The r-Whitney numbers of the second kind are a generalization of all the Stirling-type numbers of the second kind which are in line with the unified generalization of Hsu and Shuie. In this paper, asymptotic formulas for r-Whitney numbers of the second kind with integer and real parameters are obtained and the range of validity of each formula is established.
1. Introduction
The r-Whitney numbers of the second kind, denoted by Wβ,r(n,m), have been introduced by Mezo [1] to obtain a new formula for Bernoulli polynomials. These numbers are equivalent to the numbers considered by Rucinski and Voigt [2] and the (r,β)-Stirling numbers [3]. They are considered as a generalization of all the Stirling-type numbers of the second kind which satisfy
(1)1βm(m!)erz(eβz-1)m=∑n=m∞Wβ,r(n,m)znn!,
where n and m are positive integers. More properties of r-Whitney numbers of the second kind can be found in [1, 3–7]. For instance, the index K^β,r(n) for which the sequence {Wβ,r(n,k)}k=0n assumes its maximum value satisfies
(2)K^β,r(n)<nlogn-loglogn,n≥3,nβlogn-rβ<K^β,r(n),n≥max{nβ,log2βlog(1+β/r)}.
This sequence was also shown in [3] to be unimodal for fixed n≥3 with k≤n and further shown to be asymptotically normal in the sense that
(3)∑j=1xn1Gn,r,βWβ,r(n,k)⟶12π∫-∞xe-t2/2dtasn⟶∞,
where
(4)xn=Gn+2,r,βGn,r,β-(Gn+1,r,βGn,r,β)2x+(Gn+1,r,βGn,r,β-1),Gn,r,β=∑k=0nWβ,r(n,k)
represents the generalized Bell numbers.
The r-Whitney numbers of the second kind can be interpreted combinatorially as follows [5].
Consider k+1 distinct cells the first k of which each has β distinct compartments and the last cell with r distinct compartments. Suppose we distribute n distinct balls into the k+1 cells one ball at a time such that
the capacity of each compartment is unlimited;
the first k cells are nonempty.
Let Ω be the set of all possible ways of distributing n balls under restriction (A1). Then |Ω|=(βk+r)n and the number of outcomes in Ω satisfying (B1) is βkk!Wβ,r(n,k) with β, r≥0.
Recently, Cheon and Jung [8] gave certain combinatorial interpretation for the r-Whitney numbers over the Dowling lattice and derived some algebraic identities for such numbers. Moreover, they defined r-Dowling polynomials as
(5)Dm,r(n,x)=∑k=0nWβ,r(n,k)xk,
which give the above generalized Bell numbers Gn,r,β as particular case. That is, Gn,r,β=Dm,r(n,1). It is worth mentioning that Rahmani [9] obtained more combinatorial identities in relation to r-Dowling polynomials. On the other hand, Belbachir and Bousbaa [10] defined, combinatorially, certain translated r-Whitney numbers in terms of permutations and partitions under some conditions and obtained some properties parallel to those of r-Whitney numbers.
In a separate paper [11], an asymptotic formula has been obtained for r-Whitney numbers of the second kind, also called generalized Stirling numbers of the second kind, using saddle-point method. More precisely,
(6)Wβ,r(n,m)≈n!m!eμR(eR-1)m2βm-nRnπmRH[1+ImRπ],
which is valid for m>(1/4)n(r/β), n>4 such that n-m→∞ as n→∞, where μ=r/β, β≠0, and R is the unique positive solution to the equation
(7)R(μ+y1-e-R)-x=0,H=μ2y+eR(eR-R-1)2(eR-1)2.Table 1 displays the exact and approximate values of Wβ,r(n,m) for n=100, r=4, β=7.
Exact value
Approximate value
Relative error
W7,4(100,5)
5.685×10152
6.335×10152
0.11428
W7,4(100,10)
7.728×10171
8.169×10171
0.05713
W7,4(100,15)
8.411×10178
8.731×10178
0.03804
W7,4(100,30)
5.604×10174
5.706×10174
0.01824
W7,4(100,60)
7.399×10122
7.446×10122
0.00641
W7,4(100,80)
1.275×1070
1.279×1070
0.00263
W7,4(100,90)
2.208×1038
2.211×1038
0.00123
The approximation should be good for m>15 following the restriction m>(1/4)n(r/β). The computed approximate values for m=15,30,60,80,90 confirm this.
