An Asymptotic Formula for 𝑟-Bell Numbers with Real Arguments

Abstract

The 𝑟-Bell numbers are generalized using the concept of the Hankel contour. Some properties parallel to those of the ordinary Bell numbers are established. Moreover, an asymptotic approximation for 𝑟-Bell numbers with real arguments is obtained.

1. Introduction

The 𝑟-Stirling numbers of the second kind, denoted by {^𝑛_𝑘}_𝑟, are defined by Broder in [1], combinatorially, to be the number of partitions of the set {1,2,…,𝑛} into 𝑘 nonempty subsets, such that the numbers 1,2,…,𝑟 are in distinct subsets. Several properties of these numbers are established in [1–3]. Further generalization was established in [4] which is called (𝑟,𝛽)-Stirling numbers. These numbers are equivalent to the 𝑟-Whitney numbers of the second kind [5] and the Rucinski-Voigt numbers [6].

The sum of 𝑟-Stirling numbers of the second kind for integral arguments was first considered by Corcino in [7] and was called the 𝑟-Bell numbers. Corcino obtained an asymptotic approximation of 𝑟-Bell numbers using the method of Moser and Wyman. Here, we use 𝐵_𝑛,𝑟 to denote the 𝑟-Bell numbers; that is, 𝐵_𝑛,𝑟=^𝑛_𝑘=0𝑛+𝑟𝑘+𝑟_𝑟. (1) In a followup study of Mező [8], the 𝑟-Bell numbers 𝐵_𝑛,𝑟 were given more properties. One of these is the following exponential generating function: _𝑛≥0𝐵_𝑛,𝑟𝑡^𝑛𝑒𝑛!=exp𝑟𝑡+^𝑡−1. (2)

A more general form of Bell numbers, denoted by 𝐺_{𝑥,𝑟,𝛽}, was defined in [9] as 𝐺_{𝑥,𝑟,𝛽}=𝑥!2𝜋𝑖_ℋ𝑒^{𝑟𝑧+(𝑒𝛽𝑧−1)/𝛽}𝑑𝑧𝑧^𝑥+1, (3) where the parameters 𝑥, 𝑟, and 𝛽 are complex numbers with ℜ𝑟>0, ℜ𝛽>0, and 𝑥!=Γ(𝑥+1). In this paper, we define the 𝑟-Bell numbers with complex argument using the concept of Hankel contour and establish some properties parallel to those obtained by Mező in [8]. Moreover, an asymptotic formula of these numbers for real arguments will be derived using the method of Moser and Wyman [10].

2. 𝑟-Stirling Numbers of the Second Kind

Graham et al. [11] proposed another way of generalizing the Stirling numbers by extending the range of values of the parameters 𝑛 and 𝑘 to complex numbers. This problem was first considered by Flajolet and Prodinger [12] by defining the classical Stirling numbers with complex arguments using the concept of Hankel contour. Recently, the (𝑟,𝛽)-Stirling numbers with complex arguments, denoted by 𝑥𝑦𝛽,𝑟, were defined in [9] by means of the following integral representation over a Hankel contour ℋ: 𝑥𝑦_𝛽,𝑟=𝑥!𝑦!𝛽^𝑦2𝜋𝑖_ℋ𝑒^𝑟𝑧𝑒^𝛽𝑧−1^𝑦𝑑𝑧𝑧^𝑥+1, (4) where 𝑟 and 𝛽 are complex numbers with ℜ𝑟>0, ℜ𝛽>0, and 𝑥!=Γ(𝑥+1). We know that, for integral case, the 𝑟-Stirling numbers of the second kind may be obtained by taking 𝛽=1. Hence, using (4), we can define the second-kind 𝑟-Stirling numbers with complex arguments as follows.

