NUMBERS. := p n 1 + p n 2 q + p n 3 q pq n 2 + q n 1 = pn q n p q. We can write easily that [n] p,q. , where [n] q/p

Size: px

Start display at page:

Download "NUMBERS. := p n 1 + p n 2 q + p n 3 q pq n 2 + q n 1 = pn q n p q. We can write easily that [n] p,q. , where [n] q/p"

Cathleen Todd
5 years ago
Views:

1 Kragujevac Journal of Mathematic Volume 424) 2018), Page APOSTOL TYPE p, q)-frobenius-euler POLYNOMIALS AND NUMBERS UGUR DURAN 1 AND MEHMET ACIKGOZ 2 Abtract In the preent paper, we introduce p, q)-extenion of Apotol type Frobeniu-Euler polynomial and number and invetigate ome baic identitie and propertie for thee polynomial and number, including addition theorem, difference equation, derivative propertie, recurrence relation and o on Then, we provide integral repreentation, explicit formula and relation for thee polynomial and number Moreover, we dicover p, q)-extenion of Carlitz reult [L Carlitz, Mat Mag ), ] and Srivatava and Pintér addition theorem in [H M Srivatava, A Pinter, Appl Math Lett ), ] 1 Introduction Throughout thi paper, we ue the tandart notion: N 0 denote the et of nonnegative integer, N denote the et of the natural number, R denote the et of real number and C denote the et of complex number The p, q)-number are defined a := p n 1 + p n 2 q + p n 3 q pq n 2 + q n 1 = pn q n p q We can write eaily that = p n 1 [n] q/p, where [n] q/p i the q-number in q-calculu given by [n] q/p = q/p)n 1 Thereby, thi implie that p, q)-number and q-number q/p) 1 are different, that i, we can not obtain p, q)-number jut by ubtituting q by q/p in the definition of q-number In the cae of p = 1, p, q)-number reduce to q-number, ee [9, 10] Key word and phrae p, q)-calculu, Bernoulli polynomial, Euler polynomial, Genocchi polynomial, Frobeniu-Euler polynomial, generating function, Cauchy product 2010 Mathematic Subject Claification Primary: 05A30, Secondary: 11B68; 11B73 Received: November 21, 2016 Accepted: June 23,

2 556 U DURAN AND M ACIKGOZ The p, q)-derivative of a function f with repect to x i defined by 11) D ;x f x) := D f x) = f px) f qx) p q) x x 0) and D f) 0) = f 0), provided that f i differentiable at 0 derivative operator hold the following propertie 12) D f x) g x)) = g px) D f x) + f qx) D g x) and f x) 13) D g x) The p, q)-analogue of x + a) n i given by = g qx) D f x) f qx) D g x) g px) g qx) The linear p, q)- { x + a) n x + a)px + aq) p = n 2 x + aq n 2 )p n 1 x + aq n 1 ), if n 1 1, if n = 0 = p 2) q n 2 ) x a n p, q)-gau Binomial Formula), where the p, q)-gau Binomial coefficient n by n ) = [n ] [] The p, q)-exponential function, hold the identitie e x) = ) and p, q)-factorial are defined n ) and = [2] [1] n N) p n 2) x n and E x) = q n 2) x n 14) e x)e x) = 1 and e p 1,q 1x) = E x), and have the p, q)-derivative 15) D e x) = e px) and D E x) = E qx) The definite p, q)-integral i defined by a f x) d x = p q) a in conjunction with 16) b a 0 f x) d x = b 0 f x) d x a 0 p p q f +1 q a, +1 f x) d x ee [22]) A more detailed tatement of above i found in [2, 7, 9, 10, 20, 22]

