# Asymptotic Estimates for Second Kind Generalized Stirling Numbers

Authors: Cristina B. Corcino; Roberto B. Corcino
Journal: Journal of Applied Mathematics
 (2013)

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## Abstract

Asymptotic formulas for the generalized Stirling numbers of the second kind with integer and real parameters are obtained and ranges of validity of the formulas are established. The generalizations of Stirling numbers considered here are generalizations along the line of Hsu and Shuie's unified generalization.


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## Body

## 1. Introduction

Asymptotic formulas of the classical Stirling numbers have been done by many authors like Temme [1], Moser, and Wyman [2, 3] due to the importance of the formulas in computing values of the numbers under consideration when parameters become large. The Stirling numbers and their generalization, on the other hand, are important due to their applications in statistics, life science, and physics.The(r,β)-Stirling numbers [4], the r-Whitney numbers of the second kind [5], and the numbers considered by Ruciński and Voigt [6] are exactly the same numbers which can be classified as generalization of the classical Stirling numbers of the second kind. This generalization is in line with the generalization of Hsu and Shuie [7]. For brevity, we can use Sβ,r(n,m) to denote these numbers. These numbers satisfy the following exponential generating function:
(1)1βm(m!)erz(eβz-1)m=∑n=m∞‍Sβ,r(n,m)znn!,
where n and m are positive integers. When β=1, (1) reduces to a generating function of r-Stirling numbers [8] of the second kind and further reduces to a generating function of the classical Stirling numbers of the second when r=0. Combinatorial interpretation and probability distribution involving Sβ,r(n,m) are discussed in [9].The behavior of the numbersSβ,r(n,m) was shown to be asymptotically normal in [4, 6]. That means that the distribution of these numbers when the parameters n and m are large will follow a bell-shaped distribution. The unimodality of this distribution was also discussed in [4]. Moreover, the bound for the index in which the maximum value of these numbers occurs has been established in [10]. With these properties of the numbers Sβ,r(n,m), it is necessary to consider the asymptotic formula for the numbers Sβ,r(n,m) to be able to compute large values of the numbers.Applying Cauchy Integral Formula to (1), the following integral representation is obtained:
(2)Sβ,r(n,m)=n!2πiβm(m!)∫C‍erz(eβz-1)mzn+1dz,
where C is a circle about the origin.The primary purpose of the present paper is to investigate if the analysis in [3] can be extended to the generalized Stirling numbers of the second kind. The authors are motivated by the work of Chelluri et al. [11] which proved the asymptotic equivalence of the formulas obtained by Temme [1] and those obtained by Moser and Wyman in [2, 3]. Moreover, it was also shown in [11] that the formulas obtained apply not only for integral values of the parameters n and m, but for real values as well. To be able to do a similar investigation with that in [11] for the generalized Stirling numbers of the first and second kinds, it is necessary to come up with asymptotic formulas for each kind generalized Stirling numbers which may have used a method similar to that in [2, 3]. The necessary asymptotic formulas for the generalized Stirling numbers of the first kind can be found in [12, 13]. In this paper, an asymptotic formula for the generalized Stirling numbers of the second kind Sβ,r(n,m) with integral values of m and n are obtained using a similar analysis as that in [3]. The formula is proved to be valid when m>(1/4)nμ and limn→∞(n-m)=∞, where μ=r/β. Moreover, other asymptotic formulas are obtained for Sβ,r(n,m), where n and m are real numbers under the conditions n-m≥nα and n-m≥n1/3.


