Generating functions for generalized Stirling type numbers, Array type polynomials, Eulerian type polynomials and their applications

The first aim of this paper is to construct new generating functions for the generalized {\lambda}-Stirling type numbers of the second kind, generalized array type polynomials and generalized Eulerian type polynomials and numbers, attached to Dirichlet character. We derive various functional equations and differential equations using these generating functions. The second aim is provide a novel approach to deriving identities including multiplication formulas and recurrence relations for these numbers and polynomials using these functional equations and differential equations. Furthermore, by applying p-adic Volkenborn integral and Laplace transform, we derive some new identities for the generalized {\lambda}-Stirling type numbers of the second kind, the generalized array type polynomials and the generalized Eulerian type polynomials. We also give many applications related to the class of these polynomials and numbers.

http://www.fixedpointtheoryandapplications.com/content/2013/1/87 well-defined combinatorial contexts and they lead systematically to well-defined classes of functions. The concept of fixed point of a function is another useful idea in solving some functional equations. Having knowledge on the fixed points of functions frequently support to solve specific types of functional equations (cf. see, for detail, []).
Although, in the literature, one can find extensive investigations related to the generating functions for the Bernoulli, Euler and Genocchi numbers and polynomials and also their generalizations, the λ-Stirling numbers of the second kind, the array polynomials and the Eulerian polynomials, related to nonnegative real parameters, have not been studied yet. Therefore, this paper deal with new classes of generating function for generalized λ-Stirling type numbers of the second kind, generalized array type polynomials and generalized Eulerian polynomials, respectively. By using these generating functions, we derive many functional equations and differential equations. By using these equations, we investigate and introduce fundamental properties and many new identities for the generalized λ-Stirling type numbers of the second kind, the generalized array type polynomials and the generalized Eulerian type polynomials and numbers. We also derive multiplication formulas and recurrence relations for these numbers and polynomials. We derive many new identities related to these numbers and polynomials.
There are various applications of the classical Euler numbers in many branches of mathematics and mathematical physics. One of theme is related to the Brouwer fixed-point theorem, which is briefly described as follows: let D n be a unit disk in R n . It is well known that D n is a compact manifold bounded by the unit sphere S n- . Any smooth map g : D n → D n has a fixed point. The Brouwer fixed-point theorem: Any continuous function G : D n → D n as a fixed point (cf. see, for detail, []). The classical Euler numbers are related to the Brouwer fixed point theorem and vector fields (cf. see for detail []). Thus, the generalized Eulerian type numbers may be associated with the Brouwer fixed-point theorem and vector fields.
The remainder of this study is organized as follows: Section , Section  and Section  of this paper deal with new classes of generating functions which are related to generalized λ-Stirling type numbers of the second kind, generalized array type polynomials and generalized Eulerian polynomials, respectively. In Section , we derive new identities related to the generalized Bernoulli polynomials, the generalized Eulerian type polynomials, generalized λ-Stirling type numbers and the generalized array polynomials. In Section , we give relations between generalized Bernoulli polynomials and generalized array polynomials.