3 APOSTOL TYPE p, q)-frobenius-euler POLYNOMIALS AND NUMBERS 557 The claical Bernoulli polynomial and number, B n x) and B n, claical Euler polynomial and number, E n x) and E n, and claical Genocchi polynomial and number, G n x) and G n, are defined by the following generating function B n x) zn n = z e z 1 exz and E n x) zn n = 2 e z + 1 exz and G n x) zn n = 2z e z + 1 exz and B n n = z e z 1 E n n = 2 e z + 1 G n n = 2z e z + 1 z < 2π), z < π), z < π), for detailed information about thee number and polynomial, ee [5, 6, 25, 26] Apotol-type polynomial and number were firtly introduced by Apotol [1] and alo Srivatava [26] Then, Apotol-Bernoulli, Apotol-Euler and Apotol-Genocchi polynomial and number have been tudied and invetigated by many mathematician, for intance, Apotol, Araci, Acigoz, He, Jia, Mahmudov, Luo, Ozarlan, Srivatava, Wang, Wang and Yaar, in [1, 3, 4, 11, 15 18, 26 28] q-generalization of mentioned Apotol type polynomial and number have been conidered and dicued by Kurt, Mahmudov, Kelehteri, Sime, Srivatava in [14, 19, 23, 24] Recently, unification of Apotol type polynomial and number have been wored by El-Deouy, Gomaa, Kurt, Ozarlan in [8, 13, 21] Duran et al [7] defined Apotol type p, q)-bernoulli polynomial B n α) x, y; λ : p, q) of order α, the Apotol type p, q)-euler polynomial E n α) x, y; λ : p, q) of order α and the Apotol type p, q)-genocchi polynomial G α) n x, y; λ : p, q) of order α by the following generating function: B α) n x, y; λ : p, q) = z e xz) E yz) λe z) 1 E α) n x, y; λ : p, q) G α) n x, y; λ : p, q) = = z + log λ < 2π, 1 α = 1), 2 e xz) E yz) λe z) + 1 z + log λ < π, 1 α = 1), 2z e xz) E yz) λe z) + 1 z + log λ < π, 1 α = 1), where λ and α are uitable real or complex) parameter and p, q, C with 0 < q < p 1 Letting x = 0 and y = 0 above, we then have B n α) 0, 0; λ : p, q) := B n α) λ : p, q), E n α) 0, 0; λ : p, q) := E n α) λ : p, q) and G α) n 0, 0; λ : p, q) := G α) n λ : p, q) called, n-th Apotol type p, q)-bernoulli number of order α, n-th Apotol type p, q)-euler number of order α and n-th Apotol type p, q)-genocchi number of order α, repectively In

4 558 U DURAN AND M ACIKGOZ the cae α = 1, we have B 1) n x, y; λ : p, q) := B n x, y; λ : p, q), E 1) n x, y; λ : p, q) := E n x, y; λ : p, q) and G 1) n x, y; λ : p, q) := G n x, y; λ : p, q) named a n-th Apotol type p, q)-bernoulli polynomial, n-th Apotol type p, q)-euler polynomial and n-th Apotol type p, q)-genocchi polynomial Claical Frobeniu-Euler polynomial H n α) x; u) of order α are defined by the following Taylor erie expanion at z = 0: H n α) x; u) zn α n = e zt, e z u where α i uitable real or complex) parameter and u i an algebraic number, ee [4, 5, 12, 28] Apotol type Frobeniu-Euler polynomial H n α) x; u; λ) of order α are defined a follow: H n α) x; u; λ) zn α n = e zt, λe z u where λ and α are uitable real or complex) parameter and u i an algebraic number, ee [3, 23] The following definition i new and play an important role in deriving the main reult of thi paper Now we are ready to tate the following Definition 11 Definition 11 Apotol type p, q)-frobeniu-euler polynomial H n α) x, y; u; λ : p, q) of order α are defined by mean of the following Taylor erie expanion about z = 0: ) α e xz) E yz), H α) n x, y; u; λ : p, q) = λe z) u where λ and α are uitable real or complex) parameter, p, q, C with 0 < q < p 1 and u i an algebraic number Upon etting x = 0 and y = 0 in Definition 11, we have H n α) 0, 0; u; λ : p, q) := H n α) u; λ : p, q) called n-th Apotol type p, q)-frobeniu-euler number of order α We remar that H 1) n x, y; u; λ : p, q) :=H n x, y; u; λ : p, q), n x, y; u; λ : p, q) :=H α) p=1 n,q x, y; u; λ) ee [14] and [23]), H α) lim q 1 p=1 H n α) x, y; u; λ : p, q) :=H n α) x + y; u; λ) ee [3]) Thi paper i organized a follow The econd ection provide ome baic identitie and propertie for Apotol type p, q)-frobeniu-euler polynomial and number of order α, including addition theorem, difference equation, derivative propertie,