## 2. Preliminary Results

The integral representation in (2) can be written in the form
(3)Sβ,r(n,m)=n!2πiβm-nm!∫C‍eμw(ew-1)mwn+1dw,
where μ=r/β, w=βz.Using the representationw=Reiθ, for the circle C, where -π≤θ≤π and R is a positive real number, (3) becomes
(4)Sβ,r(n,m)=n!2πβm-nRn(m!)∫-ππ‍eμReiθ(eReiθ-1)meinθdθ.Multiplying and dividing the right-hand side of (4) by the correction constant
(5)(eR-1)meμR,
it reduces to
(6)Sβ,r(n,m)=A∫-ππ‍exph(θ,R)dθ,
where
(7)A=n!eμR(eR-1)m2πβm-nRnm!,
and h(θ,R) is the function,
(8)h(θ,R)=μReiθ+mlog(eReiθ-1)-mlog(eR-1)-μR-inθ.Observe thath(θ,R) can be written in the form
(9)h(θ,R)=μReiθ-μR+mg(θ,R),
where g(θ,R) is the same function g(θ,R) which appeared in [14], except for the value of R.The Maclaurin expansion ofh(θ,R) is
(10)h(θ,R)=iBθ+mRH(iθ)2+m∑k=3∞‍Dk(R)(iθ)k,
where
(11)B=R(μ+m1-e-R)-n,(12)H=12[μm+eR(eR-R-1)(eR-1)2],(13)Dk(R)=μRk!m+1k!Δklog(eR-1),
where Δ is the operator R(d/dR).Lemma 1.
There is a unique0<R<n/m such that
(14)R(μ+m1-e-R)-n=0.Proof.
Equation (14) can be written in the form
(15)R(1-e-R)-1=nm-μmR.
Letf1(R)=R(1-e-R)-1 and f2(R)=n/m-(μ/m)R. Note that f1(R) is a continuous function on (0,∞) and limR→∞f1(R)=∞ while limR→0f1(R)=1. On the other hand, f2(R) is a line with x intercept at R=n/μ and y intercept at f2(0)=n/m and is decreasing when m and μ are positive real numbers. Thus, f1 and f2, as functions of R, surely intersect at some point. The value of R at the intersection point is the desired solution. Moreover, it can be seen graphically that R<n/μ.Lemma 2.
H defined in (12) obeys the inequalities
(16)14≤H≤12+μ2.Proof.
This lemma follows from Lemma 2.2 in [3].Lemma 3.
There exists a constantM independent of k and R such that
(17)|Dk(R)R|≤M.Proof.
D
k
(
R
) defined in (13) can be written as
(18)Dk(R)=μRk!m+Ck(R),
where Ck(R) is the same as that defined in (16) in [3]. Dividing both sides of the preceding equation by R and taking the absolute value, we have
(19)|Dk(R)R|≤|μk!m|+|Ck(R)R|.
Withμ a fixed parameter and m→∞ as n→∞ and using Lemma 3.2 in [3], the desired result is obtained.
Defineϵ=(mR)-3/8 and consider the integral J defined by
(20)J=∫ϵπ‍exph(θ,R)dθ.Lemma 4.
A constantk>0 exists such that
(21)|J|≤πexp(-k(mR)1/4).Proof.
We claim that(22)|h(θ,R)|≤|(eReiθ-1)|·|(eR-1)-1|m.
Using the definition ofh(θ,R) given by (9), we have
(23)|exph(θ,R)|=|exp[μR(eiθ-1)]|·|eReiθ-1eR-1|m·|e-inθ|.
Because|einθ|=1, it remains to show that |exp[μR(eiθ-1)]|≤1.
Note that|exp[μR(eiθ-1)]|=eμRcosθ·e-μR. The claim now follows from the fact that -1≤cosθ≤1 and μR≥0.
From (9),
(24)h(θ,R)=μR(eiθ-μR)+mg(θ,R).
Thus,(25)|exph(θ,R)|=|exp[μR(eiθ-1)]||expmg(θ,R)|.
The result now follows from the previous claim and the fact thatg(θ,R) is the same function g(θ,R) that appeared in [3] except for the value of R.Observe that whenmR→∞ as n→∞, Lemma 4 will imply that J→0 as n→∞. Indeed, mR→∞ as n→∞ under some conditions, as shown in the following lemma.Lemma 5.
m
R
→
∞ as n-m→∞ provided m>(1/4)nμ, n>4.Proof.
Write (14) in the form
(26)mR1-e-R=n-μR.
Now the application of the mean value theorem to the functionf(R)=1-e-R over the interval (0,R) will yield
(27)1-e-R=Re-ς,
where ς is a number within the interval (0,R). Then,
(28)mn-μR=e-ς.
Consequently,(29)1-e-R=R(mn-μR),1-mRn-μR=e-R≤e-ς=mn-μR,1-mRn-μR≤mn-μR.
The last inequality will yield(30)mR≥(n-μR)-m,
provided n-μR>0. The previous inequality can be written in the form
(31)mR≥n-(m-μR)≥n-(m+nμm)≥n-(m+4),
when m≥(1/4)nμ. The second inequality previous follows if R<n/m. Indeed, R<n/m when m≥(1/4)nμ and n>4 because f1(n/m)>f2(n/m), where f1 and f2 are the functions in the proof of Lemma 1. Hence, R which is the x-coordinate of the point of intersection of the two functions must be less than n/m. The last inequality shows that mR→∞ as n-m→∞ under the given restriction of m.
Returning to the condition thatn-μR>0, note that R<n/m. Thus,
(32)n-μR>n-μnm>n-4,
under the restriction that m≥(1/4)nμ. So that n-μR>0 whenever n>4.For example, whenn=100 and μ=4/7, m in Lemma 5 must satisfy m≥(1/4)100(4/7)=14.2857. Thus, m≥15.