In this paper, another asymptotic formula for the r-Whitney numbers of the second kind Wβ,r(n,n-m) with integral values of m and n is obtained using a similar analysis as that in [12], which is proved to be valid when m is in the range n-o(n)≤n-m≤n. This can be considered as the final range since it covers the right most tail of the interval 0<m≤n. Since these subranges overlap, the present formula also counterchecks the other and may be used as an alternative formula for better computation. Moreover, it is shown that the formula obtained is valid in the given range when n and m are real numbers.
2. Derivation of the Asymptotic Formula
Applying Cauchy integral formula to (1), the following integral representation is obtained:
(8)Wβ,r(n,m)=n!2πiβm(m!)∫Cerz(eβz-1)mzn+1dz,
where C is a circle about the origin. Using this representation with m being replaced by n-m, we have
(9)Wβ,r(n,n-m)=(nm)m!βm2πi∫Ceνu(eu-1)n-mun+1du,
where u=βz, ν=r/β.
With f(u)=((eu-1)/u)-1, du=βdz, and eu-1=u[f(u)+1], (9) can be written as
(10)Wβ,r(n,n-m)=(nm)m!βm2πi∫Ceνu[f(u)+1]n-mum+1du.
We let q=2/(n-m) and introduce the new variable qw=u. Then du=qdw and (10) can further be written as
(11)Wβ,r(n,n-m)=(nm)m!βm2πiqm∫Ceνqw[f(qw)+1]2/qw-(m+1)dw.
Let
(12)T(q,w,ν)=exp{-w+νqw+2qlog(f(qw)+1)},
where the logarithm is to the base e. Then
(13)Wβ,r(n,n-m)=(nm)m!βm2πiqm∫CT(q,w,ν)wm+1dw.
Consider h(q,w)=eqw-1. The Maclaurin series of h(q,w) is given by
(14)h(q,w)=∑k=0∞(qw)kk!-1=∑k=1∞qkwkk!.
Let G(q,w)=log[f(qw)+1]. Then
(16)G(q,w)=w2q+w224q2+0q3-w42880q4+⋯,(17)H(q,w)=G(q,w)q=w2+w224q+0q2-w42880q3+⋯.
Note that T(q,w,ν)=exp[F(q,w,ν)], where F(q,w,ν)=-w+νqw+2H(q,w) and F(0,w,ν)=0.
Writing Tk(w,ν)=Tk(0,w,ν)/k!, we have
(18)T(q,w,ν)=∑k=0∞Tk(w,ν)qk=1+∑k=1∞Tk(w,ν)qk.
We prove the following lemma.
Lemma 1.
Tk(w,ν) is a polynomial in w whose lowest power in w is at least k.
Proof.
Let E(q,w)=e-w+2H(q,w) and L(q,w)=eνqw. Then
(19)T(q,w,ν)=e-w+νqwexp[2qlog[f(qw)+1]]=E(q,w)L(q,w).
By Leibniz Rule,
(20)[dkTdqk]q=0=[E(q,w)Lk(q,w)]q=0+[∑p=1k(kp)E(p)(q,w)L(k-p)(q,w)]q=0,
where E(p)(q,w) denotes the pth derivative of E(q,w) with respect to q and L(k-p)(q,w) denotes the (k-p)th derivative of L(q,w) with respect to q and E(0)(q,w)=E(q,w).
Denote the lowest power of w in a polynomial P(w) by η[P(w)]. From the computations above, η[f(qw)]=1; η[f(k-1)(qw)]q=0=k-1; η[H(q,w)]=1; η[H(k)(0,w)]=k+1. With h(q)=-w+2H(q,w), h(k)(q)=2H(k)(q,w). Hence, η[h(k)(0,w)]=k+1.
To find η[E(k)(0,w)], note that the concern is only the power of w, so we omit the details of the constant coefficients in the formula. With E(q,w)=eh(q,w) and applying Faa di Bruno’s formula on the mth derivative of a composite function, the following will be obtained:
(21)[E(k)(q,w)]q=0=[e-w+2H(q,w)∑ci(h′(q))b1(h′′(q))b2⋯(h(k)(q))bk]q=0,
where ci denotes the constant coefficient.