5 APOSTOL TYPE p, q)-frobenius-euler POLYNOMIALS AND NUMBERS 559 recurrence relation and o on The third ection not only include integral repreentation, explicit formula and relation for H n α) x, y; u; λ : p, q), but alo provide p, q)-extenion of Carlitz reult Eq 219) in [5] and Srivatava and Pintér addition theorem between Apotol type p, q)-frobeniu-euler polynomial of order α and Apotol type p, q)-bernoulli polynomial The lat ection deal with ome pecial cae for the reult obtained thi paper 2 Preliminarie and Lemma In thi ection, we provide ome baic formula and identite for Apotol type p, q)-frobeniu-euler polynomial of order α o that we derive the main outcome of thi paper in the next ection We now begin with the following addition theorem for H n α) x, y; u; λ : p, q) a Lemma 21 Lemma 21 Addition theorem) The following relationhip hold true: H n α) x, y; u; λ : p, q) = H α) u; λ : p, q) x + y) n = q n 2 ) α) H x, 0; u; λ : p, q) y n = p n 2 ) α) H 0, y; u; λ : p, q) x n Some pecial cae of Lemma 21 are invetigated in Corollary 21 and 22 Corollary 21 In the cae x = 0 or y = 0) in Lemma 21, we get the following formula H n α) x, 0; u; λ : p, q) = p n 2 ) α) H u; λ : p, q) x n, H n α) 0, y; u; λ : p, q) = q n 2 ) α) H u; λ : p, q) y n Corollary 22 In the cae x = 1 or y = 1) in Lemma 21, we have the following formula H n α) x, 1; u; λ : p, q) = q n 2 ) α) 21) H x, 0; u; λ : p, q), H n α) 1, y; u; λ : p, q) = p n 2 ) α) 22) H 0, y; u; λ : p, q) We remar that equation 21) and 22) are p, q)-generalization of the following familiar formula: H n α) x + 1; u; λ) = H α) x; u; λ)

6 560 U DURAN AND M ACIKGOZ Note that 23) H 0) n x, y; u; λ : p, q) = n Lemma 22 Difference equation) We have λh α) n λh α) n n p 2) q n 2 ) x y n = x + y) n 1, y; u; λ : p, q) uh n α) 0, y; u; λ : p, q) = ) H n α 1) 0, y; u; λ : p, q), x, 0; u; λ : p, q) uh n α) x, 1; u; λ : p, q) = ) H n α 1) x, 1; u; λ : p, q) Lemma 23 Derivative propertie) We have D ;x H n α) x, y; u; λ : p, q) = H α) n 1 px, y; u; λ : p, q), D ;y H n α) x, y; u; λ : p, q) = H α) n 1 x, qy; u; λ : p, q) Apotol type p, q)-frobeniu-euler polynomial H n x, y; u; λ : p, q) are related to the B n x, y; λ : p, q), E n x, y; λ : p, q) and G n x, y; λ : p, q) a in below Lemma 24 We have B n λ : p, q) = λ 1 H n 1 λ 1 ; 1 : p, q ), B n x, y; λ : p, q) = λ 1 H n 1 x, y; λ 1 ; 1 : p, q ), E n λ : p, q) = 2 λ + 1 H n λ 1 ; 1 : p, q ), E n x, y; λ : p, q) = 2 λ + 1 H n x, y; λ 1 ; 1 : p, q ), G n λ : p, q) = 2 λ + 1 H n λ 1 ; 1 : p, q ), G n x, y; λ : p, q) = 2 λ + 1 H n x, y; λ 1 ; 1 : p, q ) Lemma 25 For α, β N, Apotol type p, q)-frobeniu-euler polynomial atify the following relation H n α+β) x, y; u; λ : p, q) = H α) x, 0; u; λ : p, q) H β) n x, 0; u; λ : p, q), n uh n x, y; u; λ : p, q) = λ u 1) u 1) n x, y; u; λ : p, q) = H α β) n p n 2 ) H x, y; u; λ : p, q) + x + y) n, H α) x, 0; u; λ : p, q) H β) n 0, y; u; λ : p, q)