Generating function for generalized λ-Stirling type numbers of the second kind
The Stirling numbers are used in combinatorics, in number theory, in discrete probability distributions for finding higher order moments, etc. The Stirling number of the second kind, denoted by S(n, k), is the number of ways to partition a set of n objects into k groups. These numbers occur in combinatorics and in the theory of partitions and so on.
In this section, we construct a new generating function, related to nonnegative real parameters, for the generalized λ-Stirling type numbers of the second kind. We derive some elementary properties including recurrence relations of these numbers. The following definition provides a natural generalization and unification of the λ-Stirling numbers of the second kind. http://www.fixedpointtheoryandapplications.com/content/2013/1/87 Definition . Let a, b ∈ R + , λ ∈ C and v ∈ N  . The generalized λ-Stirling type numbers of the second kind S(n, v; a, b; λ) are defined by means of the following generating function: Remark . By setting a =  and b = e in (), we have the λ-Stirling numbers of the second kind which are defined by means of the following generating function: . Substituting λ =  into above equation, we have the Stirling numbers of the second kind By using (), we obtain the following theorem.
Theorem . Let a, b ∈ R + . Each of the following identities holds true: and Proof By using () and the binomial theorem, we can easily arrive at the desired results.
By using the formula (), for a, b ∈ R + , we can compute some values of the numbers S(n, v; a, b; λ) as follows: The above relation has been studied by Srivastava [] and Luo []. By setting λ =  in the above equation, we have the following result: By differentiating both sides of Eq. () with respect to the variable t, we obtain the following differential equations: By using Eqs. () and (), we obtain recurrence relations for the generalized λ-Stirling type numbers of the second kind by the following theorem: The generalized λ-Stirling type numbers of the second kind can also be defined by Eq. (): Proof By using (), we get ln a x-j n t n n! .
From the above equation, we obtain ln a x-j n t n n! . Therefore, Comparing the coefficients of t m m! on both sides of the above equation, we arrive at the desired result.
Remark . For a =  and b = e, the formula () can easily be shown to be reduced to the following result which is given by Luo and Srivastava [, Theorem ]: where n ∈ N  and λ ∈ C. For λ = , the above formula is reduced to [-, , ]). http://www.fixedpointtheoryandapplications.com/content/2013/1/87

Generalized array type polynomials
By using the same motivation with the λ-Stirling type numbers of the second kind, we also construct a novel generating function, related to nonnegative real parameters, of the generalized array type polynomials. We derive some elementary properties including recurrence relations of these polynomials. The following definition provides a natural generalization and unification of the array polynomials: By using the formula (), we can compute some values of the polynomials S n v (x; a, b; λ) as follows: Remark . The polynomials S n v (x; a, b; λ) may be also called generalized λ-array type polynomials. By substituting x =  into (), we arrive at (): Setting a = λ =  and b = e in (), we have a result due to Chang and Ha [, Eq. (.)] and Simsek []. It is easy to see that Generating functions for the polynomial S n v (x; a, b, c; λ) can be defined as follows.
Definition . Let a, b ∈ R + , x ∈ R, λ ∈ C and v ∈ N  . The generalized array type polynomials S n v (x; a, b; λ) are defined by means of the following generating function: (  ) http://www.fixedpointtheoryandapplications.com/content/2013/1/87 Proof By substituting () into the right-hand side of (), we obtain The right-hand side of the above equation is the Taylor series for e (ln(a v-j b x+j ))t . Thus, we get By using () and binomial theorem in the above equation, we arrive at the desired result.
Remark . If we set λ =  in (), we get a new special case of the array polynomials given by In the special case when a = λ =  and b = e, the generalized array polynomials S n v (x; a, b; λ) defined by () would lead us at once to the classical array polynomials S n v (x), which are defined by means of the following generating function: which yields the generating function for the array polynomials S n v (x) studied by Chang and The polynomials S n v (x; a, b; λ) defined by () have many interesting properties, which we give in this section.
We set Theorem . The following formula holds true: (   ) http://www.fixedpointtheoryandapplications.com/content/2013/1/87 Proof By using (), we obtain From the above equation, we get Comparing the coefficients of t n on both sides of the above equation, we arrive at the desired result.
Remark . In the special case when a = λ =  and b = e, Eq. () is reduced to By differentiating j times both sides of () with respect to the variable x, we obtain the following differential equation: From this equation, we arrive at higher order derivative of the array type polynomials by the following theorem.
Theorem . Let n, j ∈ N with j ≤ n. Then we have From (), we get the following functional equation: From this functional equation, we obtain the following identity. http://www.fixedpointtheoryandapplications.com/content/2013/1/87 Then the following identity holds true: Proof Combining () and (), we get Therefore, Comparing the coefficients of t n n! on both sides of the above equation, we arrive at the desired result.

