Generating functions for generalized Stirling type numbers, Array type polynomials, Eulerian type polynomials and their applications
aa r X i v : . [ m a t h . N T ] N ov Generating functions for generalized Stirling typenumbers, Array type polynomials, Eulerian typepolynomials and their applications
Yilmaz Simsek
Department of Mathematics, Faculty of Science University of Akdeniz TR-07058 Antalya,Turkey [email protected]
Abstract
The first aim of this paper is to construct new generating functions for thegeneralized λ -Stirling type numbers of the second kind, generalized array typepolynomials and generalized Eulerian type polynomials and numbers, attachedto Dirichlet character. We derive various functional equations and differen-tial equations using these generating functions. The second aim is providea novel approach to deriving identities including multiplication formulas andrecurrence relations for these numbers and polynomials using these functionalequations and differential equations. Furthermore, by applying p -adic Volken-born integral and Laplace transform, we derive some new identities for thegeneralized λ -Stirling type numbers of the second kind, the generalized arraytype polynomials and the generalized Eulerian type polynomials. We also givemany applications related to the class of these polynomials and numbers. Key Words.
Bernoulli polynomials; Euler polynomials; Apostol Bernoulli polynomials;generalized Frobenius Euler polynomials; Normalized Polynomials; Array polynomials; Stir-ling numbers of the second kind; p -adic Volkenborn integral; generating function; functionalequation; Laplace transform.1. Introduction, Definitions and Preliminaries
Throughout this paper, we use the following standard notations: N = { , , , . . . } , N = { , , , , . . . } = N ∪ { } and Z − = {− , − , − , . . . } . Here, Z denotes the set of integers, R denotes the set of real numbers and C denotes the set ofcomplex numbers. We assume that ln( z ) denotes the principal branch of the multi-valuedfunction ln( z ) with the imaginary part ℑ (ln( z )) constrained by − π < ℑ (ln( z )) ≤ π. Yilmaz Simsek
Furthermore, 0 n = n = 00 n ∈ N , (cid:18) xv (cid:19) = x ( x − · · · ( x − v + 1) v !and { z } = 1 and { z } j = j − Y d =0 ( z − d ) , where j ∈ N and z ∈ C cf. ([13], [29]).The generating functions have various applications in many branches of Mathematicsand Mathematical Physics. These functions are defined by linear polynomials, differentialrelations, globally referred to as functional equations . The functional equations arise inwell-defined combinatorial contexts and they lead systematically to well-defined classes offunctions (cf. see, for detail, [16]). Although, in the literature, one can find extensive inves-tigations related to the generating functions for the Bernoulli, Euler and Genocchi numbersand 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, Section 2, Section 3 and Section 4 of this paper dealwith new classes of generating functions which are related to generalized λ -Stirling typenumbers of the second kind, generalized array type polynomials and generalized Eulerianpolynomials, respectively. By using these generating functions, we derive many functionalequations and differential equations. By using these equations, we investigate and introducefundamental properties and many new identities for the generalized λ -Stirling type numbersof the second kind, the generalized array type polynomials and the generalized Eulerian typepolynomials and numbers. We also derive multiplication formulas and recurrence relationsfor these numbers and polynomials.The remainder of this study is organized as follows:In section 5, we derive new identities related to the generalized Bernoulli polynomials, thegeneralized Eulerian type polynomials, generalized λ -Stirling type numbers and the gener-alized array polynomials.In section 6, we give relations between generalized Bernoulli polynomials and generalizedarray polynomials.In section 7, We give an application of the Laplace transform to the generating functionsfor the generalized Bernoulli polynomials and the generalized array type polynomials.In section 8, by using the bosonic and the fermionic p -adic integral on Z p , we find some newidentities related to the Bernoulli polynomials, the generalized Eulerian type polynomialsand Stirling numbers. ew Generating Functions of the Stirling numbers, Frobenius-Euler and Related polynomials 3 Generating Function for generalized λ -Stirling type numbers of thesecond kind The Stirling numbers are used in combinatorics, in number theory, in discrete probabilitydistributions 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. Thesenumbers occur in combinatorics and in the theory of partitions.In this section, we construct a new generating function, related to nonnegative real pa-rameters, for the generalized λ -Stirling type numbers of the second kind. We derive someelementary properties including recurrence relations of these numbers. The following defini-tion provides a natural generalization and unification of the λ -Stirling numbers of the secondkind: Definition 1.
Let a , b ∈ R + ( a = b ), λ ∈ C and v ∈ N . The generalized λ -Stirling typenumbers of the second kind S ( n, v ; a, b ; λ ) are defined by means of the following generatingfunction: f S,v ( t ; a, b ; λ ) = ( λb t − a t ) v v ! = ∞ X n =0 S ( n, v ; a, b ; λ ) t n n ! . (2.1) Remark 1.
By setting a = 1 and b = e in (2.1), we have the λ -Stirling numbers of thesecond kind S ( n, v ; 1 , e ; λ ) = S ( n, v ; λ ) which are defined by means of the following generating function: ( λe t − v v ! = ∞ X n =0 S ( n, v ; λ ) t n n ! , cf. ( [29] , [46] ). Substituting λ = 1 into above equation, we have the Stirling numbers of thesecond kind S ( n, v ; 1) = S ( n, v ) , cf. ( [13] , [29] , [46] ). These numbers have the following well known properties: S ( n,
0) = δ n, ,S ( n,
1) = S ( n, n ) = 1 and S ( n, n −
1) = (cid:18) n (cid:19) , where δ n, denotes the Kronecker symbol (see [13] , [29] , [46] ). By using (2.1), we obtain the following theorem:
Theorem 1. S ( n, v ; a, b ; λ ) = 1 v ! v X j =0 ( − j (cid:18) vj (cid:19) λ v − j ( j ln a + ( v − j ) ln b ) n (2.2) Yilmaz Simsek and S ( n, v ; a, b ; λ ) = 1 v ! v X j =0 ( − v − j (cid:18) vj (cid:19) λ j ( j ln b + ( v − j ) ln a ) n . (2.3) Proof.
By using (2.1) and the binomial theorem, we can easily arrive at the desired results. (cid:3)
By using the formula (2.2), we can compute some values of the numbers S ( n, v ; a, b ; λ ) asfollows: S (0 , a, b ; λ ) = 1 , S (0 , a, b ; λ ) = 1 , S (1 , a, b ; λ ) = 0 , S (1 , a, b ; λ ) = ln (cid:18) b λ a (cid:19) , S (2 , a, b ; λ ) = 0 , S (2 , a, b ; λ ) = λ (ln b ) − (ln a ) , S (2 , a, b ; λ ) = λ (cid:0) ln b (cid:1) − λ ln ( ab ) + (cid:0) ln a (cid:1) , S (3 , a, b ; λ ) = 0 , S (3 , a, b ; λ ) = λ (ln b ) − (ln a ) , S (0 , v ; a, b ; λ ) = ( λ − v v ! , S ( n, a, b ; λ ) = δ n, and S ( n, a, b ; λ ) = λ (ln b ) n − (ln a ) n . Remark 2.
By setting a = 1 and b = e in the assertions (2.2) of Theorem 1, we have thefollowing result: S ( n, v ; λ ) = 1 v ! v X j =0 (cid:18) vj (cid:19) λ v − j ( − j ( v − j ) n . The above relation has been studied by Srivastava [46] and Luo [29] . By setting λ = 1 in theabove equation, we have the following result: S ( n, v ; λ ) = 1 v ! v X j =0 (cid:18) vj (cid:19) ( − j ( v − j ) n cf. ( [1] , [6] , [8] , [9] , [13] , [19] , [29] , [43] , [44] , [46] , [48] ). ew Generating Functions of the Stirling numbers, Frobenius-Euler and Related polynomials 5 By differentiating both sides of equation (2.1) with respect to the variable t , we obtainthe following differential equations: ∂∂t f S,v ( t ; a, b ; λ ) = (cid:0) λ (ln b ) b t − (ln a ) a t (cid:1) f S,v − ( t ; a, b ; λ )or ∂∂t f S,v ( t ; a, b ; λ ) = v ln( b ) f S,v ( t ; a, b ; λ ) + ln (cid:18) ba (cid:19) a t f S,v − ( t ; a, b ; λ ) . (2.4)By using equations (2.1) and (2.4), we obtain recurrence relations for the generalized λ -Stirling type numbers of the second kind by the following theorem: Theorem 2.
Let n, v ∈ N . S ( n, v ; a, b ; λ ) = n − X j =0 (cid:18) n − j (cid:19) S ( j, v − a, b ; λ ) (cid:16) λ (ln( b )) n − j − (ln( a )) n − j (cid:17) . (2.5) or S ( n, v ; a, b ; λ ) = v ln( b ) S ( n − , v ; a, b ; λ )+ ln (cid:18) ba (cid:19) n − X j =0 (cid:18) n − j (cid:19) S ( j, v − a, b ; λ ) (ln( a )) n − − j . Remark 3.
By setting a = 1 and b = e , Theorem 2 yields the corresponding results whichare proven by Luo and Srivastava [29, Theorem 11] . Substituting a = λ = 1 and b = e intoTheorem 2, we obtain the following known results: S ( n, v ) = n − X j =0 (cid:18) n − j (cid:19) S ( j, v − , and S ( n, v ) = vS ( n − , v ) + S ( n − , v − , cf. ( [1] , [10] , [13] , [29] , [43] , [44] ). The generalized λ -Stirling type numbers of the second kind can also be defined by equation(2.6): Theorem 3.
Let k ∈ N and λ ∈ C . λ x (ln b x ) m = m X l =0 ∞ X j =0 (cid:18) ml (cid:19) (cid:18) xj (cid:19) j ! S ( l, j ; a, b ; λ ) (cid:0) ln (cid:0) a ( x − j ) (cid:1)(cid:1) m − l . (2.6) Proof.
