾𝑡+𝛾∣𝛼_𝑘, (1.1) where 𝑛∈𝑁 (set of nonnegative integers), 𝛼,𝛽,and𝛾 may be real or complex numbers with (𝛼,𝛽,𝛾) ≠ (0, 0, 0), and (𝑡∣𝛼)_𝑛 denotes the generalized factorial of the form (𝑡∣𝛼)_𝑛=^𝑛−1_𝑗=0𝑡−𝑗𝛼,𝑛≥1,(𝑡∣𝛼)₀=1. (1.2) 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 relations𝑆𝑛+1,𝑘;𝛼,𝛽,𝛾=𝑆𝑛,𝑘−1;𝛼,𝛽,𝛾+𝑘𝛽−𝑛𝛼+𝛾𝑆𝑛,𝑘;𝛼,𝛽,𝛾, (1.3) the exponential generating function(1+𝛼𝑡)^𝛾/𝛼(1+𝛼𝑡)^𝛽/𝛼−1𝛽^𝑘=𝑘!_𝑛≥0𝑆𝑡𝑛,𝑘;𝛼,𝛽,𝛾^𝑛𝑛!, (1.4) the explicit formula𝑆=1𝑛,𝑘;𝛼,𝛽,𝛾𝛽^𝑘𝑘!^𝑘_𝑗=0(−1)^𝑘−𝑗𝑘𝑗𝛽𝑗+𝛾∣𝛼_𝑛, (1.5) the congruence relation, and a kind of asymptotic expansion was established. As a follow-up 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,and𝛾 real or complex numbers as follows: 𝑆_𝑛(𝑥)=^𝑛_𝑘=0𝑆𝑛,𝑘;𝛼,𝛽,𝛾𝑥^𝑘. (1.6) We may call these polynomials generalized Bell polynomials. Note that when 𝑥=1, we get 𝑊_𝑛=𝑆_𝑛(1)=^𝑛_𝑘=0𝑆𝑛,𝑘;𝛼,𝛽,𝛾, (1.7) the generalized Bell numbers. A kind of generating function of the sequence {𝑆_𝑛(𝑥)} for the generalized exponential polynomials has been established by Hsu and Shiue, viz,_𝑛≥0𝑆_𝑛(𝑥)𝑡^𝑛=()𝑛!1+𝛼𝑡^𝛾/𝛼exp(1+𝛼𝑡)^𝛽/𝛼𝑥−1𝛽, (1.8) where 𝛼,𝛽≠0. In particular, (1.8) gives the generating function for the generalized Bell numbers:_𝑛≥0𝑊_𝑛𝑡^𝑛=()𝑛!1+𝛼𝑡^𝛾/𝛼()exp1+𝛼𝑡^𝛽/𝛼−1𝛽. (1.9) Note that, when 𝛼→0, (1+𝛼𝑡)^𝛾/𝛼→exp(𝛾𝑡). Hence,(1+𝛼𝑡)^𝛾/𝛼exp(1+𝛼𝑡)^𝛽/𝛼𝑥−1𝛽⟶𝑒^𝛾𝑡exp𝑒^𝛽𝑡𝑥−1𝛽. (1.10) If we define the polynomial 𝐺_{𝑛,𝛽,𝑟}(𝑥) as 𝐺_{𝑛,𝛽,𝑟}(𝑥)=lim_𝛼→0𝑆_𝑛𝑥;𝛼,𝛽,𝑟, (1.11) then its exponential generating function is given by_𝑛≥0𝐺_{𝑛,𝛽,𝑟}(𝑥)𝑡^𝑛𝑒𝑛!=exp𝑟𝑡+^𝛽𝑡𝑥−1𝛽. (1.12) We may call 𝐺_{𝑛,𝛽,𝑟}(𝑥) the (𝑟,𝛽)-Bell polynomial. Hence, with 𝑥=1, this yields the exponential generating function for the (𝑟,𝛽)-Bell numbers. Now, if we use 𝑆(𝑛,𝑘;𝛽,𝛾) to denote the following limit: 𝑆𝑛,𝑘;𝛽,𝛾=lim_𝛼→0𝑆𝑛,𝑘;𝛼,𝛽,𝛾, (1.13) then, by (1.5),𝑆=1𝑛,𝑘;𝛽,𝛾𝛽^𝑘𝑘!^𝑘_𝑗=0(−1)^𝑘−𝑗𝑘𝑗𝛽𝑗+𝛾^𝑛, (1.14)𝐺_{𝑛,𝛽,𝑟}(𝑥)=^𝑛_𝑘=0𝑆𝑛,𝑘;𝛽,𝛾𝑥^𝑘. (1.15)

Also obtained by Hsu and Shiue is an explicit formula for 𝑆_𝑛(𝑥) of the form𝑆_𝑛(𝑥)=1𝑒^∞𝑥/𝛽_𝑘=0𝑥/𝛽^𝑘𝑘!𝑘𝛽+𝛾∣𝛼_𝑛. (1.16) Consequently, with 𝑥=1, we have 𝑊_𝑛=1𝑒^∞1/𝛽_𝑘=0𝑘𝛽+𝛾∣𝛼_𝑛𝛽^𝑘𝑘!. (1.17) Note that, by taking 𝛼=0, (1.16) gives𝐺_{𝑛,𝛽,𝑟}(𝑥)=1𝑒^∞𝑥/𝛽_𝑘=0𝑥/𝛽^𝑘𝑘!𝑘𝛽+𝛾^𝑛, (1.18) the explicit formula for (𝑟,𝛽)-Bell polynomial. When 𝑥=1, this gives 𝐺_{𝑛,𝛽,𝑟}=1𝑒^∞1/𝛽_𝑘=01/𝛽^𝑘𝑘!𝑘𝛽+𝛾^𝑛, (1.19) 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 𝐵_𝑛=1𝑒^∞_𝑘=0𝑘𝑘!, (1.20) 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.