The factor e-w+2H(q,w) in the above expression for [E(k)(q,w)]q=0 does not contribute to the resulting power of w because H(0,w)=w/2; and hence at q=0,e-w+2H(0,w)=e0=1. Thus, we only need to count the power of w in each term of the sum. Each h(j)(0), if it does occur as a factor in a term, contributes at least (j+1)bj in the power of w. Hence, the lowest power of w in E(k)(0,w) is k+i, where 1≤i≤k. The least i is 1; thus; the least power of w in Ek(0,w) is k+1. Using the greatest value of i which is k, we get 2k as the greatest power of w in E(k)(0,w). Now, we have
(22)η[E(q,w)L(k)(q,w)]=η[E(q,w)]+η[L(k)(q,w)]=0+k=k,
while
(23)η[E(p)(q,w)L(k-p)(q,w)]=p+1+k-p=k+1.
Note that
(24)[dkTdqk]q=0=[E(q,w)L(k)(q,w)]q=0+[∑p=1k(kp)E(p)(q,w)L(k-p)(q,w)]q=0.
Hence, Tk(w,ν) is a polynomial in w whose lowest power in w is at least k.
In particular, for k=1,2,3, the computation for Tk(w,ν) gives
(25)T1(w,ν)=νw+w212,T2(w,ν)=ν2w2+νw36+w4144,T3(w,ν)=ν3w3+(ν24-1240)w4+vw548+w61728.
Continuing in the derivation of the formula, we see that
(26)Wβ,r(n,n-m)=(nm)(βq)m[(dmdwm∑k=1mTk(w,ν)qkew)w=0(dmdwmew)w=0+(dmdwm∑k=1mTk(w,ν)qkew)w=0].
Note that the upper limit of the sum is replaced by m because, for k>m, the mth derivative of the sum evaluated at w=0 is 0. Hence
(27)Wβ,r(n,n-m)=(nm)(βq)m[1+∑k=1mqk(dmdwmTk(w,ν)ew)w=0].
To find the first few terms of the sum in (27), we solve ((dm/dwm)Tk(w,ν)ew)w=0 for k=1,2,3,… using the formula
(28)(dmdwmTk(w,ν)ew)w=0=∑j=0m(mj)(djdwjTk(w,ν))w=0(dm-jdwm-jew)w=0=∑j=0m(m)jj!(djdwjTk(w,ν))w=0.
It follows from the preceding lemma that, for j<k,
(29)(djdwjTk(w,ν))w=0=0.
Moreover, for j≥k,
(30)(djdwjTk(w,ν))w=0=j![wj],
where [wj] is the coefficient of wj in Tk(w,ν). Thus,
(31)(dmdwmTk(w,ν)ew)w=0=∑j=km(m)j[wj].
The first few terms of (27) are given as follows:
(32)Wβ,r(n,n-m)=(nm)(βq)m{{mν+(m)212}q+12{ν2(m)2+ν(m)36+(m)4144}q2+16{ν3(m)3+[ν24-1240](m)4+ν(m)548+(m)61728}q3+⋯}.
When ν=0, (32) will reduce to the formula obtained in [12]. Substituting q=2/(n-m) in (32) will yield
(33)Wβ,r(n,n-m)=(nm)(β(n-m)2)m×[1+1n-m{2mν+(m)26}+1(n-m)2{2ν2(m)2+ν(m)33+(m)472}+1(n-m)3{43ν3(m)3+[ν23-1180](m)4+ν(m)536+(m)61296}+⋯].
The formula in (33) gives values correct up to even the 3rd digit for m=10,6,5,4; r=1, β=2 as shown in Table 2.
Exact value
Approximate value
W2,1(100,90)
7.896×1032
7.895×1032
W2,1(100,94)
9.223×1020
9.223×1020
W2,1(100,95)
6.350×1017
6.350×1017
W2,1(100,96)
3.542×1014
3.542×1014
3. The Range of Validity of the Formula
To be able to use (33) as an exact formula beyond m=3 requires finding
(34)(dmdwmTk(w,ν)ew)w=0,
for k=4,5,…. Such computation is quite tedious considering that T(q,w,ν) is a composition of a number of functions. Hence, we need to establish the range of m for which (33) behaves as an asymptotic approximation for large n.
Write (27) in the form
(35)Wβ,r(n,m)=(nm)(βq)m×[1+∑k=1sqk(dmdwmTk(w,ν)ew)w=0+ES],
where
(36)ES=∑k=s+1mqk(dmdwmTk(w,ν)ew)w=0.
Let
(37)Amk=(dmdwmTk(w,ν)ew)w=0.
