B n (x) zn n! n=0. E n (x) zn n! n=0

UDC 517.9 Q.-M. Luo Chongqing Normal Univ., China) q-apostol EULER POLYNOMIALS AND q-alternating SUMS* q-полiноми АПОСТОЛА ЕЙЛЕРА ТА q-знакозмiннi СУМИ We establish the basic properties generating functions of the q-apostol Euler polynomials. We define q-alternating sums obtain q-extensions of some formulas in [Integral Transform. Spec. Funct. 009. 0. P. 377 391]. We also deduce an explicit relationship between the q-apostol Euler polynomials the q-hurwitz Lerch zeta-function. Встановлено основнi властивостi та твiрнi функцiї q-полiномiв Апостола Ейлера. Визначено q-знакозмiннi суми та отримано q-продовження деяких формул з [Integral Transform. Spec. Funct. 009. 0. P. 377 391]. Виведено також явне спiввiдношення мiж q-полiномами Апостола Ейлера i q-дзета-функцiєю Хурвiца Лерча. 1. Introduction definitions. Throughout this paper, we always use the following notation: N = {1,, 3,...} denotes the set of natural numbers, N 0 = {0, 1,, 3,...} denotes the set of nonnegative integers, Z 0 = {0, 1,, 3,...} denotes the set of nonpositive integers, Z denotes the set of integers, R denotes the set of real numbers, C denotes the set of complex numbers. The q-shifted factorial are defined by a; q) 0 = 1, a; q) = 1 a)1 aq)... 1 aq 1 ), = 1,,..., a; q) = 1 a)1 aq)... 1 aq )... = 1 aq ). The q-numbers are defined by [a] q = 1 qa 1 q, q 1. The above q-stard notation can be found in Gasper [1, p. 7]. The classical Bernoulli polynomials Euler polynomials are defined by means of the following generating functions see, e.g., [1, p. 804 806], [11] or [5, p. 5 3]): ze xz e z 1 = e xz e z + 1 = B n x) zn, z < π, 1.1) E n x) zn, z < π, 1.) respectively. Obviously, B n := B n 0) E n := E n 0) are the Bernoulli numbers Euler numbers respectively. Some interesting analogues of the classical Bernoulli polynomials were first investigated by Apostol. We begin by recalling here Apostol s definition as follows: * This research was supported by Natural Science Foundation Project of Chongqing, China under Grant CSTC011JJA0004, Research Project of Science Technology of Chongqing Education Commission, China under Grant KJ1065, Fund of Chongqing Normal University, China under Grant 10XLR017 011XLZ07 the PhD Program Scholarship Fund of ECNU 009 of China under Grant 009041. c Q.-M. LUO, 013 1104

q-apostol EULER POLYNOMIALS AND q-alternating SUMS 1105 Definition 1.1 []. The Apostol Bernoulli polynomials B n x; λ) are defined by means of the generating function with, of course, ze xz λe z 1 = B n x; λ) zn z < π when λ = 1; z < log λ when λ 1) B n x) = B n x; 1) B n λ) := B n 0; λ), where B n λ) denotes the so-called Apostol Bernoulli numbers in fact, it is a function in λ). Recently, Luo further extended the Euler polynomials based on the Apostol s idea [] as follows: Definition 1. cf. [18]). The Apostol Euler polynomials E n x; λ) are defined by means of the generating function with, of course, e xz λe z + 1 = 1.3) E n x; λ) zn, z < log λ), 1.4) E n x) = E n x; 1) E n λ) := E n 0; λ), 1.5) where E n λ) denote the so-called Apostol Euler numbers. Recently, M. Cenci M. Can [6] further defined the following q-extensions of the Apostol Bernoulli polynomials, i.e., the so-called q-apostol Bernoulli polynomials. Definition 1.3. The q-apostol Bernoulli numbers B n;q λ) polynomials B n x; λ) are defined by means of the generating functions U λ;q t) = t λ n q n e [n]qt = B n;q λ) tn 1.6) U x;λ;q t) = t λ n q n+x e [n+x]qt = B n;q x; λ) tn respectively. Setting λ = 1 in 1.6) 1.7), we obtain the corresponding Carlitz s definitions for the q- Bernoulli numbers B n;q q-bernoulli polynomials B n;q x) respectively. Obviously, lim B n;qx) = B n x), lim B n;qx; λ) = B n x; λ), lim B n;q = B n lim B n;q λ) = B n λ). It follows that we define the following q-extensions of the Apostol Euler numbers polynomials see [3 6, 9]). 1.7)