2. 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 by𝐵_𝑛+1=^𝑛_𝑘=0𝑛𝑘𝐵_𝑛−𝑘, (2.1) with initial condition 𝐵₀=1. Carlitz’s Bell numbers [7] also satisfy the recurrence relation:𝐴_𝑛+1(𝜆)=−𝜆𝑛𝐴_𝑛(𝜆)+^𝑛_𝑘=0𝑛𝑘𝜇𝑘𝑘!𝜆^𝑘𝐴_𝑛−𝑘(𝜆)1,𝜇=𝜆, (2.2) with 𝐴₀(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, as𝐵_𝑛,𝑟(𝑥)=𝑟𝐵_{𝑛−1,𝑟}(𝑥)+𝑥^𝑛−1_𝑘=0𝑘𝐵𝑛−1_𝑘,𝑟(𝑥). (2.3) The following theorem will generalize all of these recurrence relations.

Theorem 2.1. The generalized exponential polynomials satisfy the following recurrence relation: 𝑆_𝑛+1(𝑥)=𝛾−𝛼𝑛𝑆_𝑛(𝑥)+^𝑛_𝑘=0𝑥𝑛𝑘𝛽∣𝛼_𝑘𝑆_𝑛−𝑘(𝑥) (2.4) with 𝑆₀(𝑥)=1. Moreover, the generalized Bell numbers 𝑊_𝑛=𝑆_𝑛(1) satisfy 𝑊_𝑛+1=𝛾−𝛼𝑛𝑊_𝑛+^𝑛_𝑘=0𝑛𝑘𝛽∣𝛼_𝑘𝑊_𝑛−𝑘. (2.5)

Proof. Differentiating both sides of (1.8) with respect to 𝑡 will give _𝑛≥0𝑆_𝑛(𝑥)𝑡^𝑛−1()!=()𝑛−11+𝛼𝑡^𝛾/𝛼exp(1+𝛼𝑡)^𝛽/𝛼𝑥−1𝛽(1+𝛼𝑡)^𝛽/𝛼𝑥+𝛾1+𝛼𝑡. (2.6) Applying binomial theorem and Cauchy’s rule for product of two power series will yield (1+𝛼𝑡)_𝑛≥0𝑆_𝑛(𝑥)𝑡^𝑛−1()!=𝑛−1_𝑛≥0𝑆_𝑛(𝑥)𝑡^𝑛𝑛!_𝑛≥0𝛽𝛼𝑛𝑥𝛼^𝑛𝑡^𝑛+𝛾_𝑛≥0,𝑆_𝑛(𝑥)𝑡^𝑛−1(𝑛−1)!+_𝑛≥0𝑛𝛼𝑆_𝑛(𝑥)𝑡^𝑛𝑛!=_𝑛≥0^𝑛_𝑘=0𝑛𝑘𝛽𝑥𝑘!𝛼𝑘𝛼^𝑘𝑆_𝑛−𝑘(𝑥)𝑡^𝑛.𝑛! (2.7) Comparing the coefficients of 𝑡^𝑛/𝑛!, we obtain 𝑆_𝑛+1(𝑥)+𝛼𝑛𝑆_𝑛(𝑥)=𝛾𝑆_𝑛(𝑥)+^𝑛_𝑘=0𝑛𝑘𝛽𝑥𝑘!𝛼𝑘𝛼^𝑘𝑆_𝑛−𝑘(𝑥), (2.8) which is precisely equivalent to (1.10).

By taking 𝛼=0, Theorem 2.1 yields the recurrence relations for the (𝑟,𝛽)-Bell polynomials. More precisely, 𝐺_{𝑛+1,𝛽,𝑟}(𝑥)=𝑟𝐺_{𝑛,𝛽,𝑟}(𝑥)+^𝑛_𝑘=0𝑥𝑛𝑘𝛽^𝑘𝐺_{𝑛−𝑘,𝛽,𝑟}(𝑥). (2.9) These further give (2.3) when 𝛽=1. Surely, (2.2) can be deduced from (2.5) by letting (𝛼,𝛽,𝛾)=(𝜆,1,0). Furthermore, for (𝛼,𝛽,𝛾)=(0,0,1), (2.4) gives 𝐵_𝑛+1(𝑥)=2𝐵_𝑛(𝑥), (2.10) where 𝐵_𝑛(𝑥)=∑^𝑛_𝑘=0(^𝑛_𝑘)𝑥^𝑘. If we let 𝐵_𝑛=𝐵_𝑛(1), we get 𝐵_𝑛+1=2𝐵_𝑛, (2.11) which implies ^𝑛𝑘=0𝑘𝑛+1=2^𝑛+1−1, (2.12) the number of distinct partitions of an (𝑛+2)-set into 2 nonempty subsets, or simply 𝑆(𝑛+2,2), the classical Stirling number of the second kind.

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 𝑣_𝑘≤12𝑣_𝑘−1+𝑣_𝑘+1,𝑘∈[𝑎+1,𝑏−1]. (2.13) For instance, the sequence of binomial coefficients (^𝑛_𝑘) satisfies the convexity property since 𝑘𝑘+𝑛𝑘=𝑛+2𝑛−2𝑛+1𝑘−2>0,for𝑘≥2. (2.14) This implies that𝐵_𝑛+1≤12𝐵_𝑛+𝐵_𝑛+2, (2.15) that is, 𝐵_𝑛 is convex.