Definition 1. The 𝑟-Stirling numbers of the second kind 𝑥+𝑟𝑦+𝑟𝑟 of complex arguments 𝑥 and 𝑦 are defined by 𝑥+𝑟𝑦+𝑟_𝑟∶=𝑥!𝑦!2𝜋𝑖_ℋ𝑒^𝑟𝑧𝑒^𝑧−1^𝑦𝑑𝑧𝑧^𝑥+1, (5) where 𝑟 is complex number with ℜ𝑟>0 and 𝑥!=Γ(𝑥+1), and the logarithm involved in the functions (𝑒^𝑧−1)^𝑦 and 𝑧^𝑥+1 is taken to be the principal branch. The Hankel contour ℋ starts from −∞ below the negative axis, surrounds the origin counterclockwise, and returns to −∞ in the half plane ℑ𝑧>0, such that it has a distance 𝜖≤1 from the nonpositive real axis.

Remark 2. Since ℜ𝑟>0 and ℜ𝑧→−∞, the integral in (5) converges for all values of 𝑥 and 𝑦. Also, 𝑥+𝑟𝑦+𝑟𝑟 is a meromorphic function of 𝑥 (for any fixed 𝑦) with poles at the nonpositive integers, and it is entire as a function of 𝑦 (for any fixed 𝑥 not a negative integer).

Remark 3. By the change of variable on the integral in (4), say 𝑤=𝛽𝑧, we can express 𝑥𝑦𝛽,𝑟 in terms of 𝑥+𝑟𝑦+𝑟𝑟 as follows: 𝑥𝑦_𝛽,𝑟=𝛽^𝑥−𝑦𝑟𝑥+𝛽𝑟𝑦+𝛽_𝑟/𝛽. (6) Because of relation (6), every property of (𝑟,𝛽)-Stirling numbers with complex arguments will have a corresponding property for 𝑟-Stirling numbers with complex arguments and vice versa. For instance, the (𝑟,𝛽)-Stirling numbers in [9] satisfy the following relation: 𝑥𝑦_𝛽,𝑟=𝑥−1𝑦−1_𝛽,𝑟+𝑦𝛽𝑦+𝑟𝑥−1_𝛽,𝑟. (7) Replacing 𝑟 with 𝛽𝑟, we get 𝑥𝑦_{𝛽,𝛽𝑟}=𝑥−1𝑦−1_{𝛽,𝛽𝑟}+𝑦𝛽𝑦+𝛽𝑟𝑥−1_{𝛽,𝛽𝑟}. (8) Using (6), we obtain 𝛽^𝑥−𝑦𝑥+𝛽𝑟𝛽𝑦+𝛽𝑟𝛽_{𝛽𝑟/𝛽}=𝛽^𝑥−𝑦𝑥+𝛽𝑟𝛽−1𝑦+𝛽𝑟𝛽−1_{𝛽𝑟/𝛽}+𝛽^{𝑥−1−𝑦}𝛽𝑦+𝛽𝑟𝑥+𝛽𝑟𝛽−1𝑦+𝛽𝑟_{𝛽𝑟/𝛽}. (9) Thus, we have 𝑥+𝑟𝑦+𝑟_𝑟=𝑥+𝑟−1𝑦+𝑟−1_𝑟+𝑦+𝑟𝑥+𝑟−1𝑦+𝑟_𝑟. (10)

On the other hand, the (𝑟,𝛽)-Stirling numbers in [9] satisfy the relation 𝑥𝑦_𝛽,𝑟=1𝛽^𝑦𝑦!^∞_𝑗=0(−1)^𝑦−𝑗𝑦𝑗𝛽𝑗+𝑟^𝑥. (11) Again, replacing 𝑟 with 𝛽𝑟, we obtain 𝑥𝑦_{𝛽,𝛽𝑟}=1𝛽^𝑦𝑦!^∞_𝑗=0(−1)^𝑦−𝑗𝑦𝑗𝛽𝑗+𝛽𝑟^𝑥. (12) By (6), we get 𝛽^𝑥−𝑦𝑥+𝛽𝑟𝛽𝑦+𝛽𝑟𝛽_{𝛽𝑟/𝛽}=1𝛽^𝑦𝑦!^∞_𝑗=0(−1)^𝑦−𝑗𝑦𝑗𝛽𝑗+𝛽𝑟^𝑥. (13) Thus, we have 𝑥+𝑟𝑦+𝑟_𝑟=1𝑦!^∞_𝑗=0(−1)^𝑦−𝑗𝑦𝑗𝑗+𝑟^𝑥. (14)

Let’s state these relations formally in the following theorem.

