The Peak of Noncentral Stirling Numbers of the First Kind

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.


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, and The numbers satisfy the following recurrence relations: ( + 1, ) = ( , − 1) + ( − ) ( , ) , ( + 1, ) = ( , − 1) + ( − ) ( , ) and have initial conditions It is worth mentioning that for a given negative binomial distribution and the sum = 1 + 2 + ⋅ ⋅ ⋅ + 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 =̂+̂wherê is the sum of independent random variables following the truncated Poisson distribution away from zero and̂is a Poisson random variable. More precisely, the probability distributions of and are given, respectively, by For 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 2 International Journal of Mathematics and Mathematical Sciences 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 logconcavity 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.

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.
Then, for ≥ 3 and using (9), we get Then, we have the following lemma.

Theorem 3. The noncentral Stirling numbers of the first kind equal
Proof. We know that is 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 ∈ {1, 2, . . . , − + 1} and the combinations with ̸ = for all ∈ {0, 1, 2, . . . , − + 1}. Thus is equal to This implies that This 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 ( , ).
The next theorem contains certain convolution-type formula for ( , ) which will be proved using the combinatorics of -tableau.
Using the same argument above, we can easily obtain the convolution formula.

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 10 (see [3]). Let 1 < 2 < ⋅ ⋅ ⋅ be an infinite sequence of positive real numbers such that Denote 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 by where 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 {| ( , )|} =0 .
where [ ] is the integer part of and = − , < 0.
Proof. Using Theorem 10 and by (50), we see that | ( , )| = ∑ −1, − . Denoting by , for which ∑ , − is maximum and with 1 = + 0, 2 = + 1, . . . , −1 = + − 1 we have But using (53), we see that From this we get For the case in which > 0 we will only consider the sequence of noncentral Stirling numbers of the first kind for which ≥ .   0  1  2  3  0  1  1  1  2  2  1  3  6  6  1  4  2  Again, using (53), we get , = [log ( Example 13. The maximum element of the sequence { −1 (9, )} 9 =0 occurs at ( We know that the classical Stirling numbers of the first kind are special cases of ( , ) by taking = 0. However, formulas in Theorems 11 and 12 do not hold when = 0. 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 {1, 2, 3, . . . , − 1}. That is, 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.