7 APOSTOL TYPE p, q)-frobenius-euler POLYNOMIALS AND NUMBERS 561 Lemma 26 Recurrence relationhip) H n α) x, y; u; λ : p, q) hold the following equality: ) ) n λ p 2 u =) m H α) x, 0; u; λ : p, q) p n 2 ) m H α) p n x, 1; u; λ : p, q) 2 ) m H α 1) x, 1; u; λ : p, q) 3 Main Reult Thi ection include integral repreentation, ome identitie and explicit formula for H n α) x, y; u; λ : p, q) Alo, we preent new theorem and ome p, q)-extenion of nown reult in Carlitz [5], Kurt [14], Sime [23], Srivatava and Pintér [25] and o on We tart with the following explicit formula for Apotol type p, q)-frobeniu-euler polynomial of order α by the following theorem Theorem 31 Apotol type p, q)-frobeniu-euler polynomial of order α hold the following relation: H n α) x, y; u; λ : p, q) = 1 [λh 1, y; u; λ : p, q) Proof Indeed, H α) n x, y; u; λ : p, q) = λe z) u = q n 2) y n zn = uh 0, y; u; λ : p, q)]h α) n x, 0; u; λ : p, q) It remain to ue Lemma 22 and Eq 23) e xz) E yz) ) q 2) y H α) z n x, 0; u; λ : p, q) H α) ) n x, 0; u; λ : p, q) n The p, q)-integral repreentation of H n α) x, y; u; λ : p, q) are given by the following theorem Theorem 32 We have 31) b b H n α) x, y; u; λ : p, q) d x = p Hα) n+1, y; u; λ : p, p q) H α) a n+1, y; u; λ : p, q) p, a [n + 1]

8 562 U DURAN AND M ACIKGOZ 32) b a H n α) x, y; u; λ : p, q) d y = p Hα) n+1 x, b ; u; λ : p, q q) H α) n+1 x, a; u; λ : p, q) q [n + 1] Proof Since b a D f x) d x = f b) f a) ee [22]) in term of Lemma 23 and equation 15) and 16), we arrive at the aerted reult b H n α) p b x, y; u; λ : p, q) d x = D H α) x n+1, y; u; λ : p, q d x a [n + 1] a p b, y; u; λ : p, p q) H α) n+1 =p Hα) n+1 a, y; u; λ : p, q) p [n + 1] The other can be hown uing imilar method Therefore, we complete the proof of thi theorem The integral identitie 31) and 32) are p, q)-generalization of the formula b H n α) x; u; λ) dx = Hα) n+1 b; u; λ) H α) n+1 a; u; λ) a n + 1 Here i a recurrence relation of Apotol type p, q)-frobeniu-euler polynomial by the following theorem Theorem 33 We have H n+1 x, y; u; λ : p, q) =yp n H n q p x, q p y; u; λ : p, q ) + xp n H n x, y; u; λ : p, q) λ n n q p n H x, y; u; λ : p, q) H n 1, 0; u; λ : p, q) Proof For α = 1, applying the p, q)-derivative to H n x, y; u; λ : p, q) with repect to z yield to { } D ;z H n x, y; u; λ : p, q) = ) D e xz) E yz) ;z, λe z) u uing equation 12) and 13), it become ) λe qz) u) D ;z [e xz) E yz)] λe pz) u) λe qz) u) + u 1) e pxz) E pyz) D ;z λe z) u) λe pz) u) λe qz) u) q =y H n p x, q ) p y; u; λ : p, q p n zn + x H n x, y; u; λ : p, q) p n zn

9 APOSTOL TYPE p, q)-frobenius-euler POLYNOMIALS AND NUMBERS 563 λ H n x, y; u; λ : p, q) q n zn H n 1, 0; u; λ : p, q) p n zn By maing ue of Cauchy product, comparing the coefficient of, then we have deired reult We now preent the ymmetric identity for H n x, y; u; λ : p, q) a given below Theorem 34 We have 33) 2u 1) H u; λ : p, q)h n x, y; ; λ : p, q) =uh n x, y; u; λ : p, q) )H n x, y; ; λ : p, q) Proof If we conider the identity 2u 1 λe z) u) λe z) )) = 1 λe z) u 1 λe z) ), then 2u 1) ) e xz) 1 )) E yz) λe z) u) λe z) )) =u ) e xz) E yz) λe z) u Therefore, we deduce 2u 1) H n 0, 0; u; λ : p, q) =u H n x, y; u; λ : p, q) Checing againt the coefficient of ) e xz) 1 )) E yz) λe z) ) ) H n x, y; ; λ : p, q) H n x, y; ; λ : p, q), then we have deired reult in 33) The relation 33) i a p, q)-generalization of Carlitz reult Eq 219) in [5] Theorem 35 The following relation i valid for Apotol type p, q)-frobeniu-euler polynomial: uh n x, y; u; λ : p, q) = λ p n 2 ) H x, y; u; λ : p, q) ) x + y) n Proof From the relation 14) and the identity u λ λe z) u) e z) = 1 λe z) u 1 λe z), we write u ) e xz) E yz) λe z) u) λe z)

