Abstract
We locate the peak of the distribution of noncentral Stirling numbers of the first kind by determining the value of the index corresponding to the maximum value of the distribution.
1. Introduction
In 1982, Koutras [1] introduced the noncentral Stirling numbers of the first and second kind as a natural extension of the definition of the classical Stirling numbers, namely, the expression of the factorial in terms of powers of and vice versa. These numbers are, respectively, denoted by and which are defined by means of the following inverse relations:where , are any real numbers, is a nonnegative integer, andThe numbers satisfy the following recurrence relations:and have initial conditionsIt is worth mentioning that for a given negative binomial distribution and the sum of independent random variables following the logarithmic distribution, the numbers appeared in the distribution of the sum , while the numbers appeared in the distribution of the sum where is the sum of independent random variables following the truncated Poisson distribution away from zero and is a Poisson random variable. More precisely, the probability distributions of and are given, respectively, byFor a more detailed discussion of noncentral Stirling numbers, one may see [1].
Determining the location of the maximum of Stirling numbers is an interesting problem to consider. In [2], Mezö obtained results for the so-called -Stirling numbers which are natural generalizations of Stirling numbers. He showed that the sequences of -Stirling numbers of the first and second kinds are strictly log-concave. Using the theorem of Erdös and Stone [3] he was able to establish that the largest index for which the sequence of -Stirling numbers of the first kind assumes its maximum is given by the approximation
Following the methods of Mezö, we establish strict log-concavity and hence unimodality of the sequence of noncentral Stirling numbers of the first kind and, eventually, obtain an estimating index at which the maximum element of the sequence of noncentral Stirling numbers of the first kind occurs.
2. Explicit Formula
In this section, we establish an explicit formula in symmetric function form which is necessary in locating the maximum of noncentral Stirling numbers of the first kind.
Let , be differentiable functions and let . It can easily be verified that, for all ,Now, consider the following derivative of when : Then, for and using (9), we getThen, we have the following lemma.
Lemma 1. For any nonnegative integers and , one has
Proof. We prove by induction on . For , (12) clearly holds. For , (12) can easily be verified using (11). Suppose for , Then,where the sum has terms and its summand has terms. Therefore, the expansion of has a total of terms of the form . However, if the sum is evaluated over all possible combinations such that , then the sum has distinct terms. It follows that every term appears times in the expansion of . Thus we have
Lemma 2. Let . Then
Proof. Using Lemma 1,Note that . Hence, the expression at the right-hand side of (18) becomeswhich boils down tosincewhere denote the Stirling numbers of the second kind.
Theorem 3. The noncentral Stirling numbers of the first kind equal
Proof. We know thatis equal to the sum of the products where the sum is evaluated overall possible combinations , . These possible combinations can be divided into two: the combinations with for some and the combinations with for all . Thus is equal toThis implies thatThis is exactly the triangular recurrence relation in (4) for . This proves the theorem.
The explicit formula in Theorem 3 is necessary in locating the peak of the distribution of noncentral Stirling numbers of the first kind. Besides, this explicit formula can also be used to give certain combinatorial interpretation of .
A - tableau, as defined in [4] by de Médicis and Leroux, is a pair , whereis a partition of an integer , and is a “filling” of the cells of corresponding Ferrers diagram of shape with 0’s and 1’s, such that there is exactly one 1 in each column. Using the partition we can construct 60 distinct - tableaux. One of these - tableaux is given in the following figure with , elsewhere : Also, as defined in [4], an -tableau is a list of column of a Ferrers diagram of a partition (by decreasing order of length) such that the lengths are part of the sequence , . If is the set of -tableaux with exactly distinct columns whose lengths are in the set , then . Now, transforming each column of an -tableau in into a column of length , we obtain a new tableau which is called -tableau. If , then the -tableau is simply the -tableau. Now, we define an -tableau to be a - tableau which is constructed by filling up the cells of an -tableau with 0’s and 1’s such that there is only one 1 in each column. We use to denote the set of such -tableaux.
It can easily be seen that every combination of the set can be represented geometrically by an element in with as the length of th column of where . Hence, with , (22) may be written asThus, using (29), we can easily prove the following theorem.
Theorem 4. The number of -tableaux in where such that is equal to .