Theorem 4. The 𝑟-Stirling numbers of the second kind with complex arguments satisfy the following relations: 𝑥+𝑟𝑦+𝑟_𝑟=𝑥+𝑟−1𝑦+𝑟−1_𝑟+𝑦+𝑟𝑥+𝑟−1𝑦+𝑟_𝑟,𝑥+𝑟𝑦+𝑟_𝑟=1𝑦!^∞_𝑗=0(−1)^𝑦−𝑗𝑦𝑗𝑗+𝑟^𝑥. (15)(16)

Remark 5. We can give an alternative proof of this theorem. That is, by applying integration by parts on (5) and using the fact that 𝑒^𝛽𝑧(𝑒^𝑧−1)^𝑦−1=(𝑒^𝑧−1)^𝑦+(𝑒^𝑧−1)^𝑦−1, we obtain 𝑥+𝑟𝑦+𝑟_𝑟=(𝑥−1)!𝑦−1!2𝜋𝑖_ℋ𝑒^𝑟𝑧𝑒^𝑧−1^𝑦−1𝑑𝑧𝑧^𝑥+𝑦+𝑟(𝑥−1)!𝑦!2𝜋𝑖_ℋ𝑒^𝑟𝑧𝑒^𝑧−1^𝑦𝑑𝑧𝑧^𝑥. (17) This implies (15) immediately. On the other hand, using Newton’s Binomial Theorem, (5) can be expressed as 𝑥+𝑟𝑦+𝑟_𝑟=𝑥!1𝑦!2𝜋𝑖_ℋ𝑒^∞𝑟𝑧_𝑗=0(−1)^𝑦−𝑗𝑦𝑗𝑒^𝛽𝑧𝑗𝑑𝑧𝑧^𝑥+1=1𝑦!^∞_𝑗=0(−1)^𝑦−𝑗𝑦𝑗𝑥!2𝜋𝑖_ℋ𝑒^{(𝑗+𝑟)𝑧}𝑑𝑧𝑧^𝑥+1. (18) We know that Hankel’s contour integral is a unit for gamma function over the set of complex numbers; that is, 𝑖2𝜋_ℋ(−𝑧)^−𝑥𝑒^−𝑧𝑑𝑧Γ(𝑥)=1,|𝑧|<∞. (19) This implies that 𝑥!2𝜋𝑖_ℋ𝑒^{(𝑗+𝑟)𝑧}𝑑𝑧𝑧^𝑥+1=𝑗+𝑟^𝑥. (20) This completes the proof of (16).

The next theorem contains a property for 𝑥+𝑟𝑦+𝑟𝑟 which is analogous to the identity that usually defines Stirling-type numbers.

Theorem 6. The 𝑟-Stirling numbers of the second kind with complex arguments satisfy the following relation: (𝑡+𝑟)^𝑥=^∞_𝑘=0𝑥+𝑟𝑘+𝑟_𝑟(𝑡)_𝑘, (21) where (𝑡)_𝑘=𝑡(𝑡−1)⋯(𝑡−𝑘+1).

Proof. Using (5) with 𝑦=𝑘, a nonnegative integer, we obtain ^∞_𝑘=0𝑥+𝑟𝑘+𝑟_𝑟(𝑡)_𝑘=^∞_𝑘=0𝑥!𝑘!2𝜋𝑖_ℋ𝑒^𝑟𝑧𝑒^𝑧−1^𝑘𝑑𝑧𝑧^𝑥+1(𝑡)_𝑘=𝑥!2𝜋𝑖_ℋ𝑒^𝑟𝑧∞_𝑘=0𝑡𝑘𝑒^𝑧−1^𝑘𝑑𝑧𝑧^𝑥+1=𝑥!2𝜋𝑖_ℋ𝑒^𝑟𝑧𝑒^𝑧−1+1^𝑡𝑑𝑧𝑧^𝑥+1=𝑥!2𝜋𝑖_ℋ𝑒^𝑟𝑧𝑒^𝑡𝑧𝑑𝑧𝑧^𝑥+1. (22) Thus, by (20), we prove the theorem.