10 564 U DURAN AND M ACIKGOZ which give = ) e xz) E yz) λe z) u ) e xz) E yz), λe z) u H n x, y; u; λ : p, q) λ = H n x, y; u; λ : p, q) p n 2) λ Equating the coefficient of, we derive aerted reult x + y) n We are in a poition to provide ome relationhip for Apotol type p, q)-frobeniu- Euler polynomial of order α related to Apotol type p, q)-bernoulli polynomial, Apotol type p, q)-euler polynomial and Apotol type p, q)-genocchi polynomial in Theorem 36, 37 and 38 Theorem 36 The following recurrence relation are valid: n+1 H n α) n + 1 x, y; u; λ : p, q) = H α) n x, y; u; λ : p, q) = =0 λ ) n+1 =0 n + 1 λ ) H α) n+1 0, y; u; λ : p, q) [n + 1] B x, 0; λ : p, q) p 2) B x, 0; λ : p, q), H α) n+1 x, 0; u; λ : p, q) [n + 1] B 0, y; λ : p, q) p 2) B 0, y; λ : p, q) Proof Indeed, e xz)e yz) λe z) u z λe z) 1 = E yz) e xz) λe z) u λe z) 1 z = 1 λ B n x, 0; λ : p, q)p 2) z [n] B n x, 0; λ : p, q) H α) n 0, y; u; λ : p, q) = λ B x, 0; λ : p, q)p 2) n n B x, 0; λ : p, q) =0 =0

11 APOSTOL TYPE p, q)-frobenius-euler POLYNOMIALS AND NUMBERS 565 H α) n 0, y; u; λ : p, q) zn 1 By uing Cauchy product and comparing the coefficient of, the proof i completed Theorem 37 We have H n α) x, y; u; λ : p, q) = =0 λ H n α) x, y; u; λ : p, q) = =0 λ H α) n 0, y; u; λ : p, q) 2 E x, 0; λ : p, q) p H α) n x, 0; u; λ : p, q) 2 E 0, y; λ : p, q) p 2 ) + E x, 0; λ : p, q), 2 ) + E 0, y; λ : p, q) Proof The proof i baed on the following equalitie e xz) E yz) = e xz) λe z) u λe z) u 2 λe z) + 1 E yz), λe z) e xz) E yz) = E yz) λe z) u λe z) u and i imilar to that of Theorem 36 =0 2 λe z) + 1 λe z) + 1 e xz) 2 Theorem 38 Each of the following recurrence relation hold true: n+1 H n α) n + 1 H α) n+1 0, y; u; λ : p, q) x, y; u; λ : p, q) = 2 [n + 1] H α) n x, y; u; λ : p, q) = n+1 =0 λ n + 1 λ ) G x, 0; λ : p, q) p 2) + G x, 0; λ : p, q), H α) n+1 x, 0; u; λ : p, q) 2 [n + 1] G 0, y; λ : p, q) p 2) + G 0, y; λ : p, q)

12 566 U DURAN AND M ACIKGOZ Proof By maing ue of the following equalitie e xz) E yz) = e xz) λe z) u λe z) u 2z λe z) + 1 E yz), λe z) + 1 2z e xz) E yz) = E yz) λe z) u λe z) u 2z λe z) + 1 λe z) + 1 e xz), 2z the proof of thee theorem i completed imilar to that of Theorem 36 4 Concluion In the preent paper, we have introduced p, q)-extenion of Apotol type Frobeniu- Euler polynomial and number and invetigated ome baic identitie and propertie for thee polynomial and number, including addition theorem, difference equation, derivative propertie, recurrence relation and o on Thereafter, we have given integral repreentation, explicit formula and relation for mentioned newly defined polynomial and number Henceforth, we have obtained p, q)-extenion of Carlitz reult [5] and Srivatava and Pintér addition theorem in [25] The reult obtained here reduce to nown propertie of q-polynomial mentioned in thi paper when p = 1 Alo, in the event of q p = 1, our reult reduce to ordinary reult for Apotol type Frobeniu-Euler polynomial and number Acnowledgement The firt author i upported by TUBITAK BIDEB 2211-A Scholarhip Programme Reference [1] T M Apotol, On the Lerch zeta function, Pacific J Math ), [2] S Araci, U Duran and M Acigoz, On ρ, q)-volenborn integration, J Number Theory ), [3] A Bayad and T Kim, Identitie for Apotol-type Frobeniu-Euler polynomial reulting from the tudy of a nonlinear operator, Ru J Math Phy 232) 2016), [4] S Araci and M Acigoz, A note on the Frobeniu-Euler number and polynomial aociated with Berntein polynomial, Adv Stud Contemp Math 223) 2012), [5] L Carlitz, Eulerian number and polynomial, Math Mag ), [6] G S Cheon, A note on the Bernoulli and Euler polynomial, Appl Math Lett ), [7] U Duran and M Acigoz, Apotol type p, q)-bernoulli, p, q)-euler and p, q)-genocchi polynomial and number, Comput Math Appl 81) 2017), 7 30 [8] B S El-Deouy and R S Gomaa, A new unified family of generalized Apotol Euler, Bernoulli and Genocchi polynomial, Comput Appl Math ),