## 3. Asymptotic Formula with Integral Parameters

In the discussion that follows,R denotes the unique positive solution to (14). It will be seen later that our final asymptotic formula is expressed in terms of powers of 1/mR. Anticipating this result and in view of Lemma 4, we write
(33)Sβ,r(m,n)~A∫-ϵϵ‍exph(θ,R)dθ.Substituting (10) for h(θ,R) and noting that B(R)=0, (33) becomes
(34)Sβ,r(n,m)~A∫-ϵϵ‍exp[-mRHθ2+m∑k=3∞‍Dk(R)(iθ)k]dθ.The change of variableϕ=(mRH)1/2θ will yield dϕ=(mRH)1/2dθ,
(35)θ=ϕ(mRH)1/2,dθ=dϕ(mRH)1/2.Now, (34) becomes
(36)Sβ,r(n,m)~A(mRH)1/2∫-αα‍e-ϕ2exp[f(z,R,ϕ)]dϕ,
where α=(mRH)1/2ϵ, z=(mRH)-1/2,
(37)f(z,R,ϕ)=∑k=1∞‍Dk+2(R)(iϕ)k+2RH(k+2)/2(mR)-k/2.The equation in (37) can be written in the from
(38)f(z,R,ϕ)=∑k=1∞‍akzk,
where
(39)ak=Dk+2(R)(iϕ)k+2RH(k+2)/2.Note here thatz=(mR)-1/2 is within the radius of convergence of the series in (38). On the other hand, the Maclaurin series of expf(z,R,ϕ) is
(40)ef(z,R,ϕ)=∑k=0∞‍bk(R,ϕ)zk,
where b2k is a polynomial in ϕ containing only even powers of ϕ, and b2k+1 is a polynomial in ϕ containing only odd powers of ϕ. Now, (36) can be written in the form
(41)Sβ,r(n,m)~Q∫-αα‍e-ϕ2(∑k=0∞‍bk(R,ϕ)zk)dϕ,
where Q=A/mRH and z=(mR)-1/2.We write (41) as
(42)Sβ,r(n,m)~Q[∑k=0s-1‍zk(∫-αα‍e-ϕ2bkdϕ)+Us],
where
(43)Us=∑k=s∞‍(∫-αα‍e-ϕ2bkdϕ)zk.In view of Lemma2, H≥1/4; hence,
(44)α=(mRH)1/2ϵ=(mRH)1/2(mR)-3/8≥(mR)1/82.Sincebk is a polynomial in ϕ, we can replace α with ∞ in (41). Following the discussion in [3], it can be shown that Us=O(zs); hence, we have the asymptotic formula
(45)Sβ,r(n,m)~Q[∑k=0∞‍(∫-∞∞‍e-ϕ2b2kdϕ)(mR)-k].Note that(46)∫-∞∞‍e-ϕ2b2k+1dϕ=0.This is why no odd subscript ofbk appears in (45).An approximation is obtained by taking the first two terms of the sum in (45). Thus,
(47)Sβ,r(n,m)≈Q[∫-∞∞‍e-ϕ2b0dϕ+(mR)-1∫-∞∞‍e-ϕ2b2dϕ].It can be computed that(48)b0=1,∫-∞∞‍e-ϕ2dϕ=π,b2=a2(R,ϕ)+12[a1(R,ϕ)]2,a2=D4(R)(iϕ)4RH2,D4(R)=μR24m+C4(R),
where
(49)C4(R)=124[R+(R-7R2+6R3-R4)(eR-1)-1]+124[(18R3-7R2-7R4)(eR-1)-2kkkkkkk+(12R3-12R4)(eR-1)-3]-124[6R4(eR-1)-4],
while
(50)a1(R,ϕ)=D3(iϕ)3RH3/2=(μR3!m+C3)(iϕ)3RH3/2,
where
(51)C3=16[R+(R-3R2+R3)(eR-1)-1kkkkkk+(3R3-3R2)(eR-1)-2]+16[2R3(eR-1)-3].Thus,b2 can be written as follows:
(52)b2=[μR24m+C4]ϕ4RH2-12(μR6m+C3)2ϕ6R2H3.Let(53)I=∫-∞∞‍e-ϕ2b2dϕ.Then,(54)I=∫-∞∞‍e-ϕ2(μR/24m+C4)RH2ϕ4dϕ-12∫-∞∞‍e-ϕ2(μR/6m+C3)2R2H3ϕ6dϕ=μR/24m+C4RH2∫-∞∞‍e-ϕ2ϕ4dϕ-(μR/6m+C3)22R2H3∫-∞∞‍e-ϕ2ϕ6dϕ=μR/24m+C4RH2·3π4-(μR/6m+C3)2R2H3·15π16.Finally, we obtain the approximation formula(55)Sβ,r(n,m)≈n!m!eμR(eR-1)m2βm-nRnπmRH[1+ImRπ].In view of Lemmas4 and 5 and (33), the previous asymptotic approximation is valid for m≥(1/4)nμ. The formula obtained in the previous discussion is formally stated in the following theorem.Theorem 6.
The formula(56)Sβ,r(n,m)~Q[π+ImR]
behaves as an asymptotic approximation for the generalized Stirling numbers of the second kind Sβ,r(n,m) with positive integral parameters for m>(1/4)nμ, n>4 and such that n-m→∞ as n→∞.Table1 displays the exact values and approximate values for n=100, r=4, β=7. The exact values are obtained using recurrence formula while the approximate values are obtained using Theorem 6. In view of Lemma 5, the formula is valid for m≥15.Table 1
Exact value
Approximate value
Relative error
S
7,4
(
100,5
)
5.685
×
1
0
152
6.335
×
1
0
152
0.11428
S
7,4
(
100,10
)
7.728
×
1
0
171
8.169
×
1
0
171
0.05713
S
7,4
(
100,15
)
8.411
×
1
0
178
8.731
×
1
0
178
0.03804
S
7,4
(
100,30
)
5.604
×
1
0
174
5.706
×
1
0
174
0.01824
S
7,4
(
100,60
)
7.399
×
1
0
122
7.446
×
1
0
122
0.00641
S
7,4
(
100,80
)
1.275
×
1
0
70
1.279
×
1
0
70
0.00263
S
7,4
(
100,90
)
2.208
×
1
0
38
2.211
×
1
0
38
0.00123Values on the table affirm that the asymptotic formula in Theorem6 gives a good approximation for S7,4(100,m) when m≥15.