Let be an -tableau in with , andIf for some and , then, with ,Suppose is the set of all -tableaux corresponding to such that for each either has no column whose weight is , or has one column whose weight is , or has columns whose weights are . Then, we may write Now, if columns in have weights other than , thenwhere . Hence, (29) may be written asNote that for each , there correspond tableaux with distinct columns having weights , . Since has elements, for each , the total number of -tableaux corresponding to is elements. However, only tableaux in with distinct columns of weights other than are distinct. Hence, every distinct tableau appears times in the collection. Consequently, we obtainwhere denotes the set of all tableaux having distinct columns whose lengths are in the set . Reindexing the double sum, we get Clearly, . Thus, using (22), we obtain the following theorem.
Theorem 5. The numbers satisfy the following identity:where for some numbers and .
The next theorem contains certain convolution-type formula for which will be proved using the combinatorics of -tableau.
Theorem 6. The numbers have convolution formula
Proof. Suppose that is a tableau with exactly distinct columns whose lengths are in the set and is a tableau with exactly distinct columns whose lengths are in the set . Then and . Notice that by joining the columns of and , we obtain an -tableau with distinct columns whose lengths are in the set ; that is, . HenceNote thatAlso, using (29), we haveThus,
The following theorem gives another form of convolution formula.
Theorem 7. The numbers satisfy the second form of convolution formula
Proof. Let be a tableau with columns whose lengths are in , be a tableau with columns whose lengths are in .Then ; . Using the same argument above, we can easily obtain the convolution formula.
3. The Maximum of Noncentral Stirling Numbers of the First Kind
We are now ready to locate the maximum of . First, let us consider the following theorem on Newton’s inequality [5] which is a good tool in proving log-concavity or unimodality of certain combinatorial sequences.
Theorem 8. If the polynomial has only real zeros then
Now, consider the following polynomial:This polynomial is just the expansion of the factorial which has real roots . If we replace by , we see at once that the roots of the polynomial are . Applying Newton’s Inequality completes the proof of the following theorem.
Theorem 9. The sequence is strictly log-concave and, hence, unimodal.
By replacing with , the relation in (1) may be written aswhere . Note that, from Theorem 3 with , where . Now, we define the signless noncentral Stirling number of the first kind, denoted by , as
To introduce the main result of this paper, we need to state first the following theorem of Erdös and Stone [3].
Theorem 10 (see [3]). Let be an infinite sequence of positive real numbers such thatDenote by the sum of the product of the first of them taken at a time and denote by the largest value of for which assumes its maximum value. Then
We also need to recall the asymptotic expansion of harmonic numbers which is given bywhere is the Euler-Mascheroni constant.
The following theorem contains a formula that determines the value of the index corresponding to the maximum of the sequence .
Theorem 11. The largest index for which the sequence assumes its maximum is given by the approximation where is the integer part of and , .
Proof. Using Theorem 10 and by (50), we see that . Denoting by for which is maximum and with we haveBut using (53), we see thatFrom this we get
For the case in which we will only consider the sequence of noncentral Stirling numbers of the first kind for which .
Theorem 12. The maximizing index for which the maximum noncentral Stirling number occurs for is given by the approximation
Proof. From the definition, for , and by Theorem 3, is the sum of the products where ’s are taken from the set . By Theorem 10, . Thus with we haveAgain, using (53), we get
Example 13. The maximum element of the sequence occurs at (Table 1)
Example 14. The maximum element of the sequence occurs at (Table 2)
We know that the classical Stirling numbers of the first kind are special cases of by taking . However, formulas in Theorems 11 and 12 do not hold when . Hence, these formulas are not applicable to determine the maximum of the classical Stirling numbers. Here, we derive a formula that determines the value of the index corresponding to the maximum of the signless Stirling numbers of the first kind.
The signless Stirling numbers of the first kind [6] are the sum of all products of different integers taken from . That is, Using Theorem 10, . We use to denote the largest value of for which is maximum. With we haveUsing (53), we see thatTherefore, we have
Example 15. It is shown in Table 3 that the maximum value of when occurs at . Using (66), it can be verified that the maximum element of the sequence occurs atMoreover, when , the maximum value occurs at
Recently, a paper by Cakić et al. [7] established explicit formulas for multiparameter noncentral Stirling numbers which are expressible in symmetric function forms. One may then try to investigate the location of the maximum value of these numbers using the Erdös-Stone theorem.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgment
The authors wish to thank the referees for reading the paper thoroughly.