Remark 7. It is worth mentioning that this type of property was not established for (𝑟,𝛽)-Stirling numbers with complex arguments. However, replacing 𝑡 with 𝛽⁻¹𝑡 and 𝑟 with 𝑟/𝛽 in (21), we obtain 𝛽⁻¹𝑟𝑡+𝛽^𝑥=^∞_𝑘=0𝑟𝑥+𝛽𝑟𝑘+𝛽_𝑟/𝛽𝛽⁻¹𝑡_𝑘,(𝑡+𝑟)^𝑥=^∞_𝑘=0𝛽^𝑥−𝑘𝑟𝑥+𝛽𝑟𝑘+𝛽_𝑟/𝛽𝑡∣𝛽_𝑘. (23) Using relation (6), we get (𝑡+𝑟)^𝑥=^∞_𝑘=0𝑥𝑘_𝛽,𝑟𝑡∣𝛽_𝑘. (24) This is the corresponding property for (𝑟,𝛽)-Stirling numbers with complex arguments.

Remark 8. In a separate paper of the present authors, an asymptotic formula for 𝑟-Stirling numbers of the second kind with real arguments was established using the method of Chelluri et al. [13].

3. 𝑟-Bell Numbers

Using Cauchy’s integral formula, (2) can be transformed into the following integral representation of 𝐵_𝑛,𝑟: 𝐵_𝑛,𝑟=𝑛!2𝜋𝑖_𝛾𝑒^{𝑟𝑧+(𝑒𝑧−1)}𝑑𝑧𝑧^𝑛+1, (25) where the integral contour 𝛾 is a small contour encircling the origin. Since 𝑛 is nonnegative in (25), the contour 𝛾 can be deformed into a Hankel contour ℋ that starts from −∞ below the negative axis, surrounds the origin counterclockwise, and returns to −∞ in the half plane ℑ𝑧>0. We assume that it is at distance ≤1 from the real axis. Now, let us consider the following definition for the generalization of 𝑟-Bell numbers 𝐵_𝑛,𝑟 where 𝑛 is a complex number.

Definition 9. The 𝑟-Bell numbers 𝐵_𝑥,𝑟 of complex argument 𝑥 are defined by 𝐵_𝑥,𝑟=𝑥!2𝜋𝑖_ℋ𝑒^{𝑟𝑧+(𝑒𝑧−1)}𝑑𝑧𝑧^𝑥+1, (26) where 𝑟 is complex numbers with ℜ𝑟>0 and 𝑥!=Γ(𝑥+1).

Since ℜ𝑟>0 and ℜ𝑧→−∞, clearly, the integral in (26) converges for all values of 𝑥. Moreover, 𝐵_𝑥,𝑟 is a meromorphic function of 𝑥 with poles at the nonpositive integers.

It can easily be shown that 𝐵_𝑥,𝑟=^∞_𝑘=01𝑘!^𝑘_𝑗=0(−1)^𝑘−𝑗𝑘𝑗𝑥!2𝜋𝑖_ℋ𝑒^{(𝑗+𝑟)𝑧}𝑧^−(𝑥+1)𝑑𝑧. (27) By (16) and (20), we obtain 𝐵_𝑥,𝑟=^∞_𝑘=01𝑘!^𝑘_𝑗=0(−1)^𝑘−𝑗𝑘𝑗𝑗+𝑟^𝑥=^∞_𝑘=0𝑥+𝑟𝑘+𝑟_𝑟. (28) Hence, we have the following theorem.

Theorem 10. The 𝑟-Bell numbers are equal to 𝐵_𝑥,𝑟=^∞_𝑘=0𝑥+𝑟𝑘+𝑟_𝑟, (29) where 𝑥 and 𝑟 are complex numbers.