Then, by Leibniz’s rule,
(38)Amk=∑j=0m(mj)Tk(j)(0,ν),
where Tk(j)(0,ν) is the jth derivative of Tk(w,ν) evaluated at w=0. Because Tk(w,ν) is a polynomial in w whose lowest power in w is at least k, we may write
(39)Amk=∑j=km(mj)Tk(j)(0,ν).
Consider the Maclaurin expansion of h(q,w) in (16) and note that
(40)limk→∞|qk+1wk+1(k+1)!·k!qkwk|=limk→∞|qwk+1|=0.
Thus, by ratio test, the expansion of h(q,w) in (14) is absolutely convergent. In particular, it is absolutely convergent if |qw|<1.
Similarly, the series expansion of f(q,w) in (15) is absolutely convergent if |qw|<1. These imply that f(q,w) and h(q,w) are both analytic in the interior to the circle |w|=1 and consequently, so is T(q,w,ν) as defined in (12). Moreover, by the maximum modulus principle Tk(w,ν) takes its maximum on the circle and not inside the circle. By Cauchy’s inequality, we have
(41)|T(j)(0,ν)|≤Mj!,
where M is the maximum value of Tk(w,ν) on the circle |w|=1. Hence,
(42)|Amk|≤∑j=km(mj)Mj!=M∑j=kmm!j!j!(m-j)!≤Mm!∑n=0∞1n!≤Mm!e≤M(m)2ke,
where e denotes the natural number e=2.71828…. The last inequality above is justified by m!=(m)m≤(m)m≤(m)2k, because m≤2k and the degree of Tk(w) is at most 2k.
An estimate of ES is given by
(43)∑k=s+1m[Me2m2n-m]k.
Note that the right hand side of the last inequality is a geometric series with common ratio
(44)ρ=2m2Men-m.
If m2=o(n-m), for sufficiently large n,
(45)|Es|≤2[2Mem2n-m]s+1,
where M a finite constant. Therefore, (33) behaves as an asymptotic approximation for large values of n provided that limn→∞(m2/(n-m))=0. In other words, m=o(n-m)≤o(n). Thus, we have the following theorem.
Theorem 2.
The formula
(46)Wβ,r(n,n-m)=(nm)(β(n-m)2)m×[(m)612961+1n-m{2mν+(m)26}+1(n-m)2{2ν2(m)2+ν(m)33+(m)472}+1(n-m)3{43ν3(m)3+[ν23-1180](m)4+ν(m)536+(m)61296}+⋯]
behaves as an asymptotic approximation as n→∞ for n-m in the range n-o(n)≤n-m≤n.
Table 3 displays the exact and approximate values of Wβ,r(n,n-m) and their corresponding relative errors when β=7, r=4, and n=100.
Exact value
Approximate value
Relative error
W7,4(100,15)
8.411×10178
3.962×10168
1.00000
W7,4(100,30)
5.604×10174
3.648×10170
0.99993
W7,4(100,60)
7.399×10122
3.700×10122
0.50000
W7,4(100,80)
1.275×1070
1.269×1070
0.00539
W7,4(100,90)
2.208×1038
2.211×1038
0.00118
W7,4(100,92)
2.613×1031
2.615×1031
0.00095
W7,4(100,93)
7.249×1027
7.255×1027
0.00083
W7,4(100,95)
3.358×1020
3.360×1020
0.00058
W7,4(100,97)
6.617×1012
6.619×1012
0.00035
W7,4(100,98)
597867963
598005375
0.00023
We observe that the asymptotic formula for Wβ,r(n,m) in (6) is valid when m>(1/4)n(r/β) such that n-m→∞ as n→∞. On the other hand, the asymptotic formula in Theorem 2 for Wβ,r(n,n-m) is valid when n-o(n)≤n-m≤n as n→∞. These two asymptotic formulas are complimentary to each other since, for large values of n, the former will give a good approximation when m is not close to n, while the preceding will work efficiently when m is close to n. However, the two asymptotic formulas will fail when m≤(1/4)n(r/β). Hence, it is interesting to establish the asymptotic formula for Wβ,r(n,m) when m≤(1/4)n(r/β). Meantime, the formula in [11] which is obtained using saddle point method can be used for this range.
4. Asymptotic Formula with Real Parameters
Following Flajolet and Prodinger [13], we define
(47)Wβ,r(y,x)=y!2πiβxx!∫ℋerz(eβz-1)xzy+1dz,
where x and y are positive real numbers, y! and x! are generalized factorials defined via the gamma function as
(48)y!=Γ(y+1),x!=Γ(x+1),
and ℋ is the Hankel contour which starts at -∞, circles the origin, and goes back to -∞ subject to |ℑz|<2π. The integral in (47) may be written in the form
(49)Wβ,r(y,x)=βy-xy!2πix!∫ℋeνu(eu-1)xduuy+1.