## 4. Asymptotic Formula with Real Parameters

Recently, the(r,β)-Stirling numbers for complex arguments were defined in [15] parallel to the definition of Flajolet and Prodinger as
(57)Sβ,r(x,y)=x!y!βy2πi∫ℋ‍erz(eβz-1)ydzzx+1,
where x!:=Γ(x+1) and ℋ is a Hankel contour that starts from -∞ below the negative axis surrounds the origin counterclockwise and returns to -∞ in the half plane ℑz>0. Note that by change of variable, say w=βz, we can express Sβ,r(x,y) as follows:
(58)Sβ,r(x,y)=βx-y{x+rβy+rβ}r/β,
where
(59){x+ry+r}r=x!y!2πi∫ℋ‍erz(ez-1)ydzzx+1.The numbers{x+ry+r}r are certain generalization of r-Stirling numbers of the second kind [8] in which the parameters involved are complex numbers. These numbers satisfy the following properties.Theorem 7.
For nonnegative real numberr, one has
(60){x+ry+r}r={x+r-1y+r-1}r+(y+r){x+r-1y+r}r.Theorem 8.
For nonnegative real numbersr and β and k∈ℕ, one has
(61){x+rk+r}r=1k!∑j=0k‍(-1)k-j(kj)(j+r)x.
Furthermore, whenk=y, a complex number
(62){x+ry+r}r=1y!∑j=0∞‍(-1)y-j(yj)(j+r)x.The Bernoulli polynomial can be expressed using the explicit formula in [16] and the first formula in Theorem 8, with w=r and x=n, as
(63)B-n(r)=∑k=0∞‍(-1)kk!k+1{n+rk+r}r.Moreover, it is known that, forn=1,2,…, the Hurwitz zeta function ζ(1-n,x)=-B-n(x)/n. Note that, as y→0, (y-1)!~1/y. By Cauchy's integral formula
(64){n+ry+r-1}r~yn![zn+1]erzez-1=yB-n+1(r)n+1,
which further gives, using a result in [17], the following relation:
(65)[ddy{n+ry+r}r]y=-1=-ζ(-n,r)=n!2πi∫ℋ‍erz(ez-1)-1dzzn+1.An asymptotic formula forr-Stirling numbers of the second kind was first considered by Corcino et al. in [14]. However, the formula only holds for integral arguments. In this section, we are going to establish an asymptotic formula for r-Stirling numbers of the second kind that will hold for real arguments.Consider the integral representation in (59) where |ℑz|<2π. To see the analysis of Moser-Wyman applies to r-Stirling numbers with real arguments x and y, we deform the path ℋ into the following contour: C1∪C2∪C3∪C4∪C5 where(1)
C1 is the line ℑz=-2π+δ,δ>0 and ℜz≤ϵ, ϵ is a small positive number;(2)
C2 is the line segment ℜz=ϵ, going from ϵ+i(δ-2π) to the circle |z|=R;(3)
C5 and C4 are the reflections in the real axis of C1 and C2, respectively; and(4)
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 [11].The integrals alongC1, C2, C4, and C5 are seen to be