By Theorem 10, we have verified that identity in (1) also holds for 𝐵_𝑥,𝑟 with complex argument 𝑥. This means that the numbers 𝐵_𝑥,𝑟 in Definition 9 belong to the family of Bell numbers.

The next theorem asserts that 𝐵_𝑥,𝑟 also possess a kind of Dobiński’s formula which can easily be shown using Definiton 9 and the expansion of 𝑒^{(𝑟𝑧+𝑒𝑧−1)}.

Theorem 11. The 𝑟-Bell numbers are equal to 𝐵_𝑥,𝑟=1𝑒^∞_𝑘=0(𝑘+𝑟)^𝑥,𝑘! (30) where 𝑥 and 𝑟 are complex numbers.

Using (26) and (30), we obtain the following corollary which is a kind of extension of the integral formula obtained by Mező in [8].

Corollary 12. The following integral identity holds: ^∞_𝑘=0(𝑘+𝑟)^𝑥=𝑘!𝑥!𝑒2𝜋𝑖_ℋ𝑒^{𝑟𝑧+(𝑒𝑧−1)}𝑑𝑧𝑧^𝑥+1. (31)

The 𝑟-Bell polynomials of Mező [8] satisfy the recurrence relation 𝐵_𝑥,𝑟(𝑤)=𝑟𝐵_{𝑥−1,𝑟}(𝑤)+𝑤𝐵_{𝑥−1,𝑟+1}(𝑤), (32) where 𝑥 and 𝑟 are nonnegative integers. This will reduce to 𝐵_𝑥,𝑟=𝑟𝐵_{𝑥−1,𝑟}+𝐵_{𝑥−1,𝑟+1} (33) when 𝑤=1. Analogous to this relation, we have the following relation which can easily be shown using (28), (15), and (16).

Theorem 13. The 𝑟-Bell numbers satisfy the following relation: 𝐵_𝑥,𝑟=𝑟𝐵_{𝑥−1,𝑟}+𝐵_{𝑥−1,𝑟+1}+𝑥+𝑟−1𝑟−1_𝑟+1, (34) where 𝑥 and 𝑟 are complex numbers.

Proof. Summing up both sides of (15) gives 𝐵_𝑥,𝑟=(𝑟+1)𝐵_{𝑥−1,𝑟}+𝑥+𝑟−1𝑟−1_𝑟+^∞_𝑘=0𝑘𝑥+𝑟−1𝑘+𝑟_𝑟. (35) By applying (16), we get ^∞_𝑘=0𝑘𝑥+𝑟−1𝑘+𝑟_𝑟=−^∞_𝑘=01(𝑘−1)!^𝑘_𝑗=0(−1)^{𝑘−1−𝑗}𝑗𝑘−1𝛽𝑗+𝑟^𝑥−1−^∞_𝑘=0−1(𝑘−1)!^𝑘_𝑗=0(−1)^{𝑘−1−(𝑗−1)}𝑘−1𝑗−1𝑗−1+1+𝑟^𝑥−1. (36) Hence, we have ^∞_𝑘=0𝑘𝑥+𝑟−1𝑘+𝑟_𝑟=−𝑥+𝑟−1𝑟−1_𝑟−𝐵_{𝑥−1,𝑟}+𝐵_{𝑥−1,1+𝑟}+𝑥+𝑟−1𝑟−1_𝑟+1. (37) Substituting this to (35) completes the proof of the theorem.

Note that, when the parameters 𝑥,𝑦, and 𝑟 are nonnegative integers, Definition 1 is just equivalent to the integral representation of the 𝑟-Stirling numbers of the second kind in [1]. Hence, the value of 𝑥+𝑟−1𝑟−1𝑟+1 is equal to 0.

4. Asymptotic Formula

An asymptotic formula for 𝑟-Bell numbers 𝐵_𝑥,𝑟 was first established by Corcino in [7]. But the formula holds only when 𝑥 is a nonnegative integer. Here, we aim to obtain an asymptotic formula for 𝐵_𝑥,𝑟 when 𝑥 is a real number.