13 APOSTOL TYPE p, q)-frobenius-euler POLYNOMIALS AND NUMBERS 567 [9] V Gupta, p, q)-baaov-kantorovich operator, Appl Math Inf Sci 104) 2016), [10] V Gupta and A Aral, Berntein Durrmeyer operator and baed on two parameter, Facta Univeritati Niš), Ser Math Inform 311) 2016), [11] Y He, S Araci, H M Srivatava and M Acigoz, Some new identitie for the Apotol Bernoulli polynomial and the Apotol Genocchi polynomial, Appl Math Comput ), [12] D S Kim, T Kim, S-H Lee and J-J Seo, A note on q-frobeniu-euler number and polynomial, Adv Stud Theory Phy 718) 2013), [13] V Kurt, Some ymmetry identitie for the unified Apotol-type polynomial and multiple power um, Filomat, 303) 2016), [14] B Kurt, A note on the Apotol type q-frobeniu-euler polynomial and generalization of the Srivatava-Pinter addition theorem, Filomat, 301) 2016), [15] Q-M Luo, On the Apotol-Bernoulli polynomial, Cent Eur J Math 24) 2004), [16] Q-M Luo, Some reult for the q-bernoulli and q-euler polynomial, J Math Anal Appl ), 7 10 [17] Q-M Luo and H M Srivatava, q-extenion of ome relationhip between the Bernoulli and Euler polynomial, Taiwanee J Math 151) 2011), [18] N I Mahmudov, On a cla of q-benoulli and q-euler polynomial, Adv Differential Equation ) 2013), 11 page [19] N I Mahmudov and M E Kelehteri, q-extenion for the Apotol type polynomial, J Appl Math 2014), ID , 7 page [20] G V Milovanovic, V Gupta and N Mali, p, q)-beta function and application in approximation, Bol Soc Mat Mex 2016), 1 19 [21] M A Ozarlan, Unified Apotol-Bernoulli, Euler and Genochi polynomial, Comput Math Appl ), [22] P N Sadjang, On the fundamental theorem of p, q)-calculu and ome p, q)-taylor formula, Reulta Math to appear) [23] Y Sime, Generating function for q-apotol type Frobeniu-Euler number and polynomial, Axiom ), [24] H M Srivatava, Some generalization and baic or q-)extenion of the Bernoulli, Euler and Genocchi polynomial, Appl Math Inf Sci ), [25] H M Srivatava and A Pinter, Remar on ome relationhip between the Bernoulli and Euler polynomail, Appl Math Lett ), [26] H M Srivatava, Some formula for the Bernoulli and Euler polynomial at rational argument, Math Proc Cambridge Philo Soc ), [27] W Wang, C Jia and T Wang, Some reult on the Apotol-Bernoulli and Apotol-Euler polynomial, Comput Math Appl ), [28] B Y Yaar and M A Özarlan, Frobeniu-Euler and Frobeniu-Genocchi polynomial and their differential equation, New Trend Math Sci 32) 2015), Department of the Baic Concept of Engineering, Faculty of Engineering and Natural Science, Ienderun Technical Univerity, TR-31200, Hatay, Turey addre: duranugur@yahoocom 2 Department of Mathematic, Faculty of Science, Univerity of Gaziantep, Gaziantep, Turey addre: acigoz@gantepedutr

A Note on Carlitz s Twisted (h,q)-euler Polynomials under Symmetric Group of Degree Five

A Note on Carlitz s Twisted (h,q)-euler Polynomials under Symmetric Group of Degree Five Gen. Math. Note, Vol. 33, No. 1, March 2016, pp.9-16 ISSN 2219-7184; Copyright c ICSRS Publication, 2016 www.i-cr.org Available free online at http://www.geman.in A Note on Carlitz Twited (h,)-euler Polynomial