(66)O((2+ϵ)y(2π-δ)x).It will also be shown that these integrals go to 0 asx→∞ provided that x-y≥xα where 0<α<1. To see this, we consider C1. For the other contours, the estimate can be seen similarly.Note that forz in C1, we can choose δ so that
(67)[1-(eℜzcos(ℑz))-1]2<1.From this and the assumptions thatℜz<ϵ and ϵ<1, it follows that
(68)|ez-1|≤eℜz<ϵ<2+ϵ2-ϵ<2+ϵ.Thus,(69)|ez-1|y<(2+ϵ)y.Consequently,(70)|∫C1‍(ez-1)yzx+1dz|≤∫C1‍(2+ϵ)y(2π-δ)x+1|dz|,
where ∫C1‍|dz| is the length of C1. With C1 the horizontal line ℑz=-2π+δ, the length of C1 is a linear function of the real part of z, given by
(71)l(C1)=limt→∞(ϵ-t).Hence, we have(72)|∫C1‍(ez-1)yzx+1dz|≤(2+ϵ)y(2π-δ)x+1limt→∞(ϵ-t)<limt→∞(ϵ-t)(2π-δ)x-y.The last inequality follows from the fact that2+ϵ<2π-δ. With the condition that x-y≥xα, 0<α<1, and the fact that 2π-δ>e, it follows that the integral along C1 goes to 0 as x→∞. Thus, we have
(73){x+ry+r}r~x!y!12πi∫C3‍erz(ez-1)yzx+1dz.Using the method of Moser and Wyman [3], we obtain the following asymptotic formula.Theorem 9.
Ther-Stirling numbers of the second kind with real arguments x and y have the following asymptotic formula:
(74){x+ry+r}r=x!y!erR(eR-1)y2RxπyRH[1+O(1y)]
valid for x-y→∞ as x→∞, provided that x-y≥xα with 0<α<1, where
(75)H=r2y+eR(eR-R-1)2(eR-1)2,
and R is the unique positive solution to the equation
(76)w[r-y(1-e-w)-1]-x=0,
as a function of w.Chelluri et al. [11] has made a modification of Moser and Wyman formula and analysis. Using this analysis of Chelluri, we can restate Theorem 9 as follows.Theorem 10.
Ther-Stirling numbers of the second kind with real arguments x and y have the following asymptotic formula:
(77){x+ry+r}r=x!y!erR(eR-1)y2RxπyRH[1+O(1x)]
valid for x-y→∞ as x→∞, provided that x-y≥x1/3, where
(78)H=r2y+eR(eR-R-1)2(eR-1)2,
and R is the unique positive solution to the equation
(79)w[r-y(1-e-w)-1]-x=0,
as a function of w.As noted in (58), the (r,β)-Stirling numbers can be obtained using the r-Stirling numbers of the second kind for complex arguments x, y, r, and β. This implies, using Theorems 9 and 10, the following asymptotic formulas which agree with the formula in (55) for the integral values of x and y.Corollary 11.
The(r,β)-Stirling numbers with real arguments x and y have the following asymptotic formulas:
(80)Sβ,r(x,y)=βx-yx!y!erR/β(eR-1)y2RxπyRH[1+O(1y)]
provided that x-y≥xα with 0<α<1, and
(81)Sβ,r(x,y)=βx-yx!y!erR/β(eR-1)y2RxπyRH[1+O(1x)]
provided that x-y≥x1/3, where
(82)H=r2βy+eR(eR-R-1)2(eR-1)2
and R is the unique positive solution to the equation
(83)w[rβ-y(1-e-w)-1]-x=0
as a function of w.



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