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

Theorem 2.2. The sequence of generalized exponential polynomials 𝑆_𝑛(𝑥) with 𝑥>0,𝛼≤0, and 𝛽,𝛾≥0 possesses the convexity property, viz, 𝑆_𝑛+1(𝑥)≤12𝑆_𝑛(𝑥)+𝑆_𝑛+2(𝑥),𝑛=1,2,…. (2.16)

Proof. Since 𝛼≤0 and (𝑘𝛽+𝛾−𝑛𝛼)≥0, we have 0≤1−𝑘𝛽+𝛾−𝑛𝛼²,+−𝛼𝑘𝛽+𝛾−𝑛𝛼0≤1−2𝑘𝛽+𝛾−𝑛𝛼𝑘𝛽+𝛾−𝑛𝛼²,2.−𝛼𝑘𝛽+𝛾−𝑛𝛼𝑘𝛽+𝛾−𝑛𝛼≤1+𝑘𝛽+𝛾−𝑛𝛼𝑘𝛽+𝛾−𝑛𝛼−𝛼 (2.17) Multiplying both sides by (𝑘𝛽+𝛾∣𝛼)_𝑛, we get 2𝑘𝛽+𝛾∣𝛼_𝑛+1≤𝑘𝛽+𝛾∣𝛼_𝑛+𝑘𝛽+𝛾∣𝛼_𝑛+2. (2.18) Thus, making use of (1.16), we obtain (2.16).

Note that, for (𝛼,𝛽,𝛾,𝑥)=(0,𝛽,𝑟,1), (2.16) asserts the convexity of (𝑟,𝛽)-Bell polynomials which further imply the convexity of 𝑟-Bell polynomials when 𝛽=1. Moreover, letting (𝛼,𝛽,𝛾,𝑥)=(0,1,0,1), (2.16) yields (2.15) and implies the convexity of 𝐵_𝑛.

3. A Variation of Generalized Bell Numbers

Let us denote 𝐴(𝑛,𝑘;𝛼,𝛽,𝛾)=𝑘!𝛽^𝑘𝑆(𝑛,𝑘;𝛼,𝛽,𝛾) and define 𝐵_𝑛𝛼,𝛽,𝛾=^𝑛_𝑘=1𝐴𝑛,𝑘;𝛼,𝛽,𝛾. (3.1) The numbers 𝐴(𝑛,𝑘;𝛼,𝛽,𝛾) were given combinatorial interpretation in [2], for nonnegative integers 𝛼,𝛽,and𝛾, 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.

Illustration of (iii)
Suppose the first ball lands in compartment 3 of cell 2. The compartment numbered 4, 5, 6,…, 𝛼,𝛼+1,𝛼+2 will be closed. And suppose the second ball lands in compartment 𝛽−2 also of cell 2. Then compartments numbered 𝛽−1,𝛽,1,2,𝛼+3,𝛼+4,𝛼+5,…,2𝛼−3 of cell 2 will be closed.

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: _𝑛≥0𝐵_𝑛𝑡𝛼,𝛽,𝛾^𝑛=()𝑛!1+𝛼𝑡^𝛾/𝛼2−(1+𝛼𝑡)^𝛽/𝛼. (3.2)

Proof. Using the exponential generating function in (1.4), we get _𝑛≥0𝐵_𝑛𝑡𝛼,𝛽,𝛾^𝑛𝑛!=_{𝑛≥0𝑘≥0}𝛽^𝑘𝑡𝑘!𝑆𝑛,𝑘;𝛼,𝛽,𝛾^𝑛=()𝑛!1+𝛼𝑡^𝛾/𝛼_𝑘≥0(1+𝛼𝑡)^𝛽/𝛼−1^𝑘=(1+𝛼𝑡)^𝛾/𝛼11−(1+𝛼𝑡)^𝛽/𝛼.−1 (3.3) This is exactly the desired generating function.

Differentiating both sides of (1.9) with respect to 𝑡, we yield 𝐴=𝑑𝑛,𝑘;𝛼,𝛽,𝛾^𝑛𝑑𝑡^𝑛(1+𝛼𝑡)^𝛾/𝛼(1+𝛼𝑡)^𝛽/𝛼−1^𝑘_𝑡=0. (3.4) Since 𝐴(𝑛,𝑘;𝛼,𝛽,𝛾) vanishes when 𝑘=0 and 𝑘>𝑛, we have 𝐵_𝑛𝛼,𝛽,𝛾=^∞_𝑘=0𝑑^𝑛𝑑𝑡^𝑛(1+𝛼𝑡)^𝛾/𝛼(1+𝛼𝑡)^𝛽/𝛼−1^𝑘_𝑡=0=𝑑^𝑛𝑑𝑡^𝑛(1+𝛼𝑡)^𝛾/𝛼2−(1+𝛼𝑡)^{𝛽/𝛼−1}_𝑡=0=12𝑑^𝑛𝑑𝑡^𝑛(1+𝛼𝑡)^∞𝛾/𝛼_𝜈=012(1+𝛼𝑡)^{𝛽/𝛼𝜈}_𝑡=0=12^∞_𝜈=0𝑑^𝑛𝑑𝑡^𝑛(1+𝛼𝑡)^{(𝛾+𝛽𝜈)/𝛼}_𝑡=012^𝜈. (3.5) This result is embodied in the following theorem.

Theorem 3.2. The number 𝐵_𝑛(𝛼,𝛽,𝛾) is equal to 𝐵_𝑛=1𝛼,𝛽,𝛾2^∞_𝜈=0𝛾+𝛽𝜈∣𝛼_𝑛2^−𝜈,𝑛≥1. (3.6)

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.

Theorem 3.3. The following recurrence relation holds: 𝐵_𝑛𝛼,𝛽,𝛾=(𝛾∣𝛼)_𝑛+𝛽∣𝛼_𝑛+^𝑛−1_𝑗=1𝑛𝑗𝛽∣𝛼_𝑗𝐵_𝑛−𝑗𝛼,𝛽,𝛾, (3.7) where 𝑛≥1.