By using (2.1), we get (cid:0) λb t (cid:1) x = ∞ X j =0 (cid:18) xj (cid:19) j ! ∞ X m =0 S ( m, j ; a, b ; λ ) t m m ! ∞ X n =0 (ln a x − j ) n t n n ! . From the above equation, we obtain λ x ∞ X m =0 (ln b ) m t m m ! = ∞ X m =0 ∞ X j =0 (cid:18) xj (cid:19) j ! S ( m, j ; a, b ; λ ) t m m ! ∞ X n =0 (ln a x − j ) n t n n ! . Yilmaz Simsek
Therefore λ x ∞ X m =0 (ln b ) m t m m ! = ∞ X m =0 m X l =0 ∞ X j =0 (cid:18) ml (cid:19) (cid:18) xj (cid:19) j ! S ( l, j ; a, b ; λ ) (cid:0) ln a ( x − j ) (cid:1) m − l ! t m m ! . Comparing the coefficients of t m m ! on both sides of the above equation, we arrive at the desiredresult. (cid:3) Remark 4.
For a = 0 and b = e , the formula (2.6) can easily be shown to be reduced to thefollowing result which is given by Luo and Srivastava [29, Theorem 9] : λ x x n = ∞ X l =0 (cid:18) xl (cid:19) l ! S ( n, l ; λ ) , where n ∈ N and λ ∈ C . For λ = 1 , the above formula is reduced to x n = n X v =0 (cid:18) xv (cid:19) v ! S ( n, v ) cf. ( [1] , [10] , [13] , [19] , [29] ). Generalized array type polynomials
By using the same motivation with the λ -Stirling type numbers of the second kind, we alsoconstruct a novel generating function, related to nonnegative real parameters, of the gen-eralized array type polynomials . We derive some elementary properties including recurrencerelations of these polynomials. The following definition provides a natural generalizationand unification of the array polynomials: Definition 2.
Let a , b ∈ R + ( a = b ), x ∈ R , λ ∈ C and v ∈ N . The generalized array typepolynomials S nv ( x ; a, b ; λ ) can be defined by S nv ( x ; a, b ; λ ) = 1 v ! v X j =0 ( − v − j (cid:18) vj (cid:19) λ j (cid:0) ln (cid:0) a v − j b x + j (cid:1)(cid:1) n . (3.1)By using the formula (3.1), we can compute some values of the polynomials S nv ( x ; a, b ; λ )as follows: S n ( x ; a, b ; λ ) = (ln ( b x )) n , S v ( x ; a, b ; λ ) = (1 − λ ) v v !and S ( x ; a, b ; λ ) = − ln( ab x ) + λ ln( b x +1 ) . Remark 5.
The polynomials S nv ( x ; a, b ; λ ) may be also called generalized λ -array type poly-nomials. By substituting x = 0 into (3.1), we arrive at (2.3): S nv (0; a, b ; λ ) = S ( n, v ; a, b ; λ ) . ew Generating Functions of the Stirling numbers, Frobenius-Euler and Related polynomials 7 Setting a = λ = 1 and b = e in (3.1), we have S nv ( x ) = 1 v ! v X j =0 ( − v − j (cid:18) vj (cid:19) ( x + j ) n , a result due to Chang and Ha [12, Eq-(3.1)] , Simsek [43] . It is easy to see that S ( x ) = S nn ( x ) = 1 ,S n ( x ) = x n and for v > n , S nv ( x ) = 0 cf. [12, Eq-(3.1)] . Generating functions for the polynomial S nv ( x ; a, b, c ; λ ) can be defined as follows: Definition 3.
Let a , b ∈ R + ( a = b ), λ ∈ C and v ∈ N . The generalized array typepolynomials S nv ( x ; a, b ; λ ) are defined by means of the following generating function: g v ( x, t ; a, b ; λ ) = ∞ X n =0 S nv ( x ; a, b ; λ ) t n n ! . (3.2) Theorem 4.
Let a , b ∈ R + , ( a = b ), λ ∈ C and v ∈ N . g v ( x, t ; a, b ; λ ) = 1 v ! (cid:0) λb t − a t (cid:1) v b xt . (3.3) Proof.
By substituting (3.1) into the right hand side of (3.2), we obtain ∞ X n =0 S nv ( x ; a, b ; λ ) t n n ! = ∞ X n =0 v ! v X j =0 ( − v − j (cid:18) vj (cid:19) λ j (cid:0) ln (cid:0) a v − j b x + j (cid:1)(cid:1) n ! t n n ! . Therefore ∞ X n =0 S nv ( x ; a, b ; λ ) t n n ! = 1 v ! v X j =0 ( − v − j (cid:18) vj (cid:19) λ j ∞ X n =0 (cid:0) ln (cid:0) a v − j b x + j (cid:1)(cid:1) n t n n ! . 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 ∞ X n =0 S nv ( x ; a, b ; λ ) t n n ! = v ! v X j =0 ( − v − j (cid:18) vj (cid:19) λ j a ( v − j ) t b jt ! b xt . By using (2.1) and binomial theorem in the above equation, we arrive at the desiredresult. (cid:3)
Remark 6.
If we set λ = 1 in (3.3), we arrive a new special case of the array polynomialsgiven by f S,v ( t ; a, b ) b tx = ∞ X n =0 S nv ( x ; a, b ) t n n ! . In the special case when a = λ = 1 and b = e, Yilmaz Simsek the generalized array polynomials S nv ( x ; a, b ; λ ) defined by (3.3) would lead us at once to theclassical array polynomials S nv ( x ) , which are defined by means of the following generatingfunction: ( e t − v v ! e tx = ∞ X n =0 S nv ( x ) t n n ! , which yields to the generating function for the array polynomials S nv ( x ) studied by Changand Ha [12] see also cf. ( [6] , [43] ). The polynomials S nv ( x ; a, b ; λ ) defined by (3.3) have many interesting properties which wegive in this section.We set g v ( x, t ; a, b ; λ ) = b xt f S,v ( t ; a, b ; λ ) . (3.4) Theorem 5.
The following formula holds true: S nv ( x ; a, b ; λ ) = n X j =0 (cid:18) nj (cid:19) S ( j, v ; a, b ; λ ) (ln b x ) n − j . (3.5) Proof.
By using (3.4), we obtain ∞ X n =0 S nv ( x ; a, b ; λ ) t n n ! = ∞ X n =0 S ( n, v ; a, b ; λ ) t n n ! ∞ X n =0 (ln b x ) n t n n ! . From the above equation, we get ∞ X n =0 S nv ( x ; a, b ; λ ) t n n ! = ∞ X n =0 n X j =0 (cid:18) nj (cid:19) S ( j, v ; a, b ) (ln b x ) n − j ! t n n ! . Comparing the coefficients of t n on both sides of the above equation, we arrive at thedesired result. (cid:3) Remark 7.
In the special case when a = λ = 1 and b = e , equation (3.5) is reduced to S nv ( x ) = n X j =0 (cid:18) nj (cid:19) x n − j S ( j, v ) cf. [43, Theorem 2] . By differentiating j times both sides of (3.3) with respect to the variable x , we obtain thefollowing differential equation: ∂ j ∂x j g v ( x, t ; a, b ; λ ) = t j (ln b ) j g v ( x, t ; a, b ; λ ) . From this equation, we arrive at higher order derivative of the array type polynomials bythe following theorem:
Theorem 6.
Let n , j ∈ N with j ≤ n . Then we have ∂ j ∂x j S nv ( x ; a, b ; λ ) = { n } j (ln( b )) j S n − jv ( x ; a, b ; λ ) . ew Generating Functions of the Stirling numbers, Frobenius-Euler and Related polynomials 9 Remark 8.
By setting a = λ = j = 1 and b = e in Theorem 6, we have ddx S nv ( x ) = nS n − v ( x ) cf. [43] . From (3.3), we get the following functional equation: g v ( x , t ; a, b ; λ ) g v ( x , t ; a, b ; λ ) = (cid:18) v + v v (cid:19) g v + v ( x + x , t ; a, b ; λ ) . (3.6)From this functional equation, we obtain the following identity: Theorem 7. (cid:18) v + v v (cid:19) S nv + v ( x + x ; a, b ; λ ) = n X j =0 (cid:18) nj (cid:19) S jv ( x ; a, b ; λ ) S n − jv ( x ; a, b ; λ ) . Proof.
Combining (3.2) and (3.6), we get ∞ X n =0 S nv ( x ; a, b ; λ ) t n n ! ∞ X n =0 S nv ( x ; a, b ; λ ) t n n != (cid:18) v + v v (cid:19) ∞ X n =0 S nv + v ( x + x ; a, b ; λ ) t n n ! . Therefore ∞ X n =0 n X j =0 (cid:18) nj (cid:19) S jv ( x ; a, b ; λ ) S n − jv ( x ; a, b ; λ ) ! t n n != (cid:18) v + v v (cid:19) ∞ X n =0 S nv + v ( x + x ; a, b ; λ ) t n n ! .Comparing the coefficients of t n n ! on both sides of the above equation, we arrive at the desiredresult. (cid:3) 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 generalizedApostol type Frobenius Euler polynomials and numbers . We derive fundamental properties,recurrence relations and many new identities for these polynomials and numbers based onthe 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: Yilmaz Simsek
Definition 4.
Let a, b ∈ R + ( a = b ) , x ∈ R , λ ∈ C and u ∈ C(cid:31) { } . The generalizedEulerian type polynomials H n ( x ; u ; a, b, c ; λ ) are defined by means of the following generatingfunction: F λ ( t, x ; u, a, b, c ) = ( a t − u ) c xt λb t − u = ∞ X n =0 H n ( x ; u ; a, b, c ; λ ) t n n ! . (4.1)By substituting x = 0 into (4.1), we obtain H n (0; u ; a, b, c ; λ ) = H n ( u ; a, b, c ; λ ) , where H n ( u ; a, b, c ; λ ) denotes generalized Eulerian type numbers . Remark 9.