Using Definition 9, 𝐵_𝑥,𝑟=𝑥!2𝜋𝑖_ℋ𝑒^{𝑟𝑧+(𝑒𝑧−1)}𝑧^𝑥+1𝑑𝑧. (38) To obtain an asymptotic formula, we deform the path ℋ into the following contour: 𝐶₁∪ 𝐶₂∪ 𝐶₃∪𝐶₄∪𝐶₅, where (i)𝐶₁ is the line ℑ𝑧=−2𝜋+𝛿,𝛿>0, and ℜ𝑧≤𝜆, 𝜆 is a small positive number; (ii)𝐶₂ is the line segment ℜ𝑧=𝜖, going from 𝜆+𝑖(𝛿−2𝜋) to the circle |𝑧|=𝑅; (iii)𝐶₅ and 𝐶₄ are the reflections in the real axis of 𝐶₁ and 𝐶₂, respectively;(iv)𝐶₃ is the portion of the circle |𝑧|=𝑅, meeting 𝐶₂ and 𝐶₄. The new contour is 𝐶₁∪𝐶₂∪𝐶₃∪𝐶₄∪𝐶₅ in the counterclockwise sense. This idea of deforming the contour ℋ is also done in [13]. The integrals along 𝐶₁, 𝐶₂, 𝐶₄, and 𝐶₅ are seen to be 𝑂(2+𝜆)^𝑦(2𝜋−𝛿)^𝑥. (39) It will also be shown that these integrals go to 0 as 𝑥→∞ provided that 𝑥−𝑦≥𝑥^1/3. To see this, we consider 𝐶₁. For the other contours, the estimate can be seen similarly. We look at _𝐶1𝑒^{𝑟𝑧+(𝑒𝑧−1)}𝑧^𝑥+1𝑑𝑧≤_𝐶1𝑒^{𝑟𝑧+(𝑒𝑧−1)}𝑧^𝑥+1|𝑑𝑧|. (40) Note that 𝑒^{𝑟𝑧+(𝑒𝑧−1)}=𝑒^𝑟𝑧𝑒^{(𝑒𝑧−1)}=𝑒^𝑟ℜ𝑧𝑒^{ℜ(𝑒𝑧−1)}. (41) Turning to 𝑒^𝑧−1, 𝑒^𝑧−1=𝑒^ℜ𝑧𝑒^𝑖ℑ𝑧=𝑒−1_ℜ𝑧cosℑ𝑧−1+𝑖𝑒^ℜ𝑧sinℑ𝑧. (42) Thus, 𝑒^{𝑟𝑧+(𝑒𝑧−1)}=𝑒^𝑟ℜ𝑧𝑒^{𝑒ℜ𝑧cosℑ𝑧}𝑒⁻¹<𝑒^𝑟ℜ𝑧𝑒^𝑒ℜ𝑧𝑒⁻¹. (43) Consider 𝑟ℜ𝑧+𝑒^ℜ𝑧−1<𝑟𝜆+𝑒^𝜆𝜆−1=𝑟𝜆+𝜆1++𝜆2!²+𝜆3!³𝜆4!+⋯<𝑟𝜆+𝜆1+2+𝜆²2²+𝜆³2³2+⋯=𝑟𝜆+𝜆.2−𝜆 (44) Choose 𝜆<1, so that 2−𝜆>1. Then 𝑟ℜ𝑧+𝑒^ℜ𝑧−1<𝜆(2+𝑟). (45) This implies that 𝑒^{𝑟𝑧+(𝑒𝑧−1)}<𝑒^𝜆(2+𝑟). (46)

Consequently, we obtain _𝐶1𝑒^{𝑟𝑧+(𝑒𝑧−1)}𝑧^𝑥+1𝑑𝑧<𝑒^𝜆(2+𝑟)1(2𝜋−𝛿)^𝑥+1𝑙𝐶₁, (47) where 𝑙𝐶₁=lim_{𝑡→−∞}𝜆−𝑡. (48)