Proof. Making use of (3.6), we have 𝛽𝛼𝑗𝐵_𝑛−𝑗𝛼,𝛽,𝛾𝛼^𝑛−𝑗!=1𝑛−𝑗2^∞_𝜈=0𝛽𝛼𝑗𝛽𝜈+𝛾𝛼2𝑛−𝑗^−𝜈. (3.8) Summing up both sides from 𝑗=0 to 𝑛−1 and using Vandermonde’s formula, we get ^𝑛−1_𝑗=0𝑛𝑗𝛽∣𝛼_𝑗𝐵_𝑛−𝑗𝛼,𝛽,𝛾𝛼^𝑛=1𝑛!2^∞_𝜈=0^𝑛−1_𝑗=0𝛽𝛼𝑗𝛽𝜈+𝛾𝛼2𝑛−𝑗^−𝜈=12^∞_𝜈=0𝛽+𝜈𝛽+𝛾𝛼𝑛2^−𝜈−12^∞_𝜈=0𝛽𝛼𝑛2^−𝜈. (3.9) Hence, we have ^𝑛−1_𝑗=0𝑛𝑗𝛽∣𝛼_𝑗𝐵_𝑛−𝑗=1𝛼,𝛽,𝛾2^∞_𝜈=0(𝜈+1)𝛽+𝛾∣𝛼_𝑛2^−𝜈−𝛽∣𝛼_𝑛12^∞_𝜈=02^−𝜈. (3.10) Now, by (3.6), 12^∞_𝜈=0𝛽(𝜈+1)+𝛾∣𝛼_𝑛2^−𝜈=^∞_𝜈=0𝛽(𝜈+1)+𝛾∣𝛼_𝑛2^−(𝜈+1)=^∞_𝜈=0𝛽𝜈+𝛾∣𝛼_𝑛2^−𝜈−𝛾∣𝛼_𝑛=2𝐵_𝑛𝛼,𝛽,𝛾−𝛾∣𝛼_𝑛 (3.11) and (1/2)∑^∞_𝜈=02^−𝜈=1. Thus, ^𝑛−1_𝑗=1𝑛𝑗𝛽∣𝛼_𝑗𝐵_𝑛−𝑗𝛼,𝛽,𝛾=𝐵_𝑛𝛼,𝛽,𝛾−𝛾∣𝛼_𝑛−𝛽∣𝛼_𝑛 (3.12) which is precisely equivalent to (3.7).

Note that when 𝑛=1, (3.7) gives 𝐵₁𝛼,𝛽,𝛾=𝛾+𝛽, (3.13) while (3.6) gives 𝐵₁𝛼,𝛽,𝛾=𝛾+𝛽^∞_𝜈=1𝜈2^𝜈+1. (3.14) This implies that ^∞_𝜈=1𝜈2^𝜈+1=1. (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: 𝐺_{𝑛,𝛽,𝑟}=lim_𝛼→0𝐵_𝑛𝛼,𝛽,𝛾. (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: 𝐺_{𝑛+𝑝−1,𝛽,𝑟}−𝐺_{𝑛,𝛽,𝑟}≡0mod2𝑝. (3.17)

Proof. Note that the explicit formula in (1.14) can be expressed in terms of a 𝑘th difference operator. That is, Δ^𝑘𝛽𝑡+𝑟^𝑛_𝑡=0=^𝑘_𝑗=0(−1)^𝑘−𝑗𝑘𝑗𝛽𝑗+𝑟^𝑛, (3.18) where Δ^𝑘 denotes the 𝑘th difference operator. Hence, 𝐺_{𝑛+𝑝−1,𝛽,𝑟}=^∞_𝑘=0Δ^𝑘𝛽𝑡+𝑟^{𝑛+𝑝−1}_𝑡=0. (3.19) Thus, 𝐺_{𝑛+𝑝−1,𝛽,𝑟}−𝐺_{𝑛,𝛽,𝑟}=^∞_𝑘=0Δ^𝑘𝛽𝑡+𝑟^𝑛−1𝛽𝑡+𝑟^𝑝−𝛽𝑡+𝑟_𝑡=0. (3.20) Since, by Fermat’s little theorem, (𝛽𝑡+𝑟)^𝑝−(𝛽𝑡+𝑟) is divisible by p, 𝛽𝑡+𝑟^𝑛−1𝛽𝑡+𝑟^𝑝−𝛽𝑡+𝑟=𝑝𝑥, (3.21) for some integer 𝑥. Also, since (𝛽𝑡+𝑟)^𝑛 and (𝛽𝑡+𝑟)^𝑝−1−1 are of different parity, (𝛽𝑡+𝑟)^𝑛[(𝛽𝑡+𝑟)^𝑝−1−1] is divisible by 2. Hence, 𝛽𝑡+𝑟^𝑛𝛽𝑡+𝑟^𝑝−1−1=2𝑝𝑦, (3.22) for some integer 𝑦. Thus, we have 𝛽𝑡+𝑟^𝑛𝛽𝑡+𝑟^𝑝−1−1≡0mod2𝑝. (3.23) This completes the proof of the theorem.

4. 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_𝑛≥𝑘𝑆𝑛,𝑘;𝛽,𝑟𝑡^𝑛=𝑡^𝑘∏^𝑘_𝑗=01−𝛽𝑗+𝑟𝑡, (4.1) we can have _𝑛≥0𝑆𝑛,𝑘;𝛽,𝑟𝑡^𝑛=1𝛽^𝑘+1𝑡1∏^𝑘_𝑖=0)/=1(1−𝑟𝑡𝛽𝑡−𝑖𝛽^𝑘+1𝑡1(1−𝑟𝑡)/𝛽𝑡∏^𝑘_𝑖=1()/=1−𝑟𝑡𝛽𝑡−𝑖−1𝛽^𝑘()()𝑟𝑡−1−1^𝑘∏^𝑘_𝑖=1()/.𝑟𝑡−1𝛽𝑡+𝑖 (4.2) It can easily be shown that ^𝑘_𝑖=1𝑟𝑡−1=𝛽𝑡−𝑖𝛽+𝑟𝑡−1𝛽𝑡_𝑘. (4.3)