Substituting a = 1 into (4.1), we have (1 − u ) c xt λb t − u = ∞ X n =0 H n ( x ; u ; 1 , b, c ; λ ) t n n ! a result due to Kurt and Simsek [25] . In their special case when λ = 1 and b = c = e , the generalized Eulerian type polynomials H n ( x ; u ; 1 , b, c ; λ ) are reduced to the Eulerianpolynomials or Frobenius Euler polynomials which are defined by means of the followinggenerating function: (1 − u ) e xt e t − u = ∞ X n =0 H n ( x ; u ) t n n ! , (4.2) with, of course, H n (0; u ) = H n ( u ) denotes the so-called Eulerian numbers cf. ( [8] , [7] , [9] , [10] , [22] , [45] , [21] , [39] , [40] , [47] , [51] ). Substituting u = − , into (4.2), we have H n ( x ; −
1) = E n ( x ) where E n ( x ) denotes Euler polynomials which are defined by means of the following generatingfunction: e xt e t + 1 = ∞ X n =0 E n ( x ) t n n ! (4.3) where | t | < π cf. [1] - [53] . The following elementary properties of the generalized Eulerian type polynomials andnumbers are derived from their generating functions in (4.1).
Theorem 8. (Recurrence relation for the generalized Eulerian type numbers): For n = 0 ,we have H ( u ; a, b ; λ ) = − uλ − u if a = 1 , uλ − u if a = 1 . For n > , following the usual convention of symbolically replacing ( H ( u ; a, b ; λ )) n by H n ( u ; a, b ; λ ) ,we have λ (ln b + H ( u ; a, b ; λ )) n − u H n ( u ; a, b ; λ ) = (ln a ) n . ew Generating Functions of the Stirling numbers, Frobenius-Euler and Related polynomials 11 Proof.
By using (4.1), we obtain ∞ X n =0 (ln a ) n t n n ! − u = ∞ X n =0 ( λ (ln b + H ( u ; a, b ; λ )) n − u H n ( u ; a, b ; λ )) t n n ! . Comparing the coefficients of t n n ! on both sides of the above equation, we arrive at the desiredresult. (cid:3) By differentiating both sides of equation (4.1) with respect to the variable x , we obtainthe following higher order differential equation: ∂ j ∂x j F λ ( t, x ; u, a, b, c ) = (cid:0) ln (cid:0) c t (cid:1)(cid:1) j F λ ( t, x ; u, a, b, c ) . (4.4)From this equation, we arrive at higher order derivative of the generalized Eulerian typepolynomials by the following theorem: Theorem 9.
Let n , j ∈ N with j ≤ n . Then we have ∂ j ∂x j H n ( x ; u ; a, b, c ; λ ) = { n } j (ln ( c )) j H n − j ( x ; u ; a, b, c ; λ ) . Proof.
Combining (4.1) and (4.4), we have ∞ X n =0 ∂ j ∂x j H n ( x ; u ; a, b, c ; λ ) t n n ! = (ln c ) j ∞ X n =0 H n ( x ; u ; a, b, c ; λ ) t n + j n ! . From the above equation, we get ∞ X n =0 ∂ j ∂x j H n ( x ; u ; a, b, c ; λ ) t n n ! = (ln c ) j ∞ X n =0 { n } j H n − j ( x ; u ; a, b, c ; λ ) t n n ! . Comparing the coefficients of t n n ! on both sides of the above equation, we arrive at the desiredresult. (cid:3) Remark 10.
Setting j = 1 in Theorem 9, we have ∂∂x H n ( x ; u ; a, b, c ; λ ) = n H n − ( x ; u ; a, b, c ; λ ) ln ( c ) . In their special case when a = λ = 1 and b = c = e, Theorem 9 is reduced to the following well known result: ∂ j ∂x j H n ( x ; u ) = n !( n − j )! H n − j ( x ; u ) cf. [8, Eq-(3.5)] . Substituting j = 1 into the above equation, we have ∂∂x H n ( x ; u ) = nH n − ( x ; u ) cf. ( [8, Eq-(3.5)] , [25] ). Yilmaz Simsek
Theorem 10.
The following explicit representation formula holds true: ( x ln c + ln a ) n − ux n (ln c ) n = λ ( x ln c + ln b + H ( u ; a, b ; λ )) n − u ( x ln c + H ( u ; a, b ; λ )) n . Proof.
By using (4.1) and the umbral calculus convention , we obtain a t − uλb t − u = e H ( u ; a,b ; λ ) t . From the above equation, we get ∞ X n =0 ((ln a + x ln c ) n − u ( x ln c )) t n n != ∞ X n =0 ( λ ( H ( u ; a, b ; λ ) + ln b + x ln c ) n − u ( H n ( u ; a, b ; λ ) + x ln c ) n ) t n n ! . Comparing the coefficients of t n n ! on both sides of the above equation, we arrive at the desiredresult. (cid:3) Remark 11.
By substituting a = λ = 1 and b = c = e into Theorem 10, we have (1 − u ) x n = H n ( x + 1; u ) − uH n ( x ; u ) (4.5) cf. ( [8, Eq-(3.3)] , [51] ). By setting u = − in the above equation, we have x n = E n ( x + 1) + E n ( x ) a result due to Shiratani [38] . By using (4.5), Carlitz [8] studied on the Mirimonoff poly-nomial f n (0 , m ) which is defined by f n ( x, m ) = m − X j =0 ( x + j ) n u m − j − = H n ( x + m ; u ) − u m H n ( x ; u )1 − u . By applying Theorem 10, one may generalize the Mirimonoff polynomial.
Theorem 11.
The following explicit representation formula holds true: H n ( x ; u ; a, b, c ; λ ) = n X j =0 (cid:18) nj (cid:19) ( x ln c ) n − j H j ( u ; a, b, c ; λ ) . (4.6) Proof.
By using (4.1), we get ∞ X n =0 H n ( u ; a, b, c ; λ ) t n n ! ∞ X n =0 (ln c ) n t n n ! = ∞ X n =0 H n ( x ; u ; a, b, c ; λ ) t n n ! . From the above equation, we obtain ∞ X n =0 n X j =0 (cid:18) nj (cid:19) ( x ln c ) n − j H j ( u ; a, b, c ; λ ) ! t n n ! = ∞ X n =0 H n ( x ; u ; a, b, c ; λ ) t n n ! . ew Generating Functions of the Stirling numbers, Frobenius-Euler and Related polynomials 13 Comparing the coefficients of t n n ! on both sides of the above equation, we arrive at the desiredresult. (cid:3) Remark 12.
Substituting a = λ = 1 and b = c = e into (4.6), we have H n ( x ; u ) = n X j =0 (cid:18) nj (cid:19) x n − j H j ( u ) cf. ( [8] , [7] , [9] , [10] , [22] , [45] , [21] , [25] , [39] , [40] , [47] , [51] ). Remark 13.
From (4.6), we easily get H n ( x ; u ; a, b, c ; λ ) = ( H ( u ; a, b, c ; λ ) + x ln c ) n , where after expansion of the right member, H n ( u ; a, b, c ; λ ) is replaced by H n ( u ; a, b, c ; λ ) , weuse this convention frequently throughout of this paper. Theorem 12. H n ( x + y ; u ; a, b, c ; λ ) = n X j =0 (cid:18) nj (cid:19) ( y ln c ) n − j H j ( x ; u ; a, b, c ; λ ) . (4.7) Proof.
By using (4.1), we have ∞ X n =0 H n ( x + y ; u ; a, b, c ; λ ) t n n ! = ∞ X n =0 ( y ln c ) n t n n ! . ∞ X n =0 H n ( x ; u ; a, b, c ; λ ) t n n ! . Therefore ∞ X n =0 H n ( x + y ; u ; a, b, c ; λ ) t n n ! = ∞ X n =0 n X j =0 (cid:18) nj (cid:19) ( y ln c ) n − j H j ( x, u ; a, b, c ; λ ) t n n ! . Comparing the coefficients of t n n ! on both sides of the above equation, we arrive at the desiredresult. (cid:3) Remark 14.
In the special case when a = λ = 1 and b = c = e , equation (4.7) is reduced tothe following result: H n ( x + y ) = n X j =0 (cid:18) nj (cid:19) y n − j H j ( x, u ) cf. [8, Eq-(3.6)] . Substituting u = − into the above equation, we get the following well-known result: E n ( x + y ) = n X j =0 (cid:18) nj (cid:19) y n − j E j ( x ) . (4.8)By using (4.1), we define the following functional equation: F λ ( t, x ; u , a , b , c ) c yt = F λ ( t, x ; u, a, b, c ) F λ ( t, y ; − u, a, b, c ) . (4.9) Theorem 13. H n ( x + y ; u ; a, b, c ; λ ) = ( H ( x ; u ; a, b, c ; λ ) + H ( y ; − u ; a, b, c ; λ )) n . (4.10) Yilmaz Simsek
Proof.
Combining (4.9) and (4.7), we easily arrive at the desired result. (cid:3)
Remark 15.
In the special case when a = λ = 1 and b = c = e , equation (4.10) is reducedto the following result: H n ( x + y ; u ) = n X j =0 (cid:18) nj (cid:19) H j ( x ; u ) H n − j ( y ; − u ) cf. [8, Eq-(3.17)] . Theorem 14. ( − n H n (1 − x ; u − ; a, b, c ; λ − ) = λ n X j =0 (cid:18) nj (cid:19) (cid:18) ln (cid:18) ba (cid:19)(cid:19) n − j H j ( x − , u ; a, b, c ; λ ) . Proof.