Since 𝑙(𝐶₁)/(2𝜋−𝛿)^𝑥+1→0 as 𝑥→∞, we have the integral along 𝐶₁ goes to 0 as 𝑥→∞. So what remains is the integral along 𝐶₃. That is, 𝐵_𝑥,𝑟∼𝑥!2𝜋𝑖_𝐶3𝑒^{𝑟𝑧+(𝑒𝑧−1)}𝑧^𝑥+1𝑑𝑧, (49) where 𝐶₃ is a semicircle 𝑧=𝑅𝑒^𝑖𝜃, −𝛼≤𝜃≤𝛼. Hence, by Laplace method or following the analysis in [7], 𝐵_𝑥,𝑟∼𝑥!2𝜋𝑖𝑅^𝑛𝛼_−𝛼exp𝑒^{𝑅𝑒𝑖𝜃}+𝑟𝑅𝑒^𝑖𝜃∼−𝑖𝑛𝜃𝑑𝜃𝑥!2𝜋𝑖𝑅^𝑛𝜖_−𝜖exp𝑒^{𝑅𝑒𝑖𝜃}+𝑟𝑅𝑒^𝑖𝜃−𝑖𝑛𝜃𝑑𝜃. (50) In [7], the integration is along a circle about zero with radius 𝑅. This number 𝑅 is shown to be the unique solution to 𝜇𝑥−𝑟𝜇=𝑒^−𝜇 (51) as a function of 𝜇 (see Lemma 3 [7]). We see that the asymptotic formula for the 𝑟-Bell numbers obtained in [7] holds for real argument 𝑥. Thus, we have the following asymptotic formula.

Theorem 14. The 𝑟-Bell numbers 𝐵_𝑥,𝑟 with real arguments 𝑥 and 𝑟 have the following asymptotic formula: 𝐵_𝑥,𝑟∼𝐶√𝜋1+𝐷−2𝑅⁴+9𝑅³+16𝑅²+6𝑅+224𝑒^𝑅𝑅𝑅+1+𝑟𝑒^−𝑅3, (52) where 𝐶=𝑥!exp𝑟𝑅+𝑒^𝑅−1𝜋𝑅^𝑥2𝑟²𝑒^𝑅+𝑅𝑒^𝑅+𝑟𝑅^1/2,𝐷=3𝑅³+8𝑅²−6𝑅−4−2𝑟𝑒^−𝑅𝑟𝑒^−𝑅, (53) and 𝑅 is the unique positive solution to 𝜇𝑒^𝜇+𝑟𝜇−𝑥=0 (54) as a function of 𝜇.

An asymptotic formula for (𝑟,𝛽)-Bell numbers 𝐺_{𝑥,𝑟,𝛽} has already been established in [14]. However, the formula holds only when 𝑥 is a nonnegative integer. Here, using the same method as employed above, we can show that this asymptotic formula will also work for the case in which the parameters 𝑥, 𝑟, and 𝛽 are real numbers.

Now, for real parameters 𝑥, 𝑟, and 𝛽, we have 𝐺_{𝑥,𝑟,𝛽}=𝑥!2𝜋𝑖_ℋ𝑒^{𝑟𝑧+(𝑒𝛽𝑧−1)/𝛽}𝑑𝑧𝑧^𝑥+1, (55) where the path ℋ can also be deformed into the following contour: 𝐶₁∪𝐶₂∪𝐶₃∪𝐶₄∪𝐶₅, such that (i)𝐶₁ is the line ℑ𝑧=−2𝜋+𝛿, 𝛿>0, and ℜ𝑧≤𝜆, 𝜆 is a small positive number; (ii)𝐶₂ is the line segment ℜ𝑧=𝜆, going from 𝜆+𝑖(𝛿−2𝜋) to the circle |𝑧|=𝑅; (iii)𝐶₅ and 𝐶₄ are the reflections in the real axis of 𝐶₁ and 𝐶₂, respectively; (iv)𝐶₃ is the portion of the circle |𝑧|=𝑅, meeting 𝐶₂ and 𝐶₄.