Thus, _{𝑘≥0𝑛≥0}𝑆𝑛,𝑘;𝛽,𝑟𝑡^𝑛𝑥^𝑘=_𝑘≥0−1𝛽^𝑘()()𝑟𝑡−1−1^𝑘𝛽+𝑟𝑡−1/𝛽𝑡_𝑘𝑥^𝑘,_{𝑛≥0𝑛𝑘=0}𝑆𝑛,𝑘;𝛽,𝑟𝑥^𝑘𝑡^𝑛=−1𝑟𝑡−1_𝑘≥0(1)_𝑘𝛽+𝑟𝑡−1/𝛽𝑡_𝑘−𝑥/𝛽^𝑘.𝑘! (4.4)

This can be expressed further as _𝑛≥0𝐺_{𝑛,𝛽,𝑟}(𝑥)𝑡^𝑛=−1⋅𝑟𝑡−1¹𝐹₁1𝛽+𝑟𝑡−1𝛽𝑡−𝑥𝛽, (4.5)

where ₁𝐹₁ is the hypergeometric function which is defined by _𝑝𝐹_𝑞𝑎₁,𝑎₂,…,𝑎_𝑝𝑏₁,𝑏₂,…,𝑏_𝑞𝑡=^∞_𝑘=0𝑎_1𝑘𝑎_2𝑘⋯𝑎_𝑝𝑘𝑏_1𝑘𝑏_2𝑘⋯𝑏_𝑞𝑘𝑡^𝑘𝑘!, (4.6)

where (𝑎_𝑖)_𝑗=𝑎_𝑖(𝑎_𝑖+1)(𝑎_𝑖+2)⋯(𝑎_𝑖+𝑗−1). Applying Kummer’s formula [8], 𝑒^𝑥₁𝐹₁𝑎𝑏−𝑥=₁𝐹₁𝑏𝑥𝑏−𝑎, (4.7)

we obtain the following generating function.

Theorem 4.1. The (𝑟,𝛽)-Bell polynomials satisfy the following generating function: _𝑛≥0𝐺_{𝑛,𝛽,𝑟}(𝑥)𝑡^𝑛=−1⋅1𝑟𝑡−1𝑒^𝑥/𝛽⋅₁𝐹₁𝑟𝑡−1𝛽𝑡𝛽𝑡+𝑟𝑡−1𝑥𝛽𝑡𝛽. (4.8)

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^𝜋₀𝑒^{𝑗𝑒𝑖𝜃}()𝜋sin𝑛𝜃𝑑𝜃=2𝑗^𝑛𝑛!, (4.9) and the explicit formula in (1.14), we get 𝜋𝑆=12𝑛!𝑛,𝑘;𝛽,𝑟𝛽^𝑘𝑘!^𝑘_𝑗=0(−1)^𝑘−𝑗𝑘𝑗Im^𝜋₀𝑒^{(𝛽𝑗+𝑟)𝑒𝑖𝜃}()=1sin𝑛𝜃𝑑𝜃𝛽^𝑘𝑘!Im^𝜋₀^𝑘_𝑗=0(−1)^𝑘−𝑗𝑘𝑗𝑒^{𝛽𝑒𝑖𝜃𝑗}𝑒^{𝑟𝑒𝑖𝜃}sin(𝑛𝜃)𝑑𝜃=Im^𝜋₀𝑒^{𝛽𝑒𝑖𝜃}−1/𝛽^𝑘𝑒𝑘!^{𝑟𝑒𝑖𝜃}sin(𝑛𝜃)𝑑𝜃. (4.10) Hence, ^∞_𝑘=0𝑆𝑛,𝑘;𝛽,𝑟𝑥^𝑘=2𝑛!𝜋Im^𝜋₀^∞_𝑘=0𝑒^{𝛽𝑒𝑖𝜃}−1/𝛽^𝑘𝑥𝑘!^𝑘𝑒^{𝑟𝑒𝑖𝜃}()=sin𝑛𝜃𝑑𝜃2𝑛!𝜋Im^𝜋₀𝑒^{𝑥(𝑒𝛽𝑒𝑖𝜃−1)/𝛽}𝑒^{𝑟𝑒𝑖𝜃}sin(𝑛𝜃)𝑑𝜃. (4.11) Thus, 𝐺_{𝑛,𝛽,𝑟}(𝑥)=2𝑛!𝜋𝑒^𝑥/𝛽Im^𝜋₀𝑒^{𝑥𝛽−1𝑒𝛽𝑒𝑖𝜃}𝑒^{𝑟𝑒𝑖𝜃}sin(𝑛𝜃)𝑑𝜃, (4.12) where 𝛽≠0. By simple algebraic manipulation, this can further be expressed as follows.

Theorem 4.2. The (𝑟,𝛽)-Bell polynomials have the following integral representation: 𝐺_{𝑛,𝛽,𝑟}(𝑥)=2𝑛!𝜋𝑒^{𝑥/𝛽𝜋}₀𝑒^𝐽1(𝜃)sin𝐽₂(𝜃)sin(𝑛𝜃)𝑑𝜃, (4.13) where 𝐽₁(𝜃)=𝑟cos𝜃+𝑥𝑒^𝛽cos𝜃cos𝛽sin𝜃𝛽,𝐽₂(𝜃)=𝑟sin𝜃+𝑥𝑒^𝛽cos𝜃sin𝛽sin𝜃𝛽. (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 𝐵_𝑛(𝜆)=𝐸_𝜆𝑍^𝑛. (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)(𝑒𝛽𝑡−1)}=𝑒^{(𝑟/𝛽)𝛽𝑡}𝐸_𝑥/𝛽𝑒^{(𝛽𝑡)𝑍}=_𝑛≥0^𝑛_𝑘=0𝑛𝑘𝑟^𝑛−𝑘𝛽^𝑘𝐸_𝑥/𝛽𝑍^𝑘𝑡^𝑛.𝑛! (4.16) Hence, we can also express the (𝑟,𝛽)-Bell polynomials in terms of the following moment:𝐺_{𝑛,𝛽,𝑟}(𝑥)=𝐸_𝑥/𝛽𝛽𝑍+𝑟^𝑛. (4.17) Now, 𝐺_{𝑛,𝛽,𝑟}(𝑥)=^𝑛_𝑘=0𝑛𝑘𝑟^𝑛−𝑘𝛽^𝑘𝐸_𝑥/𝛽𝑍^𝑘=^𝑛_𝑘=0𝑛𝑘𝑟^𝑛−𝑘𝛽^𝑘𝐵_𝑘𝑥𝛽=^𝑛_𝑘=0𝑛𝑘𝑟^𝑛−𝑘𝛽^𝑘𝑘_𝑗=0𝑆𝑥𝑘,𝑗𝛽^𝑗. (4.18) Thus, we have the following theorem.