By using (4.1), we obtain( a − t − u − ) c − (1 − x ) t λ − b − t − u − = λ (cid:18) ba (cid:19) t ∞ X n =0 H n ( x − u ; a, b, c ; λ ) t n n ! . From the above equation, we get ∞ X n =0 H n (1 − x ; u − ; a, b, c ; λ − ) ( − n t n n != λ ∞ X n =0 H n ( x − u ; a, b, c ; λ ) t n n ! ! ∞ X n =0 (cid:18) ln (cid:18) ba (cid:19)(cid:19) n t n n ! ! . Therefore ∞ X n =0 ( − n H n (1 − x ; u − ; a, b, c ; λ − ) t n n != ∞ X n =0 λ n X j =0 (cid:18) nj (cid:19) (cid:18) ln (cid:18) ba (cid:19)(cid:19) n − j H j ( x − , u ; a, b, c ; λ ) ! t n n ! . Comparing the coefficients of t n n ! on both sides of the above equation, we arrive at the desiredresult. (cid:3) Remark 16.
In their special case when a = λ = 1 and b = c = e , Theorem 14 is reduced tothe following result: ( − n H n (1 − x ; u − ) = H n ( x − , u ) cf. [8, Eq-(3.7)] . Substituting u = − into the above equation, we get the following well-known result: ( − n E n (1 − x ) = E n ( x ) cf. ( [8, Eq-(3.7)] , [15] , [36] , [38] , [46] ). ew Generating Functions of the Stirling numbers, Frobenius-Euler and Related polynomials 15 Theorem 15. H n (cid:18) x + y u ; a, b, c ; λ (cid:19) = n X j =0 (cid:18) nj (cid:19) H j ( x ; u ; a, b, c ; λ ) H n − j ( y ; − u ; a, b, c ; λ )2 n . Proof.
By using (4.1), we get ∞ X n =0 H n (cid:18) x + y u ; a, b, c ; λ (cid:19) n t n n != ∞ X n =0 n X j =0 (cid:18) nj (cid:19) H j ( x ; u ; a, b, c ; λ ) H n − j ( y ; − u ; a, b, c ; ! t n n ! . Comparing the coefficients of t n n ! on both sides of the above equation, we arrive at the desiredresult. (cid:3) Remark 17.
When a = λ = 1 and b = c = e , Theorem 15 is reduced to the following result: H n (cid:18) x + y u (cid:19) = 2 − n n X j =0 (cid:18) nj (cid:19) H j ( x ; u ) H n − j ( y ; − u ) , cf. [8, Eq-(3.17)] . Multiplication formulas for normalized polynomials.
In this section, using gen-erating functions, we derive multiplication formulas in terms of the normalized polynomialswhich are related to the generalized Eulerian type polynomials, the Bernoulli and the Eulerpolynomials.
Theorem 16. (Multiplication formula) Let y ∈ N . Then we have H n ( yx ; u ; a, b, b ; λ ) (4.11)= y n n X k =0 y − X j =0 (cid:18) nk (cid:19) λ j (ln a ) n − k u j +1 − y − u j +1 H k (cid:18) x + jy ; u y ; a, b, b ; λ y (cid:19) × (cid:18) H n − k (cid:18) y ; u y (cid:19) − uH n − k ( u y ) (cid:19) , where H n ( x ; u ) and H n ( u ) denote the Eulerian polynomials and numbers, respectively.Proof. Substituting c = b into (4.1), we have ∞ X n =0 H n ( x ; u ; a, b, b ; λ ) t n n ! = ( a t − u ) b xt λb t − u = (cid:18) a t − u − u (cid:19) ( a t − u ) b xt − λb t u . (4.12)By using the following finite geometric series y − X j =0 (cid:18) λb t u (cid:19) j = 1 − (cid:16) λb t u (cid:17) y − λb t u , Yilmaz Simsek on the right-hand side of (4.12), we obtain ∞ X n =0 H n ( x ; u ; a, b, b ; λ ) t n n ! = ( a t − u ) b xt − u (cid:0) − (cid:0) λb t u (cid:1) y (cid:1) y − X j =0 (cid:18) λb t u (cid:19) j . From this equation, we get ∞ X n =0 H n ( x ; u ; a, b, b ; λ ) t n n ! = ( a t − u )( a yt − u y ) y − X j =0 λ j u j +1 − y ( a yt − u y ) b yt ( x + jy )( λb yt − u y ) . Now by making use of the generating functions (4.1) and (4.2) on the right-hand side of theabove equation, we obtain ∞ X n =0 H n ( x ; u ; a, b, b ; λ ) t n n != 11 − u y y − X j =0 λ j u j +1 − y ∞ X n =0 H n (cid:18) x + jy ; u y ; a, b, b ; λ y (cid:19) y n t n n ! ! × ∞ X n =0 (cid:18) H n (cid:18) y ; u y (cid:19) − uH n ( u y ) (cid:19) ( y ln a ) n t n n ! ! . Therefore ∞ X n =0 H n ( x ; u ; a, b, b ; λ ) t n n != ∞ X n =0 n X k =0 y − X j =0 (cid:18) nk (cid:19) y n λ j (ln a ) n − k u j +1 − y − u j +1 H k (cid:18) x + jy ; u y ; a, b, b ; λ y (cid:19) × (cid:18) H n − k (cid:18) y ; u y (cid:19) − uH n − k ( u y ) (cid:19) t n n ! . By equating the coefficients of t n n ! on both sides, we get H n ( x ; u ; a, b, b ; λ )= n X k =0 y − X j =0 (cid:18) nk (cid:19) y n λ j (ln a ) n − k u j +1 − y − u j +1 H k (cid:18) x + jy ; u y ; a, b, b ; λ y (cid:19) × (cid:18) H n − k (cid:18) y ; u y (cid:19) − uH n − k ( u y ) (cid:19) . Finally, by replacing x by yx on both sides of the above equation, we arrive at the desiredresult. (cid:3) Remark 18.
By substituting a = 1 into Theorem 16, for n = k , we obtain H n ( yx ; u ; 1 , b, b ; λ ) = y n u y − − u − u y y − X j =0 λ j u j H n (cid:18) x + jy ; u y ; 1 , b, b ; λ y (cid:19) . (4.13) ew Generating Functions of the Stirling numbers, Frobenius-Euler and Related polynomials 17 By substituting b = e and λ = 1 into the above equation, we arrive at the multiplicationformula for the Eulerian polynomials H n ( yx ; u ) = y n u y − (1 − u )1 − u y y − X j =0 u j H n (cid:18) x + jy ; u y (cid:19) , (4.14) cf. ( [9] , [8, Eq-(3.12)] ). If u = − , then the above equation reduces to the well knownmultiplication formula for the Euler polynomials: for y is an odd positive integer, we have E n ( yx ) = y n y − X j =0 ( − j E n (cid:18) x + jy (cid:19) , (4.15) where E n ( x ) denotes the Euler polynomials in the usual notation. If y is an even positiveinteger, we have E n ( yx ) = 2 y n − n y − X j =0 ( − j B n (cid:18) x + jy (cid:19) , (4.16) where B n ( x ) and E n ( x ) denote the Bernoulli polynomials and Euler polynomials, respectively,cf. ( [7] , [50] ). To prove the multiplication formula of the generalized Apostol Bernoulli polynomials, weneed the following generating function which is defined by Srivastava et al. [48, pp. 254, Eq.(20)]:
Definition 5.
Let a, b, c ∈ R + with a = b, x ∈ R and n ∈ N . Then the generalized Bernoullipolynomials B ( α ) n ( x ; λ ; a, b, c ) of order α ∈ C are defined by means of the following generatingfunctions: f B ( x, a, b, c ; λ ; α ) = (cid:18) tλb t − a t (cid:19) α c xt = ∞ X n =0 B ( α ) n ( x ; λ ; a, b, c ) t n n ! , (4.17) where (cid:12)(cid:12)(cid:12) t ln( ab ) + ln λ (cid:12)(cid:12)(cid:12) < π and α = 1 . Observe that if we set λ = 1 in (4.17), we have (cid:18) tb t − a t (cid:19) α c xt = ∞ X n =0 B ( α ) n ( x ; a, b, c ) t n n ! . (4.18)If we set x = 0 in (4.18), we obtain (cid:18) tb t − a t (cid:19) α = ∞ X n =0 B ( α ) n ( a, b ) t n n ! , (4.19) Yilmaz Simsek with of course, B ( α ) n ( x ; a, b, c ) = B ( α ) n ( a, b ), cf. ([30]-[31], [21], [45], [26], [32], [34], [35], [47],[49], [46], [48]). If we set α = 1 in (4.19) and (4.18), we have tb t − a t = ∞ X n =0 B n ( a, b ) t n n ! (4.20)and (cid:18) tb t − a t (cid:19) c xt = ∞ X n =0 B n ( x ; a, b, c ) t n n ! , (4.21)which have been studied by Luo et al. [30]-[31]. Moreover, by substituting a = 1 and b = c = e into (4.17), then we arrive at the Apostol-Bernoulli polynomials B n ( x ; λ ), whichare defined by means of the following generating function (cid:18) tλe t − (cid:19) e xt = ∞ X n =0 B n ( x ; λ ) t n n ! , These polynomials B n ( x ; λ ) have been introduced and investigated by many Mathematicianscf. ([3], [23], [18], [21], [25], [28], [35], [41], [47]). When a = λ = 1 and b = c = e into (4.20)and (4.21), B n ( a, b ) and B n ( x ; a, b, c ) reduce to the classical Bernoulli numbers and theclassical Bernoulli polynomials, respectively, cf. [1]-[53]. Remark 19.
The constraints on | t | , which we have used in Definition 5 and (4.3), aremeant to ensure that the generating function in (4.18)and (4.3) are analytic throughoutthe prescribed open disks in complex t -plane (centred at the origin t = 0 ) in order to havethe corresponding convergent Taylor-Maclaurin series expansion (about the origin t = 0 )occurring on the their right-hand side (with a positive radius of convergence) cf. [49] . Theorem 17.