Generalized Eulerian type numbers and polynomials
In this section, we provide generating functions, related to nonnegative real parameters, for the generalized Eulerian type polynomials and numbers, that is, the so-called generalized Apostol type Frobenius Euler polynomials and numbers. We derive fundamental properties, recurrence relations and many new identities for these polynomials and numbers based on the generating functions, functional equations and differential equations.
These polynomials and numbers have many applications in many branches of mathematics.
The following definition gives us a natural generalization of the Eulerian polynomials.
By substituting x =  into (), we obtain By substituting t =  into (), we have From the above equation, we find that The generalized Eulerian type polynomials of order m, H (m) n (x; u; a, b, c; λ) are defined by means of the following generating function: Throughout this paper, we assume that a, b, c ∈ R + , x ∈ R, λ ∈ C and u ∈ C {λ}.
The following elementary properties of the generalized Eulerian type polynomials and numbers are derived from their generating functions in () and ().
Theorem . (Recurrence relation for the generalized Eulerian type numbers) For n = , we have H  (u; a, b; λ) = -u λ-u . For n > , following the usual convention of symbolically replacing (H(u; a, b; λ)) n by H n (u; a, b; λ), we have Proof By using (), we obtain Comparing the coefficients of t n n! on both sides of the above equation, we arrive at the desired result.
By differentiating both sides of Eq. () with respect to the variable x, we obtain the following higher order differential equation: x; u, a, b, c). () From this equation, we arrive at higher order derivative of the generalized Eulerian type polynomials by the following theorem. In their special case when a = λ =  and b = c = e, Theorem . is reduced to the following well-known result: Theorem . The following explicit representation formula holds true: Proof By using () and the umbral calculus convention, we obtain From the above equation, we get By using (), we define the following functional equation: Theorem . The following formula holds true: Proof Combining () and (), we easily arrive at the desired result. Theorem . The following formula holds true: Proof By using (), we obtain From the above equation, we get Comparing the coefficients of t n n! on both sides of the above equation, we arrive at the desired result. Comparing the coefficients of t n n! on both sides of the above equation, we arrive at the desired result.

Multiplication formulas for normalized polynomials
In this section, using generating functions, we derive multiplication formulas in terms of the normalized polynomials which are related to the generalized Eulerian type polynomials, the Bernoulli and the Euler polynomials.
where H n (x; u) and H n (u) denote the Eulerian polynomials and numbers, respectively.
By using the following finite geometric series, on the right-hand side of (), we obtain From this equation, we get Now by making use of the generating functions () and () on the right-hand side of the above equation, we obtain × H n-k  y ; u y -uH n-k u y t n n! .
By equating the coefficients of t n n! on both sides, we get Finally, by replacing x by yx on both sides of the above equation, we arrive at the desired result.
Remark . By substituting a =  into Theorem ., for n = k, we obtain By substituting b = e and λ =  into the above equation, we arrive at the multiplication formula for the Eulerian polynomials where E n (x) denotes the Euler polynomials in the usual notation. If y is an even positive integer, we have Definition . Let a, b, c ∈ R + with a = b, x ∈ R and n ∈ N  . Then the generalized Bernoulli polynomials B (α) n (x; λ; a, b, c) of order α ∈ C are defined by means of the following generating functions: It is easily observe that By replacing x by yx on both sides of the above equation, we arrive at the desired result.
where g n- (x) and f n (x) denote the normalized polynomials of degree n - and n, respectively. More precisely, as Carlitz has pointed out [, p.], if y is a fixed even integer ≥  and f n (x) is an arbitrary normalized polynomial of degree n, then () determines g n- (x) as a normalized polynomial of degree n -. Thus, for a single value y, () does not suffice to determine the normalized polynomials g n- (x) and f n (x).
Remark . According to (), the set of normalized polynomials {f n (x)} is an Appell set, (cf. []). We now modify () as follows: where ξ r =  (ξ = λ, r ∈ N). The polynomial H n (x; ξ ; a, b, c; λ) is a normalized polynomial of degree m in x. The polynomial H n (x; ξ ; , e, e; ) may be called Eulerian polynomials with parameter ξ . In particular we note that  If we substitute x =  and a =  into (), then we obtain Y n (λ; ) =  u .
By using the above partial differential equations, we obtain recurrence relations for the generalized Eulerian type polynomials by the following theorem:

New identities involving families of polynomials
In this section, we derive some new identities related to the generalized Bernoulli polynomials and numbers of order , the Eulerian type polynomials and the generalized array type polynomials.