Theorem 4.3. The (𝑟,𝛽)-Bell polynomials equal 𝐺_{𝑛,𝛽,𝑟}(𝑥)=^𝑛_𝑘=0𝑛𝑘𝑟^{𝑘𝑛−𝑘}_𝑗=0𝛽^𝑘−𝑗𝑆𝑘,𝑗𝑥^𝑗. (4.19)

An extension of the Bell polynomials 𝐵_𝑛(𝑦,𝜆), defined by Privault [10] as ^∞_𝑛=0𝐵_𝑛𝑡𝑦,𝜆^𝑛𝑛!=𝑒^{𝑦𝑡−𝜆(𝑒𝑡−𝑡−1)}, (4.20) can be expressed in terms of the (𝑟,𝛽)-Bell polynomials as 𝐵_𝑛𝑦,𝜆=𝐺_{𝑛,1,𝜆+𝑦}(−𝜆). (4.21) Using Theorem 4.3, we obtain 𝐵_𝑛𝑦,−𝜆=𝐺_{𝑛,1,−𝜆+𝑦}(𝜆)=^𝑛_𝑘=0𝑛𝑘𝑦−𝜆^{𝑘𝑛−𝑘}_𝑗=0𝑆𝑘,𝑗𝜆^𝑗. (4.22) This is exactly the identity obtained by Privault in [10].

5. 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 𝐺_{𝑛,𝑟,𝛽}=𝑛!2𝜋𝑖_𝛾exp𝑟𝑧+𝑒^𝛽𝑧−1/𝛽𝑧^𝑛+1𝑑𝑧, (5.1) where 𝛾 is the circle 𝑧=𝑅𝑒^𝑖𝜃,−𝜋≤𝜃≤𝜋. Contour integration yields 𝐺_{𝑛,𝑟,𝛽}=𝑛!2𝜋𝑖𝑅^𝑛𝜋_−𝜋exp𝛽⁻¹𝑒^{𝛽𝑅𝑒𝑖𝜃}+𝑟𝑅𝑒^𝑖𝜃−𝑖𝑛𝜃−𝛽⁻¹𝑑𝜃, (5.2) which can be written into the compact form𝐺_{𝑛,𝑟,𝛽}=𝐴^𝜋_−𝜋exp(𝐹(𝜃))𝑑𝜃, (5.3) where 𝐴=𝑛!exp𝑟𝑅+𝛽⁻¹𝑒^𝛽𝑅−𝛽⁻¹2𝜋𝑅^𝑛,𝐹(𝜃)=𝛽⁻¹𝑒^{𝛽𝑅𝑒𝑖𝜃}+𝑟𝑅𝑒^𝑖𝜃−𝑖𝑛𝜃−𝑟𝑅+𝛽⁻¹𝑒^𝛽𝑅. (5.4) Define 𝜖=𝑒^−3𝑅/8 and let 𝐽₁=^𝜖_−𝜋exp(𝐹(𝜃))𝑑𝜃,𝐽₂=^𝜋_𝜖exp(𝐹(𝜃))𝑑𝜃. (5.5) Thus (5.3) can be written as 𝐺_{𝑛,𝑟,𝛽}=𝐴𝐽₁+𝐴^𝜖_𝜖exp(𝐹(𝜃))𝑑𝜃+𝐴𝐽₂. (5.6)

Lemma 5.1. There exists a constant 𝑘>0 such that 𝐽₂<𝑒^{−𝑘𝛽−1𝑒𝛽𝑅}(𝜋−𝜖). (5.7)

Proof. It can be shown that exp(𝐹(𝜃))=𝑒^{−[(𝑟𝑅+𝛽−1𝑒𝛽𝑅)+𝛽−1cos(𝛽𝑅sin𝜃)𝑒𝛽𝑅cos𝜃]}. (5.8) Since cos𝜃<1 for 0<𝜖<𝜃≤𝜋, we have exp(𝐹(𝜃))=𝑒^{−𝛽−1𝑒𝛽𝑅}1−cos𝛽𝑅sin𝜃. (5.9) Since [1−cos(𝛽𝑅sin𝜃)]>0 for cos𝜃<1 for 0<𝜖<𝜃≤𝜋, there exists a constant 𝑘>0 such that [1−cos(𝛽𝑅sin𝜃)]<𝑘. Hence 𝐽₂<𝑒^{−𝑘𝛽−1𝑒𝛽𝑅}(𝜋−𝜖). (5.10)