Let y ∈ N . Then we have B n ( yx ; λ ; a, b, b ) = n X l =0 y − X j =0 (cid:18) nl (cid:19) λ j y l − (( y − − j ) ln a ) n − l B l (cid:18) x + jy ; λ y ; a, b, b (cid:19) . Proof.
Substituting c = b and α = 1 into (4.17), we get ∞ X n =0 B n ( x ; λ ; a, b, c ) t n n ! = 1 y y − X j =0 λ j ytλ y b yt − a yt b ( x + jy ) yt a t ( y − j − . Therefore ∞ X n =0 B n ( x ; λ ; a, b, c ) t n n != ∞ X n =0 n X l =0 y − X j =0 (cid:18) nl (cid:19) λ j (( y − − j ) ln a ) n − l y l − B l (cid:18) x + jy ; λ y ; a, b, b (cid:19) t n n ! . ew Generating Functions of the Stirling numbers, Frobenius-Euler and Related polynomials 19 Comparing the coefficients of t n n ! on both sides of the above equation, we get B ( α ) n ( x ; λ ; a, b, c ) = n X l =0 y − X j =0 (cid:18) nk (cid:19) λ j (( k − − j ) ln a ) n − l y l − B l (cid:18) x + jk ; λ y ; a, b, b (cid:19) . By replacing x by yx on both sides of the above equation, we arrive at the desired result. (cid:3) Remark 20.
Kurt and Simsek [26] proved multiplication formula for the generalized Bernoullipolynomials of order α . When a = λ = 1 and b = c = e into Theorem 17, we have the mul-tiplication formula for the Bernoulli polynomials given by B n ( yx ) = y n − y − X j =0 B n (cid:18) x + jy (cid:19) , (4.22) cf. ( [3] , [7] , [8] , [14] , [21] , [31] , [28] , [27] , [32] , [29] , [47] , [48] ). If f is a normalized polynomial such that it satisfies the formula f n ( yx ) = y n − y − X j =0 f n (cid:18) x + jy (cid:19) , (4.23)then f is the y th degree Bernoulli polynomial due to (4.22) cf. ([7], [53]). According toNielsen [7], if a normalized polynomial satisfies (4.22) for a single value of y >
1, then it isidentical with B m ( x ). Consequently, if a normalized polynomial satisfies (4.13) for a singlevalue of y >
1, then it is identical with H n ( x ; u ; 1 , b, b ; λ ). The formula (4.16) is different.Therefore, for y is an even positive integer, Carlitz [7, Eq-(1.4)] considered the followingequation: g n − ( yx ) = − y n − n y − X j =0 ( − j f n (cid:18) x + jy (cid:19) , where g n − ( x ) and f n ( x ) denote the normalized polynomials of degree n − n , respec-tively. More precisely, as Carlitz has pointed out [7, p. 184], if y is a fixed even integer ≥ f n ( x ) is an arbitrary normalized polynomial of degree n , then (4.16) determines g n − ( x )as a normalized polynomial of degree n −
1. Thus, for a single value y , (4.16) does not sufficeto determine the normalized polynomials g n − ( x ) and f n ( x ). Remark 21.
According to (4.23), the set of normalized polynomials { f n ( x ) } is an Appellset, cf. [7] . We now modify (4.1) as follows:( a t − ξ ) c xt λb t − ξ = ∞ X n =0 H n ( x ; ξ ; a, b, c ; λ ) t n n ! (4.24)where ξ r = 1 , ξ = 1 . Yilmaz Simsek
The polynomial H n ( x ; ξ ; a, b, c ; λ ) is a normalized polynomial of degree m in x . The poly-nomial H n ( x ; ξ ; 1 , e, e ; 1) may be called Eulerian polynomials with parameter ξ . In particularwe note that H n ( x ; −
1; 1 , e, e ; 1) = E n ( x )since for a = λ = 1, b = c = e , equation (4.24) reduces to the generating function for theEuler polynomials.By means of equation (4.11), it is easy to verify the following multiplication formulas:If y is an odd positive integer, then we have H n − ( yx ; ξ ; a, b, b ; λ ) = y n − n y − X j =0 (cid:18) λξ (cid:19) j B n (cid:18) x + jy ; b ; λ y (cid:19) (4.25) − ξn n X k =0 y − X j =0 (cid:18) λξ (cid:19) j y k − (ln a ) n − k B k (cid:18) x + jy ; b ; λ y (cid:19) , where H k (cid:18) x + jy ; ξ y ; 1 , b, b ; λ y (cid:19) = B n (cid:18) x + jy ; b ; λ y (cid:19) .If y is an even positive integer, then we have H n ( yx ; ξ ; a, b, b ; λ ) = y n y − X j =0 (cid:18) λξ (cid:19) j E n (cid:18) x + jy ; b ; λ y (cid:19) (4.26) − ξ n X k =0 y − X j =0 (cid:18) λξ (cid:19) j y k (ln a ) n − k E k (cid:18) x + jy ; b ; λ y (cid:19) , where H k (cid:18) x + jy ; ξ y ; 1 , b, b ; λ y (cid:19) = E n (cid:18) x + jy ; b ; λ y (cid:19) ,where E n ( x ; a, b, c ) denotes the generalized Euler polynomials, which are defined by meansof the following generating function: (cid:18) tb t − a t (cid:19) c xt = ∞ X n =0 E n ( x ; a, b, c ) t n n !cf. ([30]-[31], [24], [26], [35], [47], [49], [46], [48]). Remark 22.
If we set a = λ = 1 and b = e , then (4.25) and (4.26) reduce to the followingmultiplication formulas, respectively: H n − ( yx ; ξ ) = y n − n (cid:18) − ξ (cid:19) y − X j =0 ξ j B n (cid:18) x + jy (cid:19) cf. [7, Eq. (3.3)] and H n ( yx ; ξ ) = y n (cid:18) − ξ (cid:19) y − X j =0 ξ j E n (cid:18) x + jy (cid:19) . ew Generating Functions of the Stirling numbers, Frobenius-Euler and Related polynomials 21 Let f n ( x ) and g n ( x ) be normalized polynomials in the usual way. Carlitz [7, Eq. (3.4)] definedthe following equation: g n − ( yx ) = (1 − ρ ) y n − n y − X j =0 ρ j f n (cid:18) x + jy (cid:19) , where ρ is a fixed primitive r th root of unity, r > , y ≡ r ) . Remark 23.
If we set a = λ = 1 , b = c = e and ξ = − , then (4.25) and (4.26) reduce to(4.16) and (4.15). Remark 24.
Walum [53] defined multiplication formula for periodic functions as follows: ϑ ( y ) f ( yx ) = X j ( y ) f (cid:18) x + jy (cid:19) , (4.27) where f is periodic with period and j ( y ) under the summation sign indicates that j runsthrough a complete system of residues mod y .Formulas (4.23), (4.27) and other multiplication formulas related to periodic functions andnormalized polynomials occur in Franel’s formula, in the theory of the Dedekind sums andHardy-Berndt sums, in the theory of the zeta functions and L -functions and in the theory ofperiodic bounded variation, cf. ( [4] , [5] , [53] ). Generalized Eulerian type numbers and polynomials attached to Dirichletcharacter.
In this section, we construct generating function, related to nonnegative realparameters, for the generalized Eulerian type numbers and polynomials attached to Dirichletcharacter. We also give some properties of these polynomials and numbers.
Definition 6.
Let χ be the Dirichlet character of conductor f ∈ N . Let x ∈ R , a, b ∈ R + , ( a = b ) , λ ∈ C and u ∈ C(cid:31) { } . The generalized Eulerian type polynomials H n,χ ( x ; u ; a, b, c ; λ ) are defined by means of the following generating function: F λ,χ ( t, x ; u, a, b, c ) = f − X j =0 (cid:0) a ft − u f (cid:1) χ ( j ) u f − j − c ( x + jf ) ft λ f b ft − u f = ∞ X n =0 H n,χ ( x ; u ; a, b, c ; λ ) t n n ! (4.28) with, of course H n,χ (0; u ; a, b, c ; λ ) = H n,χ ( u ; a, b, c ; λ ) , where H n,χ ( u ; a, b, c ; λ ) denotes generalized Eulerian type numbers. Remark 25.
In the special case when a = λ = 1 and b = c = e , the generalized Euleriantype polynomials H n,χ ( x ; u ; a, b, c ; λ ) are reduced to the Frobenius Euler polynomials whichare defined by means of the following generating function: f − X j =0 (cid:0) − u f (cid:1) χ ( j ) u f − j − e ( x + jf ) ft e ft − u f = ∞ X n =0 H n,χ ( x ; u ) t n n ! , cf. ( [51] , [21] , [45] , [39] , [40] , [47] ). Substituting u = − into the above equation, we havegenerating function of the generalized Euler polynomials attached to Dirichlet character with Yilmaz Simsek odd conductor: f − X j =0 χ ( j )( − j e ( x + jf ) ft e ft + 1 = ∞ X n =0 E n,χ ( x ) t n n ! , cf. ( [51] , [39] , [40] , [47] ). Combining (4.1) and (4.28), we obtain the following functional equation: F λ,χ ( t, x ; u, a, b, c ; ) = f − X j =0 χ ( j ) u f − j − F λ f ( f t, x + jf ; u f , a, b, c ) . By using the above functional equation we arrive at the following Theorem:
Theorem 18. H n,χ ( x ; u ; a, b, c ; λ ) = f n f − X j =0 χ ( j ) u f − j − H n ( x + jf ; u f ; a, b, c ; λ f ) . Theorem 19. H n,χ ( x ; u ; a, b, c ; λ ) = n X j =0 (cid:18) nj (cid:19) ( x ln c ) n − j H j,χ ( u ; a, b, c ; λ ) . Proof.
