Article Generalized q-Stirling Numbers and Their Interpolation Functions OPEN ACCESS

In this paper, we define the generating functions for the generalized q-Stirling numbers of the second kind. By applying Mellin transform to these functions, we construct interpolation functions of these numbers at negative integers. We also derive some identities and relations related to q-Bernoulli numbers and polynomials and q-Stirling numbers of the second kind.

In [16], Simsek studied the generating functions of the fermionic and deformic Stirling numbers.By applying the derivative operator d n dt n | t=0 to these functions, he constructed interpolation functions of these numbers at negative integers.
It is well known that the Stirling numbers of the second kind S(n, k) are defined by means of the generating function [16][17][18][19][20][21]: It is also well known that the usual Stirling numbers of the second kind S (α) (n, k) are defined by means of the generating function [16][17][18][19][20][21]: and Let q ∈ C with | q |< 1.Some well known results related to the q-integers are given by (see for detail ): Generating functions of the q-Stirling numbers of the second kind were defined in [8]: and By the above equation, we have [1,8] 2. New Generating Functions for q-Stirling Numbers of the Second Kind Here, by using the same method of Simsek [16,18], we construct interpolation functions for the generalized q-Stirling numbers of the second kind.We shall define new functions to interpolate the second kind q-Stirling numbers.We define q-version of Equations ( 1) and (2) functions.Generalized q-Stirling numbers of the second kind are defined by means of the following generating functions: and By comparing the coefficients of t n n! on both sides of the above equations, we easily obtain that Observe that when q → 1, Equations ( 5) and ( 6) reduce to Equation (2).When q → 1 in Equation ( 5), we have Here we use the binomial expansion and the fact that By comparing the coefficients of t n n! on both sides of the above equations, we easily obtain that We also see that We also define the following generating function which is generalized Equation ( 6): By Equation ( 7), we obtain By using Pb.189 in [24], we can write We give the q-version of the above equation as follows

Some Special Zeta Functions
Throughout this section, let s ∈ C with Res > 1.By using the same method of Simsek [16,18], we construct interpolation functions for the generalized q-Stirling numbers of the second kind.By applying the Mellin transform to Equation (4), we have Thus we define the following zeta function: By substituting s = 1 − n into above definition, we have Using the above relation, we arrive at the following result: Theorem 1 Let n and k be positive integers.Then By applying the Mellin transform to the Equation (2), we have So we have the following definition: For s = 1 − n, n ∈ Z + above equation, we obtain Therefore, we arrive at the following result: By using Equation ( 8), we have By applying the Mellin transform to Equations ( 6) and ( 7), we define the following functions, respectively: The above functions interpolate the numbers S (α) (n, k, q) and S ( α r ) (n, k, q) at negative integers, respectively.

Relations between Bernoulli Numbers of Order k and Stirling Numbers of the Second Kind
Let where the coefficients n are called Bernoulli numbers of order k [19,20,28].By Equation (1), we have By using Equations ( 9) and (10), relation between F B (t) and F S (t) is given by By using the above relation, we have By using Cauchy product above, we get By comparing coefficients of t k in both sides of the above equation, we arrive at the following theorem: The Barnes' type multiple Changhee q-Bernoulli polynomials are defined by means of the following generating function (see for details [28]): with as usual, It follows from Equation (11) that This gives the generating function of Barnes' type multiple Bernoulli numbers.Thus we get the following limit relationship: This gives the Barnes' type multiple Bernoulli numbers as a limit when q approaches 1.
If w = 0 and w 1 = w 2 = w k = 1 in Equation ( 12), we have Using Equation ( 12), we define By Equations ( 13) and ( 14), we have By using the above equation, we have By applying the Cauchy product to the above, we arrive at the following theorem, which is the generalized form of Theorem 2: Observe that F Thus we have Y (1) (n, k | 1, 1, ..., 1) = S(n, k)

Conclusions
q-Stirling numbers of the second kind arise in many different generating functions for various statistical partitions.The theory of q-Stirling numbers is enriched by combinatorial interpretations.By using these numbers, one can investigate orthogonality relations, recurrences, explicit expressions, and generating functions for the generalized (q-) Stirling numbers.Recently, many authors have generalized the Stirling numbers by differential operators.The Stirling numbers are related to Newton's interpolation, (q-) Lah numbers, exponential generating functions, q-calculus and related topics, combinatorial enumeration problems, Binomial coefficients and Bell numbers.