It will be seen later that 𝑅→∞ as 𝑛→∞. With the result in Lemma 5.1 we see that 𝐽₁ and 𝐽₂ will tend to zero as 𝑛→∞. Hence𝐺_{𝑛,𝑟,𝛽}∼𝐴^𝜖_−𝜖exp(𝐹(𝜃))𝑑𝜃. (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 form𝐹(𝜃)=𝑅𝑒^𝛽𝑅1+𝑟𝑅−𝑛𝑖𝜃+2𝛽𝑅²+𝑅𝑒^𝛽𝑅+𝑟𝑅𝑖²𝜃+^∞_𝑘=3𝛽⁻¹𝜌^𝑘𝑒^𝛽𝑅+𝑟𝑅(𝑖𝜃)^𝑘, (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 𝑥𝑛−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 𝐹(𝜃)1=−2𝛽𝑅²+𝑅𝑒^𝛽𝑅+𝑟𝑅𝜃+^∞_𝑘=3𝛽⁻¹𝜌^𝑘𝑒^𝛽𝑅+𝑟𝑅(𝑖𝜃)^𝑘. (5.14) We now introduce the following notations:𝜙=(1/2)𝛽𝑅²𝑒^𝛽𝑅+𝑅𝑒^𝛽𝑅+𝑟𝑅^1/2𝑎𝜃,_𝑘=𝛽⁻¹𝑒^−𝛽𝑅𝜌^𝑘+2𝑒^𝛽𝑅+𝑟𝑅𝑒^−𝛽𝑅𝑖𝜙^𝑘+2(𝑘+1)!1/2𝛽𝑅²+𝑅+𝑟𝑅𝑒^{−𝛽𝑅𝑘+2/2},𝑧=𝑒^{−𝛽𝑅/2},𝑓(𝑧)=^∞_𝑘=1𝑎_𝑘𝑧^𝑘. (5.15) Then 𝐹(𝜃)=−𝜙²+𝑓(𝑧) and𝐺_{𝑛,𝑟,𝛽}∼𝐶^ℎ_−ℎexp−𝜙²+𝑓(𝑧)𝑑𝑧, (5.16) where ℎ=(1/2)(𝛽𝑅²𝑒^𝛽𝑅+𝑅𝑒^𝛽𝑅+𝑟𝑅)^1/2𝑒^−3𝑅/8 and 𝐶=𝐴/[(1/2)(𝛽𝑅²𝑒^𝛽𝑅+𝑟𝑅)]^1/2.

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 form𝑒^𝑓(𝑧)=^∞_𝑘=0𝑏_𝑘𝑧^𝑘, (5.17) where 𝑏₀=𝑒^𝑓(0)=1,𝑏₁=𝑒^𝑓(0)𝑓^′(0)=𝑎₁,and𝑏₂=𝑎₂+(𝑎²₁/2).

Lemma 5.3. There is a constant 𝑅_𝑜 such that for all 𝑅>𝑅_𝑜, 𝑎_𝑘<2𝜙^𝑘+2. (5.18)

Proof. We see that 𝑎_𝑘=𝑅^𝑘+21+𝑜𝑅^𝑘+2(2)^(𝑘+2)/2(𝑘+2)!𝛽𝑅^2(𝑘+2)/21+𝑜𝑅²𝜙^𝑘+2 (5.19) which tends to 2^𝑘+2/2(𝑘+2)!<2^𝑘+2𝜙^𝑘+2 (5.20) 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.

With 𝑧=𝑒^{−𝛽𝑅/2},𝐺_{𝑛,𝑟,𝛽}∼𝐶^𝑠−1_𝑘=0^ℎ_−ℎ𝑒^−𝜙2𝑏_𝑘𝑑𝜙𝑧^𝑘+𝑄_𝑠, (5.21) where 𝑄_𝑠=^ℎ_−ℎ^∞_𝑘=𝑠𝑒^−𝜙2𝑏_𝑘𝑧^𝑘𝑑𝜙. (5.22)

Note that 𝑅→∞ as 𝑛→∞. Furthermore with 1ℎ=2𝛽𝑅²𝑒^𝛽𝑅+𝑅𝑒^𝛽𝑅+𝑟𝑅^1/2𝑒^−3𝑅/8=12𝛽𝑅²+𝑅+𝑟𝑅𝑒^{−𝛽𝑅1/2}𝑒^{(𝑅(4𝛽−3))/8}, (5.23)ℎ→∞ as 𝑅→∞. From these facts and the known asymptotic expansion of the function of the form ^ℎ_−ℎ𝑒^−𝜙2polynomialin𝜙𝑑𝜙, (5.24) the replacement of ℎ by ∞ in (5.16) is easily justified (see [11]). Hence 𝐺_{𝑛,𝑟,𝛽}∼𝐶^𝑠−1_𝑘=0^∞_−∞𝑒^−𝜙2𝑏_𝑘𝑑𝜙𝑧^𝑘+𝑄_𝑠. (5.25) It remains to show that 𝑄_𝑠=𝑜(|𝑧|^𝑠) as 𝑅→∞, that is, 𝑧→0. From a lemma in [12], |𝑏_𝑘|≤|2𝜙|^𝑘+2(1+|2𝜙|²)^𝑘−1. Thus, ^∞_𝑘=𝑠𝑏_𝑘𝑧^𝑘≤2𝜙^𝑠+21+2𝜙^2𝑠−1|𝑧|^𝑠1+𝜇+𝜇²+⋯, (5.26) where 𝜇=|2𝜙|(1+|2𝜙|²)|𝑧|.

Now, for 𝜇<1, we have^∞_𝑘=𝑠𝑏_𝑘𝑧^𝑘≤2𝜙^𝑠+21+2𝜙^2𝑠−1|𝑧|^𝑠1−|𝑧|2𝜙1+2𝜙². (5.27)

Let 𝑀 and 𝑃_𝑠(|𝜙|)|𝑧|^𝑠 denote the denominator and the numerator, respectively, in (5.27). Since |𝜙|≤ℎ and 𝑧=𝑒^{−𝛽𝑅/2}, we have 𝜙³𝑧|≤1|8𝛽𝑅²+𝑅+𝑟𝑅𝑒^{−𝛽𝑅3/2}𝑒^−3𝑅/8⟶0as𝑅⟶∞. (5.28) Hence for sufficiently large 𝑅, 𝑀≥1/2. Moreover,^∞_−∞𝑒^−𝜙2𝑃_𝑠𝜙𝑑𝜙 (5.29) exists and tends to zero as 𝑅→∞. Therefore, 𝑄_𝑠|𝑧|^𝑠≤^∞_−∞𝑒^−𝜙2𝑃_𝑠𝜙𝑀𝑑𝜙. (5.30) Thus, |𝑄_𝑠|=𝑜(|𝑧|^𝑠). Consequently,𝐺_{𝑛,𝑟,𝛽}∼𝐶^∞_𝑘=0^∞_−∞𝑒^−𝜙2𝑏_𝑘𝑑𝜙𝑒^{(−𝑘𝛽𝑅)/2}. (5.31) Since ∫^∞_−∞𝑒^−𝑥2𝑥^𝑛=0 for odd 𝑛, and 𝑏_2𝑘+1, as a polynomial in 𝜙, contain only odd powers of 𝜙, it follows that𝐺_{𝑛,𝑟,𝛽}∼𝐶^∞_𝑘=0^∞_−∞𝑒^−𝜙2𝑏_2𝑘𝑑𝜙𝑒^{−𝑘𝛽𝑅}. (5.32)