By using (4.28), we get ∞ X n =0 H n,χ ( u ; a, b, c ; λ ) t n n ! ∞ X n =0 ( x ln c ) n t n n ! = ∞ X n =0 H n,χ ( x ; u ; a, b, c ; λ ) t n n ! . From the above equation, we obtain ∞ X n =0 n X j =0 (cid:18) nj (cid:19) ( x ln c ) n − j H j,χ ( u ; a, b, c ; λ ! t n n ! = ∞ X n =0 H n,χ ( x ; u ; a, b, c ; λ ) t n n ! . Comparing the coefficients of t n n ! on both sides of the above equation, we arrive at the desiredresult. (cid:3) Recurrence relation for the generalized Eulerian type polynomials.
In thissection we are going to differentiate (4.1) with respect to the variable t to derive a recurrencerelation for the generalized Eulerian type polynomials. Therefore, we obtain the followingdifferential equation: ∂∂t F λ ( t, x ; u, a, b, c ) = (ln a ) F λ ( t, x ; u, a, b, c ) + ln at f B ( x, , b, c ; λu ; 1) − ln (cid:0) b λ (cid:1) ut F λ ( t, x ; u, a, b, c ) f B (1 , , b, b ; λu ; 1)+ ln ( c x ) F λ ( t, x ; u, a, b, c ) . By using this equation, we obtain a recurrence relation for the generalized Eulerian typepolynomials by the following theorem: ew Generating Functions of the Stirling numbers, Frobenius-Euler and Related polynomials 23
Theorem 20.
Let n ∈ N . We have n H n ( x ; u ; a, b, c ; λ ) = (ln a ) (cid:18) n H n − ( x ; u ; a, b, c ; λ ) + B n ( x ; λu ; a, b, c ) (cid:19) − λ ln bu n X j =0 (cid:18) nj (cid:19) H j ( x ; u ; a, b, c ; λ ) B n − j (1; λu ; 1 , b, b )+ (ln ( c nx )) H n − ( x ; u ; a, b, c ; λ ) , where B n ( x ; λ ; a, b, c ) denotes the generalized Bernoulli polynomials of order . Remark 26.
When a = λ = 1 and b = c = e , the recurrence relation for the generalizedEulerian type polynomials is reduced to nH n ( x ; u ) = nxH n − ( x ; u ) − u n X j =0 (cid:18) nj (cid:19) H j ( x ; u ) B n − j (1; 1 u ) . New identities involving families of polynomials
In this section, we derive some new identities related to the generalized Bernoulli polyno-mials and numbers of order 1, the Eulerian type polynomials and the generalized array typepolynomials.
Theorem 21.
The following relationship holds true: B n ( x ; λ ; a, b, b ) = n X j =0 (cid:18) nj (cid:19) H j ( x ; λ − ; a, ba , ba ; 1) B n − j ( x − λ ; 1 , a, a ) . Proof. ∞ X n =0 B n ( x ; λ ; a, b, b ) t n n ! = (cid:18) ta ( x − t λa t − (cid:19) (cid:0) a t − λ − (cid:1) (cid:0) ba (cid:1) xt (cid:0) ba (cid:1) t − λ − ! . Combining (4.17) and (4.1) with the above equation, we get ∞ X n =0 B n ( x ; λ ; a, b, b ) t n n ! = ∞ X n =0 B n ( x − λ ; 1 , a, a ) t n n ! ∞ X n =0 H n ( x ; λ − ; a, ba , ba ; 1) t n n ! . Therefore ∞ X n =0 B n ( x ; λ ; a, b, b ) t n n ! = ∞ X n =0 n X j =0 (cid:18) nj (cid:19) H j ( x ; λ − ; a, ba , ba ; 1) B n − j ( x − λ ; 1 , a, a ) ! t n n ! . Comparing the coefficients of t n n ! on both sides of the above equation, we arrive at the desiredresult. (cid:3) Relationship between the generalized Bernoulli numbers and the Frobenius Euler numbersis given by the following result: Yilmaz Simsek
Theorem 22.
The following relationship holds true: B n ( λ ; a, b ) = 1 λ − n X j =0 (cid:18) nj (cid:19) j (cid:0) ln a − (cid:1) n − j (cid:18) ln (cid:18) ba (cid:19)(cid:19) j H j − (cid:0) λ − (cid:1) . (5.1) Proof.
By using (4.17), we obtain ∞ X n =0 B n ( λ ; a, b ) t n n ! = ta − t λ − (cid:18) − λ − e t ln ( ba ) − λ − (cid:19) . From the above equation, we get ∞ X n =0 B n ( λ ; a, b ) t n n ! = 1 λ − ∞ X n =0 (cid:18) ln (cid:18) a (cid:19)(cid:19) n t n n ! ∞ X n =0 n H n ( λ − ) (cid:18) ln (cid:18) ba (cid:19)(cid:19) n t n n ! . Therefore ∞ X n =0 B n ( λ ; a, b ) t n n ! = ∞ X n =0 n X j =0 (cid:18) nj (cid:19) j (ln a − ) n − j (cid:0) ln (cid:0) ba (cid:1)(cid:1) j λ − H j − (cid:0) λ − (cid:1)! t n n ! . Comparing the coefficients of t n n ! on both sides of the above equation, we arrive at thedesired result. (cid:3) Remark 27.
By substituting a = 1 and b = e into (5.1), we have B n ( λ ) = nλ − H n − ( λ − ) , cf. [21] . Relationship between the generalized Eulerian type polynomials and generalized arraytype polynomials are given by the following theorem:
Theorem 23.
The following relationship holds true: H n ( x ; u ; a, b, b ; λ ) = ∞ X k =0 ∞ X m =0 n X d =0 (cid:18) m + k − m (cid:19) (cid:18) nd (cid:19) k ! (ln a m ) n − d u m + k S dk ( x ; a, b ; λ ) . Proof.
From (4.1), we obtain ∞ X n =0 H n ( x ; u ; a, b, c ; λ ) t n n ! = ∞ X k =0 (cid:18) λb t − a t u − a t (cid:19) k b xt . Combining (3.3) with the above equation, we get ∞ X n =0 H n ( x ; u ; a, b, b ; λ ) t n n ! = ∞ X k =0 k !( u − a t ) k ∞ X n =0 S nk ( x ; a, b ; λ ) t n n ! . From the above equation, we get ∞ X n =0 H n ( x ; u ; a, b, b ; λ ) t n n ! = ∞ X n =0 ∞ X k =0 k ! S nk ( x ; a, b ; λ ) u k (cid:0) − a t u (cid:1) k t n n ! . ew Generating Functions of the Stirling numbers, Frobenius-Euler and Related polynomials 25 Now we assume (cid:12)(cid:12)(cid:12) a t u (cid:12)(cid:12)(cid:12) < ∞ X n =0 H n ( x ; u ; a, b, b ; λ ) t n n != ∞ X n =0 ∞ X k =0 ∞ X m =0 (cid:18) m + k − m (cid:19) k ! S nk ( x ; a, b ; λ ) u k + m a mt t n n ! . Therefore ∞ X n =0 H n ( x ; u ; a, b, b ; λ ) t n n != ∞ X n =0 ∞ X k =0 ∞ X m =0 n X d =0 (cid:18) m + k − m (cid:19) (cid:18) nd (cid:19) k ! (ln a m ) n − d u m + k S dk ( x ; a, b ; λ ) . ! t n n ! . Comparing the coefficients of t n n ! on both sides of the above equation, we arrive at the desiredresult. (cid:3) Remark 28.
Substituting a = 1 into the above Theorem and noting that d = n , we deducethe following identity: H n ( x ; u ; 1 , b, b ; λ ) = ∞ X k =0 k !( u − k S nk ( x ; 1 , b ; λ ) which upon setting λ = 1 and b = e , yields H n ( x ; u ) = n X k =0 k !( u − k S nk ( x ) which was found by Chang and Ha [12, Lemma 1] . Relationship between the generalized Bernoulli polynomials and thegeneralized array type polynomials
In this section, we give some applications related to the generalized Bernoulli polynomials,generalized array type polynomials. We derive many identities involving these polynomials.By using same method with Agoh and Dilcher’s [1], we give the following Theorem:
Theorem 24. (cid:18) λb t − a t t (cid:19) k b xt = ∞ X n =0 S n + kk ( x ; a, b ; λ ) (cid:0) n + kk (cid:1) t n n ! . (6.1) Proof.
Combining (3.3) and (3.2), we get (cid:18) λb t − a t t (cid:19) k b xt = 1 t k ∞ X n =0 k ! n ! S nk ( x, a, b ; λ ) t n = ∞ X n =0 k ! n ! S n + kk ( x, a, b ; λ ) t n − k . Yilmaz Simsek
From the above equation, we arrive at the desired result. (cid:3)
Remark 29.
By setting x = 0 , a = λ = 1 and b = e , Theorem 24 yields the correspondingresult which is proven by Agoh and Dilcher [1] . Theorem 25. ( n + k ) S n + kk ( x ; a, b ; λ ) (cid:0) n + kk (cid:1) − xn S n + k − k ( x ; a, b ; λ ) (cid:0) n + k − k (cid:1) = n X j =0 (cid:18) nj (cid:19)(cid:18) j + k − k − (cid:19) S j + k − k − ( x ; a, b ; λ ) (cid:16) ln (cid:0) b λk (cid:1) (ln( b )) n − j − ln (cid:0) a k (cid:1) (ln( a )) n − j (cid:17) . Proof.
By differentiating both sides of equation (6.1) with respect to the variable t , aftersome elementary calculations, we get the formula asserted by Theorem 25. (cid:3) Theorem 26.
The following relationship holds true: S nk − ( x + y ; a, b ; λ ) = n X j =0 (cid:18) nj (cid:19) (cid:18) n + k − k − (cid:19)(cid:18) j + kk (cid:19) S j + kk ( x ; a, b ; λ ) B n − j ( y ; λ ; a, b, b ) . Proof.