Consequently,
(50)Wβ,r(y,y-x)=(yx)x!βx2πi∫ℋeνu(eu-1)xuy+1du.
Then the computations from (10) up to the lemma are valid. Equation (50) becomes
(51)Wβ,r(y,y-x)=(yx)x!βx2πiqx∫ℋT(q,w,ν)wx+1dw=(yx)x!βx2πiqx∫ℋ1+∑k=1∞Tk(w,ν)wx+1dw,
where Tk(w,ν) is a polynomial in w in Lemma 1. Then
(52)Wβ,r(y,y-x)=βx(yx)(y-x2)x[∑k=1∞x!2πi∫ℋewwx+1+x!2πi∫ℋ∑k=1∞Tk(w,ν)qkewwx+1dw].
To compute the first few terms of the integrals in (52), we use the computed value of T1, T2, and T3 obtained in Section 2 and apply the following classical identity due to Hankel:
(53)12πi∫ℋeww-x-1dw=1Γ(x+1).
Computation yields
(54)12πi∫ℋT1(w,ν)q1ewwx+1dw=q[ν(x-1)!+112+1(x-2)!],12πi∫ℋT2(w,ν)q2ewwx+1dw=q2[ν2(x-2)!+ν6·1(x-3)!+1144·1(x-4)!],12πi∫ℋT3(w,ν)q3ewwx+1dw=q3[ν3(x-3)!+(ν24-1240)1(x-4)!+ν48·1(x-5)!+11728·1(x-6)!ν3(x-3)!].
Substituting to (52) gives the following asymptotic formula:
(55)Wβ,r(y,y-x)=(yx)(β(y-x)2)x×[1+1y-x{2xν+(x)26}+1(y-x)2{2ν2(x)2+ν(x)33+(x)472}+1(y-x)3{43ν3(x)3+[ν23-1180](x)4+ν(x)536+(x)61296}+⋯],
which is analogous to the asymptotic formula in Theorem 2.
For the range of validity of this formula, we observe that, in (52), we can let
(56)Es=x!2πi∫ℋ∑k=s+1∞Tk(w,ν)qkewwx+1dw.
Since the series is convergent, we can interchange the order of integration and summation. Thus,
(57)Es=x!2πi∑k=s+1∞qk∫ℋTk(w,ν)ewwx+1dw.
Note that
(58)x!2πi∫ℋTk(w,ν)qkewwx+1dw=x!2πi∫ℋqkew∑j=k2kajwjwx+1dw=x!qk∑j=k2kaj12πi∫ℋw-(x+1-j)ewdw.
Using the classical identity of Hankel, we obtain
(59)x!2πi∫ℋTk(w,ν)qkewwx+1dw=x!qk∑j=k2kaj1Γ(x+1-j)=qk∑j=k2kajΓ(x+1)Γ(x+1-j).
Since aj is finite for all j=k,k+1,…,2k, there is a constant M such that |aj|≤M. Thus, we have
(60)|x!2πi∫ℋTk(w,ν)qkewwx+1dw|≤M|∑j=k2kqkΓ(x+1)Γ(x+1-j)|.
It is known that Γ(x+1)/Γ(x+1-j)~xj. Hence,
(61)|x!2πi∫ℋTk(w,ν)qkewwx+1dw|≤M|qk∑j=k2kxj|=M|qkx2k∑j=1k1xj|≤M|qkx2k11-1/x|.
Then,
(62)|Es|≤∑k=s+1∞M|qkx2k11-1/x|≤2kM|11-1/x|∑k=s+1∞|(x2y-x)k|≤2kM|11-1/x|(x2y-x)s+1∑k=0∞|(x2y-x)k|.
The series
(63)∑k=0∞|(x2y-x)k|
is bounded provided x2=o(y-x). Moreover, the factor 1/(1-1/x) is bounded for x≠1. Hence, the expansion for Wβ,r(y,y-x) behaves as an asymptotic formula when x=o(y-x)≤o(y), that is, when y-o(y)≤y-x≤y.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgment
The authors would like to thank Mindanao State University-Main Campus, Marawi City, Philippines for partially funding this research. The authors also wish to thank the referees for their corrections and suggestions that improved the clarity of this paper.
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