Calculation yields 𝑎₁=𝛽𝑅³+3𝑅²+𝛽⁻¹𝑅+𝑟𝑅𝑒^−𝛽𝑅3!(1/2)𝛽𝑅²+𝑅+𝑟𝑅𝑒^{−𝛽𝑅3/2}𝑖𝜙³,𝑎₂=𝛽𝑅⁴+6𝛽𝑅³+7𝑅²+𝛽⁻¹𝑅+𝑟𝑅𝑒^−𝛽𝑅4!(1/2)𝛽𝑅²+𝑅+𝑟𝑅𝑒^−𝛽𝑅2𝑖𝜙⁴. (5.33) Taking the first two terms of the asymptotic expansion of (5.32), we have 𝐺_{𝑛,𝑟,𝛽}∼𝐶^∞_−∞𝑒^−𝜙2𝑏_𝑜𝑑𝜙+𝐶𝑧^2∞_−∞𝑒^−𝜙2𝑏₂𝑑𝜙. (5.34) Since 𝑏₂=𝑎₂+𝑎²₁/2 and 𝑏_𝑜=1,𝐺_{𝑛,𝑟,𝛽}∼𝐶^∞_−∞𝑒^−𝜙2𝑑𝜙+𝐶𝑧^2∞_−∞𝑎₂𝑒^−𝜙2𝑧𝑑𝜙+𝐶²2^∞_−∞𝑒^−𝜙2𝑎²₁𝑑𝜙. (5.35) Let 𝐼₁,𝐼₂,and𝐼₃ denote, respectively, the integrals in (5.35). Then evaluating the last two integrals by parts and since ∫^∞_−∞𝑒^−𝜙2𝑑𝜙=√𝜋, we obtain 𝐼₁=𝐶√𝐼𝜋,₂=𝐶𝑒^−𝑅√𝜋𝛽𝑅³+6𝛽𝑅²+𝛽⁻¹+𝑟𝑒^−𝛽𝑅8𝑅𝛽𝑅+1+𝑟𝑒^−𝛽𝑅2,𝐼₃=−5𝐶𝑒^−𝑅√𝜋𝛽𝑅²+3𝛽⁻¹𝑅²+𝑟𝑒^−𝛽𝑅224𝑅𝛽𝑅+1+𝑟𝑒^−𝛽𝑅3. (5.36) Substituting the results in (5.35) and simplifying, we obtain 𝐺_{𝑛,𝑟,𝛽}∼𝐶√𝜋1+𝐷+𝐸𝐹, (5.37) where 𝐷=3𝛽²𝑅³+8𝛽𝑅³+3𝛽𝑅+3−10𝛽⁻¹−2𝑟𝑒^−𝛽𝑅𝑟𝑒^−𝛽𝑅,𝐸=3𝛽³−5𝛽²𝑅⁴+21𝛽²−30𝛽𝑅³+39𝛽−55𝑅²+24−30𝛽⁻¹𝑅+3𝛽⁻¹−5𝛽⁻²𝐹=24𝑅𝑒^𝛽𝑅𝛽𝑅+1+𝑟𝑒^−𝛽𝑅3., (5.38) Since 𝑅𝑒^𝛽𝑅=(𝑛−𝑟𝑅)𝛽⁻¹ and 𝑅^𝑛=𝑛^𝑛(𝛽𝑒^𝛽𝑅+𝑟)^−𝑛,𝐶=𝑛!exp𝑟𝑅+𝛽⁻¹𝑒^𝛽𝑅−𝛽𝜋𝑛^𝑛𝛽𝑒^𝛽𝑅+𝑟^−𝑛2(𝑛−𝑟𝑅)𝛽^−11/2𝛽𝑅+1+𝑟𝑒^{−𝛽𝑅1/2}. (5.39) Using Stirling's approximation for 𝑛!, viz, 𝑛!∼(2𝜋)𝑒^−𝑛𝑛^𝑛+(1/2)11+,12𝑛 (5.40) we obtain 𝑛𝐶∼^1/2(1+(1/12𝑛))exp𝑟𝑅+𝛽⁻¹𝑒^𝛽𝑅−𝛽𝛽^𝛽𝑅+𝑟^𝑛𝜋^1/2(𝑛−𝑟𝑅)𝛽^−11/2𝛽𝑅+1+𝑟𝑒^{−𝛽𝑅1/2}𝑒^𝑛. (5.41) Using (5.37), we obtain 𝐺_{𝑛,𝑟,𝛽}∼𝑛^1/2(1+(1/12𝑛))exp𝑟𝑅+𝛽⁻¹𝑒^𝛽𝑅−𝛽−𝑛𝛽^𝛽𝑅+𝑟^𝑛(𝑛−𝑟𝑅)𝛽^−11/2𝛽𝑅+1+𝑟𝑒^{−𝛽𝑅1/2}1+𝐷+𝐸𝐹. (5.42)

Acknowledgments

The authors wish to thank the referee for reading and evaluating the manuscript. They would also like to thank the Office of the President and the Office of the Vice Chancellor for Research and Extension of Mindanao State University-Main Campus for the support extended to this research.