We set (cid:18) λb t − a t t (cid:19) k b xt (cid:18) tb yt λb t − a t (cid:19) = (cid:18) λb t − a t t (cid:19) k − b ( x + y ) t . Combining (6.1) and (4.21) with the above equation, we get ∞ X n =0 S n + k − k − ( x + y ; a, b ; λ ) (cid:0) n + k − k − (cid:1) t n n ! = ∞ X n =0 B n ( y ; λ ; a, b, b ) t n n ! ∞ X n =0 S n + kk ( x ; a, b ; λ ) (cid:0) n + kk (cid:1) t n n ! . Therefore ∞ X n =0 S n + k − k − ( x + y ; a, b ; λ ) (cid:0) n + k − k − (cid:1) t n n ! = ∞ X n =0 n X j =0 (cid:18) nj (cid:19)(cid:18) j + kk (cid:19) S j + kk ( x ; a, b ; λ ) B n − j ( y ; λ ; a, b, b ) t n n ! . Comparing the coefficients of t n n ! on both sides of the above equation, we arrive at the desiredresult. (cid:3) Remark 30.
By setting x = y = 0 , a = λ = 1 and b = e , Theorem 26 yields the correspond-ing result which is proven by Agoh and Dilcher [1] . Theorem 27.
The following relationship holds true: B ( u − v ) n ( x + y ; λ ; a, b, b ) = n X j =0 (cid:18) nj (cid:19)(cid:18) n + vv (cid:19) S j + vv ( x ; a, b ; λ ) B ( u ) n − j ( y ; λ ; a, b, b ) . ew Generating Functions of the Stirling numbers, Frobenius-Euler and Related polynomials 27 Proof.
We set (cid:18) λb t − a t t (cid:19) v b xt (cid:18) tλb t − a t (cid:19) u b yt = (cid:18) tλb t − a t (cid:19) u − v b ( x + y ) t . (6.2)Combining (6.1) and (4.17) with the above equation, by using same calculations with theproof of Theorem 26, we arrive at the desired result. (cid:3) Application of the Laplace transform to the generating functions forthe generalized Bernoulli polynomials and the generalized array typepolynomials
In this section, we give an application of the Laplace transform to the generating functionfor the generalized Bernoulli polynomials and the generalized array type polynomials. Weobtain interesting series representation for the families of these polynomials.By using (6.2), we obtain ∞ X n =0 B ( u − v ) n ( λ ; a, b, b ) t n n ! e − t ( y − x ) ln b = ∞ X n =0 n X j =0 (cid:18) nj (cid:19)(cid:18) n + vv (cid:19) S j + vv ( x ; a, b ; λ ) B ( u ) n − j ( λ ; a, b, b ) t n n ! e − ty ln b . Integrate this equation (by parts) with respect to t from 0 to ∞ , we get ∞ X n =0 B ( u − v ) n ( λ ; a, b, b ) n ! ∞ Z t n e − t ( y − x ) ln b dt = ∞ X n =0 n ! n X j =0 (cid:18) nj (cid:19)(cid:18) n + vv (cid:19) S j + vv ( x ; a, b ; λ ) B ( u ) n − j ( λ ; a, b, b ) ∞ Z t n e − ty ln b dt. By using Laplace transform in the above equation, we arrive at the following Theorem:
Theorem 28.
The following relationship holds true: ∞ X n =0 B ( u − v ) n ( λ ; a, b, b )(ln b y − x ) n +1 = ∞ X n =0 n X j =0 (cid:18) nj (cid:19)(cid:18) n + vv (cid:19) S j + vv ( x ; a, b ; λ ) B ( u ) n − j ( λ ; a, b, b )(ln b y ) n +1 . Remark 31.
When a = λ = 1 and b = e , Theorem 28 is reduced to the following result: ∞ X n =0 B ( u − v ) n ( y − x ) n +1 = ∞ X n =0 n X j =0 (cid:18) nj (cid:19)(cid:18) n + vv (cid:19) S j + vv ( x ) B ( u ) n − j y n +1 . Yilmaz Simsek Applications the p -adic integral to the family of the normalizedpolynomials and the generalized λ -Stirling type numbers By using the p -adic integrals on Z p , we derive some new identities related to the Bernoullinumbers, the Euler numbers, the generalized Eulerian type numbers and the generalized λ -Stirling type numbers.In order to prove the main results in this section, we recall each of the following knownresults related to the p -adic integral.Let p be a fixed prime. It is known that µ q ( x + p N Z p ) = q x [ p N ] q is a distribution on Z p for q ∈ C p with | − q | p <
1, cf. [19]. Let
U D ( Z p ) be the set ofuniformly differentiable functions on Z p . The p -adic q -integral of the function f ∈ U D ( Z p )is defined by Kim [19] as follows: Z Z p f ( x ) dµ q ( x ) = lim N →∞ p N ] q p N − X x =0 f ( x ) q x , where [ x ] = 1 − q x − q . From this equation, the bosonic p -adic integral ( p -adic Volkenborn integral) was consideredfrom a physical point of view to the bosonic limit q →
1, as follows ([19]): Z Z p f ( x ) dµ ( x ) = lim N →∞ p N p N − X x =0 f ( x ) , (8.1)where µ (cid:0) x + p N Z p (cid:1) = 1 p N . The p -adic q -integral is used in many branch of mathematics, mathematical physics andother areas cf. ([2], [19], [21], [37], [38], [41], [42], [47], [52]).By using (8.1), we have the Witt’s formula for the Bernoulli numbers B n as follows: Z Z p x n dµ ( x ) = B n (8.2)cf. ([2], [19], [20], [22], [37], [52]).We consider the fermionic integral in contrast to the convential bosonic, which is calledthe fermionic p -adic integral on Z p cf. [20]. That is Z Z p f ( x ) dµ − ( x ) = lim N →∞ p N − X x =0 ( − x f ( x ) (8.3) ew Generating Functions of the Stirling numbers, Frobenius-Euler and Related polynomials 29 where µ (cid:0) x + p N Z p (cid:1) = ( − x p N cf. [20]. By using (8.3), we have the Witt’s formula for the Euler numbers E n as follows: Z Z p x n dµ − ( x ) = E n , (8.4)cf. ([20], [22], [42], [47]).The Volkenborn integral in terms of the Mahler coefficients is given by the followingTheorem: Theorem 29.
Let f ( x ) = ∞ X j =0 a j (cid:18) xj (cid:19) ∈ U D ( Z p ) . Then Z Z p f ( x ) dµ ( x ) = ∞ X j =0 a j ( − j j + 1 . Proof of Theorem 29 was given by Schikhof [37].
Theorem 30. Z Z p (cid:18) xj (cid:19) dµ ( x ) = ( − j j + 1 . Proof of Theorem 30 was given by Schikhof [37] . Theorem 31.
The following relationship holds true: B m = 1ln m b m X j =0 ( − j j ! j + 1 S ( m, j ; 1 , b ; 1) . (8.5) Proof.
If we substitute a = λ = 1 in Theorem 3, we have(ln b x ) m = m X j =0 (cid:18) xj (cid:19) j ! S ( m, j ; 1 , b ; 1) . By applying the p -adic Volkenborn integral with Theorem 30 to the both sides of the aboveequation, we arrive at the desired result. (cid:3) Remark 32.
By substituting b = 1 into (8.5), we have B m = m X j =0 ( − j j ! j + 1 S ( m, j ) where S ( m, j ) denotes the Stirling numbers of the second kind cf. ( [11] , [15] , [23] ). Yilmaz Simsek
Theorem 32.
The following relationship holds true: n X j =0 (cid:18) nj (cid:19) (ln a ) n − j (ln c ) j B j − u (ln c ) n B n = n X j =0 (cid:18) nj (cid:19) (ln c ) j (cid:16) λ ( H ( u ; a, b, c ; λ ) + ln b ) n − j − u H n − j ( u ; a, b, c ; λ ) (cid:17) B j . Proof.
By using Theorem 10, we have n X j =0 (cid:18) nj (cid:19) (ln a ) n − j (ln c ) j x j − u (ln c ) n x n (8.6)= n X j =0 (cid:18) nj (cid:19) (ln c ) j x j (cid:16) λ ( H ( u ; a, b, c ; λ ) + ln b ) n − j − u H n − j ( u ; a, b, c ; λ ) (cid:17) . By applying Volkenborn integral in (8.1) to the both sides of the above equation, we get n X j =0 (cid:18) nj (cid:19) (ln a ) n − j (ln c ) j Z Z p x j dµ ( x ) − u (ln c ) n Z Z p x n dµ ( x )= n X j =0 (cid:18) nj (cid:19) (ln c ) j (cid:16) λ ( H ( u ; a, b, c ; λ ) + ln b ) n − j − u H n − j ( u ; a, b, c ; λ ) (cid:17) Z Z p x j dµ ( x ) . By substituting (8.2) into the above equation, we easily arrive at the desired result. (cid:3)
Remark 33.
By substituting b = c = e and a = λ = 1 into Theorem 32, we arrive at thefollowing nice identity: B n = 11 − u n X j =0 (cid:18) nj (cid:19) (cid:16) ( H ( u ) + 1) n − j − uH n − j ( u ) (cid:17) B j . Theorem 33.
The following relationship holds true: n X j =0 (cid:18) nj (cid:19) (ln a ) n − j (ln c ) j E j − u (ln c ) n E n = n X j =0 (cid:18) nj (cid:19) (ln c ) j (cid:16) λ ( H ( u ; a, b, c ; λ ) + ln b ) n − j − u H n − j ( u ; a, b, c ; λ ) (cid:17) E j . Proof.
Proof of Theorem 33 is same as that of Theorem 32. Combining (8.3), (8.6) and (8.4),we easily arrive at the desired result. (cid:3)
Remark 34.
