Abstract
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 -Whitney numbers of the second kind, denoted by , 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 -Stirling numbers [3]. They are considered as a generalization of all the Stirling-type numbers of the second kind which satisfy where and are positive integers. More properties of -Whitney numbers of the second kind can be found in [1, 3–7]. For instance, the index for which the sequence assumes its maximum value satisfies This sequence was also shown in [3] to be unimodal for fixed with and further shown to be asymptotically normal in the sense that where represents the generalized Bell numbers.
The -Whitney numbers of the second kind can be interpreted combinatorially as follows [5].
Consider distinct cells the first of which each has distinct compartments and the last cell with distinct compartments. Suppose we distribute distinct balls into the cells one ball at a time such that(A1)the capacity of each compartment is unlimited;(B1)the first cells are nonempty.
Let be the set of all possible ways of distributing balls under restriction (A1). Then and the number of outcomes in satisfying (B1) is with , .
Recently, Cheon and Jung [8] gave certain combinatorial interpretation for the -Whitney numbers over the Dowling lattice and derived some algebraic identities for such numbers. Moreover, they defined -Dowling polynomials as which give the above generalized Bell numbers as particular case. That is, . It is worth mentioning that Rahmani [9] obtained more combinatorial identities in relation to -Dowling polynomials. On the other hand, Belbachir and Bousbaa [10] defined, combinatorially, certain translated -Whitney numbers in terms of permutations and partitions under some conditions and obtained some properties parallel to those of -Whitney numbers.
In a separate paper [11], an asymptotic formula has been obtained for -Whitney numbers of the second kind, also called generalized Stirling numbers of the second kind, using saddle-point method. More precisely, which is valid for , such that as , where , , and is the unique positive solution to the equation Table 1 displays the exact and approximate values of for , , .
The approximation should be good for following the restriction . The computed approximate values for confirm this.
In this paper, another asymptotic formula for the -Whitney numbers of the second kind with integral values of and is obtained using a similar analysis as that in [12], which is proved to be valid when is in the range . This can be considered as the final range since it covers the right most tail of the interval . 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 and are real numbers.
2. Derivation of the Asymptotic Formula
Applying Cauchy integral formula to (1), the following integral representation is obtained: where is a circle about the origin. Using this representation with being replaced by , we have where , .
With , , and , (9) can be written as We let and introduce the new variable . Then and (10) can further be written as Let where the logarithm is to the base . Then
Consider . The Maclaurin series of is given by
Thus,
Let . Then Note that , where and .
Writing , we have We prove the following lemma.
Lemma 1. is a polynomial in whose lowest power in is at least .
Proof. Let and . Then
By Leibniz Rule,
where denotes the th derivative of with respect to and denotes the th derivative of with respect to and .
Denote the lowest power of in a polynomial by . From the computations above, ; ; ; . With , . Hence, .
To find , note that the concern is only the power of , so we omit the details of the constant coefficients in the formula. With and applying Faa di Bruno’s formula on the th derivative of a composite function, the following will be obtained:
where denotes the constant coefficient.
The factor in the above expression for does not contribute to the resulting power of because ; and hence at . Thus, we only need to count the power of in each term of the sum. Each , if it does occur as a factor in a term, contributes at least in the power of . Hence, the lowest power of in is , where . The least is 1; thus; the least power of in is . Using the greatest value of which is , we get as the greatest power of in . Now, we have
while
Note that
Hence, is a polynomial in whose lowest power in is at least .
In particular, for , the computation for gives
Continuing in the derivation of the formula, we see that Note that the upper limit of the sum is replaced by because, for , the th derivative of the sum evaluated at is 0. Hence
To find the first few terms of the sum in (27), we solve for using the formula It follows from the preceding lemma that, for , Moreover, for , where is the coefficient of in . Thus, The first few terms of (27) are given as follows: When , (32) will reduce to the formula obtained in [12]. Substituting in (32) will yield
The formula in (33) gives values correct up to even the 3rd digit for ; , as shown in Table 2.
3. The Range of Validity of the Formula
To be able to use (33) as an exact formula beyond requires finding for . Such computation is quite tedious considering that is a composition of a number of functions. Hence, we need to establish the range of for which (33) behaves as an asymptotic approximation for large .
Write (27) in the form where Let Then, by Leibniz’s rule, where is the th derivative of evaluated at . Because is a polynomial in whose lowest power in is at least , we may write
Consider the Maclaurin expansion of in (16) and note that
Thus, by ratio test, the expansion of in (14) is absolutely convergent. In particular, it is absolutely convergent if .
Similarly, the series expansion of in (15) is absolutely convergent if . These imply that and are both analytic in the interior to the circle and consequently, so is as defined in (12). Moreover, by the maximum modulus principle takes its maximum on the circle and not inside the circle. By Cauchy’s inequality, we have where is the maximum value of on the circle . Hence, where denotes the natural number . The last inequality above is justified by , because and the degree of is at most .
An estimate of is given by Note that the right hand side of the last inequality is a geometric series with common ratio If , for sufficiently large , where a finite constant. Therefore, (33) behaves as an asymptotic approximation for large values of provided that . In other words, . Thus, we have the following theorem.
Theorem 2. The formula behaves as an asymptotic approximation as for in the range .
Table 3 displays the exact and approximate values of and their corresponding relative errors when , , and .
We observe that the asymptotic formula for in (6) is valid when such that as . On the other hand, the asymptotic formula in Theorem 2 for is valid when as . These two asymptotic formulas are complimentary to each other since, for large values of , the former will give a good approximation when is not close to , while the preceding will work efficiently when is close to . However, the two asymptotic formulas will fail when . Hence, it is interesting to establish the asymptotic formula for when . 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 where and are positive real numbers, and are generalized factorials defined via the gamma function as and is the Hankel contour which starts at , circles the origin, and goes back to subject to . The integral in (47) may be written in the form Consequently, Then the computations from (10) up to the lemma are valid. Equation (50) becomes where is a polynomial in in Lemma 1. Then
To compute the first few terms of the integrals in (52), we use the computed value of , , and obtained in Section 2 and apply the following classical identity due to Hankel: Computation yields Substituting to (52) gives the following asymptotic formula: 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 Since the series is convergent, we can interchange the order of integration and summation. Thus,
Note that Using the classical identity of Hankel, we obtain Since is finite for all , there is a constant such that . Thus, we have It is known that . Hence, Then, The series is bounded provided . Moreover, the factor is bounded for . Hence, the expansion for behaves as an asymptotic formula when , that is, when .
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.