Also, the new contour is 𝐶₁∪𝐶₂∪𝐶₃∪𝐶₄∪𝐶₅ in the counterclockwise sense. It can easily be shown that, along the path 𝐶₁, 𝑒^{𝑟𝑧+(𝑒𝛽𝑧−1)/𝛽}<𝑒^{𝜆((2/𝛽)+𝑟)}. (56) Hence, as 𝑥 goes to ∞, _𝐶1𝑒^{𝑟𝑧+(𝑒𝛽𝑧−1)/𝛽}𝑑𝑧𝑧^𝑥+1→0. (57) This can be done similarly along 𝐶₂,𝐶₄, and 𝐶₅. Thus, we have 𝐺_{𝑥,𝑟,𝛽}∼𝑥!2𝜋𝑖_𝐶3𝑒^{𝑟𝑧+(𝑒𝛽𝑧−1)/𝛽}𝑧^𝑥+1𝑑𝑧, (58) where 𝐶₃ is a semicircle 𝑧=𝑅𝑒^𝑖𝜃, −𝛼≤𝜃≤𝛼. Then 𝐺_{𝑥,𝑟,𝛽}∼𝑥!2𝜋𝑖𝑅^{𝑛𝛼−𝛼}exp𝛽⁻¹𝑒^{𝛽𝑅𝑒𝑖𝜃}+𝑟𝑅𝑒^𝑖𝜃−𝑖𝑥𝜃−𝛽⁻¹∼𝑑𝜃𝑥!2𝜋𝑖𝑅^{𝑛𝜖−𝜖}exp𝛽⁻¹𝑒^{𝛽𝑅𝑒𝑖𝜃}+𝑟𝑅𝑒^𝑖𝜃−𝑖𝑥𝜃−𝛽⁻¹𝑑𝜃. (59)

This implies that the asymptotic formula for the (𝑟,𝛽)-Bell numbers obtained in [14] holds for real argument. Thus, from [14], we have the following asymptotic formula for (𝑟,𝛽)-Bell numbers 𝐺_{𝑥,𝑟,𝛽} with real arguments 𝑥, 𝑟, and 𝛽: 𝐺_{𝑥,𝑟,𝛽}∼𝑥^1/2(1+1/12𝑥)exp𝑟𝑅+𝛽⁻¹𝑒^𝛽𝑅−𝛽−𝑥𝛽^𝛽𝑅+𝑟^𝑥𝑥−𝑟𝑅𝛽^−11/2𝛽𝑅+1+𝑟𝑒^{−𝛽𝑅1/2}×1+𝐷+𝐸𝐹, (60) where 𝐷=3𝛽²𝑅³+8𝛽𝑅³+3𝛽𝑅+3−10𝛽⁻¹−2𝑟𝑒^−𝛽𝑅𝑟𝑒^−𝛽𝑅,𝐸=3𝛽³−5𝛽²𝑅⁴+21𝛽²−30𝛽𝑅³+39𝛽−55𝑅²+24−30𝛽⁻¹𝑅+3𝛽⁻¹−5𝛽⁻²,𝐹=24𝑅𝑒^𝛽𝑅𝛽𝑅+1+𝑟𝑒^−𝛽𝑅3, (61) and 𝑅 is the unique positive solution to 𝜇𝑒^𝛽𝜇+𝑟𝜇−𝑥=0 (62) as a function of 𝜇.

5. Summary and Recommendation

In this paper, we have defined 𝑟-Stirling numbers of the second kind and 𝑟-Bell numbers with complex arguments using the concept of Hankel contour and established some properties parallel to those of the classical Stirling and Bell numbers. Moreover, we have derived an asymptotic formula for 𝑟-Bell numbers as well as for (𝑟,𝛽)-Bell numbers for real arguments using the method of Chelluri and that of Moser and Wyman.

We observe that, by employing those methods, one can possibly establish an asymptotic formula for 𝑟-Stirling numbers of the second kind with real arguments and, consequently, using relation (6), an asymptotic formula for (𝑟,𝛽)-Stirling numbers with real arguments.

Acknowledgment

The authors would like to acknowledge the support from the Office of the Vice Chancellor for Research andDevelopment of the University of the Philippines Diliman for this research project. The authors also wish tothank the referees for reading and evaluating the paper thoroughly.