By substituting b = c = e and a = λ = 1 into Theorem 33, we arrive at thefollowing nice identity: E n = 11 − u n X j =0 (cid:18) nj (cid:19) (cid:16) ( H ( u ) + 1) n − j − uH n − j ( u ) (cid:17) E j . ew Generating Functions of the Stirling numbers, Frobenius-Euler and Related polynomials 31 Acknowledgement 1.
The present investigation was supported by the Scientific ResearchProject Administration of Akdeniz University.
References [1] T. Agoh and K. Dilcher, Shortened recurrence relations for Bernoulli numbers, Discrete Math. 309(2009), 887-898.[2] Y. Amice, Integration p -adique, selon A. Volkenborn, Seminaire Delange-Pisot-Poitou, Theorie desNombres 13(2) (1971-1972), G4, p. G1-G9.[3] T. M. Apostol, On the Lerch zeta function, Pacific J. Math. 1 (1951), 161-167.[4] B. C. Berndt and L. Schoenfeld, Periodic analogues of the Euler-Maclaurin and Poisson summationformulas with applications to number theory, Acta Arith. 28 (1975), 23-68.[5] B. C. Berndt, Periodic Bernoulli numbers, summation formulas and applications, in Theory and Appli-cation of Special Functions (Proc. Advanced Sem., Math. Res. Center, Univ. Wisconsin, Madison, Wis.,(1975), 143-189.[6] N. P. Cakic and G. V. Milovanovic, On generalized Stirling numbers and polynomials, Math. Balkanica18(3-4) (2004), 241-248.[7] L. Carlitz, A note on the multiplication formulas for the Bernoulli and Euler polynomials, Proc. Amer.Math. Soc. 4 (1953), 184-188.[8] L. Carlitz, Eulerian Numbers and Polynomials, Math. Mag. 32 (1959), 247-260.[9] L. Carlitz, Generating functions, Fibonacci Quart. 7 (1969), 359-393.[10] L. Carlitz, Some numbers related to the Stirling numbers of the first and second kind, Univ. BeogradPubl. Elektrotehn. Fak. Ser. Mat. Fiz. 544-576 (1976), 49-55.[11] O. Y. Chan and D. Manna, A new q -analogue for Bernoulli numbers, preprint.[12] C.-H. Chang and C.-W. Ha, A multiplication theorem for the Lerch zeta function and explicit repre-sentations of the Bernoulli and Euler polynomials, J. Math. Anal. Appl. 315 (2006), 758-767.[13] L. Comtet, Advanced Combinatorics: The Art of Finite and Infinite Expansions, Reidel, Dordrecht andBoston, 1974 (Translated from the French by J. W. Nienhuys).[14] R. Dere and Y. Simsek, Unification of the three families of generalized Apostol type polynomials on theUmbral algebra, arXiv:1110.2047v1 [math.CA].[15] The Digital Library of Mathematical Function 24.6; http://dlmf.nist.gov/24.15.[16] P. Flajolet and R. Sedgewick, Analytic combinatorics: Functional equations, rational and algebraicfunctions, INRIA No 4103, January 2001.[17] K.-W. Hwang, Y.-H. Kim, and T. Kim, Interpolation functions of q -extensions of Apostol’s type Eulerpolynomials, Journal of Inequalities and Applications, vol. 2009, Article ID 451217, 12 pages, 2009.[18] Y.-H. Kim, W. Kim, and L.-C. Jang, On the q -Extension of Apostol-Euler Numbers and Polynomials,Abstr. Appl. Anal. Volume 2008 (2008), 10 pages.[19] T. Kim, q -Volkenborn integration, Russian J. Math. Phys. 19 (2002), 288-299.[20] T. Kim, q -Euler numbers and polynomials associated with p -adic q -integral and basic q -zeta function,Trend Math. Information Center mathematical Sciences 9 (2006), 7-12.[21] T. Kim, S.-H. Rim, Y. Simsek and D. Kim, On the analogs of Bernoulli and Euler numbers, relatedidentities and zeta and L -functions, J. Korean Math. Soc. 45 (2008), 435-453.[22] T. Kim, M. S. Kim and L. C. Jang, New q -Euler numbers and polynomials associated with p -adic q -integrals, Adv. Stud. Contemp. Math. 15 (2007), 140-153.[23] T. Kim, J. Choi and Y.-H. Kim, Some identities on the q -Stirling numbers and q -Bernoulli numbers,arXiv:1006.2033v1.[24] T. Kim, L.-C. Jang and C.-S. Ryoo, Note on q -extensions of Euler numbers and polynomials of higherorder, J. Inequal. Appl. 2008, Art. ID 371295, 9 pp.[25] B. Kurt and Y. Simsek, Frobenius-Euler type polynomials related to Hermite-Bernoulli polynomials,Proceedings of the International Conference on Numerical Analysis and Applied Mathematics, Amer.Inst. Phys. Conf. Proc. 1389 (2011), 385-388. Yilmaz Simsek [26] B. Kurt and Y. Simsek, Notes on generalization of the Bernoulli type polynomials, Appl. Math. Comput.218 (2011), 906-911.[27] Q.-M. Luo, B.-N. Guo, F. Qi, and L. Debnath, Generalizations of Bernoulli numbers and polynomials,IJMMS 2003:59, 3769-3776, PII. S0161171203112070, http://ijmms.hindawi.com.[28] Q. M. Luo and H. M. Srivastava, Some relationships between the Apostol-Bernoulli and Apostol-Eulerpolynomials, Comput. Math. Appl. 51 (2006), 631-642.[29] Q.-M. Luo and H. M. Srivastava, Some generalizations of the Apostol-Genocchi polynomials and theStirling numbers of the second kind, Appl. Math. Comput. 217 (2011), 5702-5728.[30] Q. M. Luo, B. N. Guo, F. Qi and L. Debnath, Generalizations of Bernoulli numbers and polynomials,Int. J. Math. Sci. 59 (2003), 3769-3776.[31] Q. M. Luo B. N. Guo, F. Qi and L. Debnath, Generalizations of Euler numbers and polynomials, Int.J. Math. Sci. 61 (2003), 3893-3901.[32] M. A. Ozarslan, Unified Apostol–Bernoulli, Euler and Genocchi polynomials, Comput. Math. Appl. 62(2011), 2452-2462.[33] H. Ozden, Unification of generating function of the Bernoulli, Euler and Genocchi numbers and poly-nomials, Proceedings of the International Conference on Numerical Analysis and Applied Mathematics(2010), Amer. Inst. Phys. Conf. Proc. 1281, (2010), 1125-1128.[34] H. Ozden and Y. Simsek, A new extension of q -Euler numbers and polynomials related to their inter-polation functions, Appl. Math. Lett. 21 (2008), 934-939.[35] H. Ozden, Y. Simsek and H. M. Srivastava, A unified presentation of the generating functions of thegeneralized Bernoulli, Euler and Genocchi polynomials, Comput. Math. Appl. 60 (2010), 2779-2787.[36] B. P. Parashar, On generalized exponential Euler polynomials, Indian J. Pure. Appl. Math. 15 (1984),1332-1339.[37] W. H. Schikhof, Ultrametric Calculus: An Introduction to p -Adic Analysis, Cambridge Studies inAdvanced Mathematics 4 (Cambridge University Press, Cambridge), 1984.[38] K. Shiratani, On Euler numbers, Mem. Fac. Kyushu Univ. Series A, Mathematics 27 (1973), 1-5.[39] Y. Simsek, On twisted generalized Euler numbers, Bull. Korean Math. Soc. 41 (2004), 299-306.[40] Y. Simsek, q -analogue of the twisted l -Series and q -twisted Euler numbers, J. Number Theory 100(2005), 267-278.[41] Y. Simsek, Twisted ( h, q )-Bernoulli numbers and polynomials related to twisted ( h, q )-zeta function and L -function, J. Math. Anal. Appl. 324 (2006), 790-804.[42] Y. Simsek, Complete sum of products of ( h, q )-extension of Euler polynomials and numbers, J. Differ.Equ. Appl. 16 (2010), 1331-1348.[43] Y. Simsek, Interpolation function of generalized q -Bernstein type polynomials and their application,Curve and Surface, Springer Verlag Berlin Heidelberg 2011, LNCS 6920, (2011), 647-662.[44] On q -deformed Stirling numbers, to appear in Int. J. Math. Stat. 2012.[45] Y. Simsek, T. Kim, D.W. Park, Y.S. Ro, L.-J. Jang and S.H. Rim, An explicit formula for the multipleFrobenius-Euler numbers and polynomials, JP J. Algebra Number Theory Appl. 4 (2004), 519-529.[46] H. M. Srivastava, Some generalizations and basic (or q -) extensions of the Bernoulli, Euler and Genocchipolynomials, Appl. Math. Inform. Sci. 5 (2011), 390-444.[47] H. M. Srivastava, T. Kim and Y. Simsek, q -Bernoulli numbers and polynomials associated with multiple q -zeta functions and basic L -series, Russian J. Math. Phys. 12 (2005), 241-268.[48] H. M. Srivastava, M. Garg and S. Choudhary, A new generalization of the Bernoulli and related poly-nomials, Russian J. Math. Phys .
17 (2010), 251-261.[49] H. M. Srivastava, M. Garg and S. Choudhary, Some new families of the generalized Euler and Genocchipolynomials, Taiwanese J. Math. 15 (2011), 283-305.[50] H. M. Srivastava, B. Kurt and Y. Simsek, Some families of Genocchi type polynomials and their inter-polation functions, preprint.[51] H. Tsumura, On a p -adic interpolation of generalized Euler numbers and its applications, Tokyo J.Math. 10 (1987), 281-293.[52] A. Volkenborn, On generalized p -adic integration, Memoires de la S. M. F. 39-40 (1974), 375-384. ew Generating Functions of the Stirling numbers, Frobenius-Euler and Related polynomials 33ew Generating Functions of the Stirling numbers, Frobenius-Euler and Related polynomials 33