1106 Q.-M. LUO Definition 1.4. The q-apostol Euler numbers E n;q λ) polynomials E n x; λ) are defined by means of the generating functions V λ;q t) = λ) n q n e [n]qt = E n;q λ) tn 1.8) V x;λ;q t) = λ) n q n+x e [n+x]qt = E n;q x; λ) tn 1.9) respectively. When λ = 1, then the above definitions 1.8) 1.9) will become the corresponding definitions of the q-euler numbers E n;q q-euler polynomials E n;q x). Clearly, lim E n;qx) = E n x), lim E n;q = E n lim E n;qx; λ) = E n x; λ), lim E n;q λ) = E n λ). There are numerous recent investigations on this subject by, among many other authors, Ceni et al. [6 8], Choi et al. [9, 10], Kim [14 16], Luo Srivastava [17 4], Ozden [6] Simse [7 9]. The aim of the present paper is to investigate the basic properties, generating functions, Raabe s multiplication theorem alternating sums for the q-apostol Euler polynomials to obtain some q-extensions of some formulas in [Integral Transform. Spec. Funct. 009. 0. P. 377 391]. We also derive some interesting formulas relationships between the q-apostol Euler polynomials, the q-apostol Euler polynomials q-hurwitz Lerch zeta function.. The properties of the q-apostol Euler polynomials. The following elementary properties of the q-apostol Euler polynomials E n;q x; λ) are readily derived from 1.8) 1.9). We, therefore, choose to omit the details involved.

q-apostol EULER POLYNOMIALS AND q-alternating SUMS 1107 Proposition.1 the several values). E 0;q x; λ) = λq + 1 qx, E 1;q x; λ) = λ λq + 1 qx [x] q λq + 1)λq + 1) qx+1, E 0;q λ) = λq + 1, λq E 1;q λ) = λq + 1)λq + 1),.1) Proposition.. E ;q λ) = λqλq 1) λq + 1)λq + 1)λq 3 + 1), λqλ q 5 4λq + 1) E 3;q λ) = λq + 1)λq + 1)λq 3 + 1)λq 4 + 1). Proposition.3 difference equation). Proposition.4 differential relation). An expansion formula of q-apostol Euler polynomials n E n;q x; λ) = E ;q λ)q +1)x [x] n q..) λe n;q x + 1; λ) + E n;q x; λ) = q x [x] n q..3) E n;q x; λ) = E n;q x; λ) log q + n log q x q 1 qx E n 1;q x; λq)..4) Proposition.5 integral formula). b a q x E n;q x; λq) dx = 1 q b E n+1;q x; λ) dx + q 1 log q a Proposition.6 addition theorem). E n;q x + y; λ) = n Proposition.7 theorem of complement). E n+1;q b; λ) E n+1;q a; λ)..5) E ;q x; λ)q +1)y [y] n q..6) E n;q 1 x; λ) = 1)n λq n E n;q 1x; λ 1 ),.7) E n;q 1 + x; λ) = 1)n λq n E n;q 1 x; λ 1 )..8)

1108 Q.-M. LUO Remar.1. If q 1, then the formulas.1).8) become the corresponding formulas for the Apostol Euler polynomials see [18, p. 918, 919], Eqs. 3) 11) when α = 1). So the above formulas are q-extensions of the corresponding formulas of the Apostol Euler polynomials respectively. Remar.. When λ = 1, then the formulas.1).8) become the corresponding formulas for the q-euler polynomials see [3 5]). 3. The generating functions of q-apostol Euler polynomials. By 1.8) 1.9) yields that V x;λ;q t) = λ) n q n+x e [n+x]qt = = e t = e t = e t λ) n q n+x e qn+x 1) q +1)x t 1 q)! 1) q +1)x 1 + λq +1 t = λq +1 ) n = ) 1 t 1 q!. 3.1) Therefore, we obtain the generating function of E n;q x; λ) as follows: V x;λ;q t) = e t 1) q +1)x ) 1 t 1 + λq +1 1 q! = E n;q x; λ) tn. 3.) Clearly, setting x = 0 in 3.) we have the generating function of E n;q λ): V λ;q t) = e t 1) ) 1 t 1 + λq +1 1 q! = E n;q λ) tn. 3.3) Putting λ = 1 in 3.) 3.3), we deduce the generating function of E n;q x) E n;q V x;q t) = e t 1) q +1)x ) 1 t 1 + q +1 1 q! = E n;q x) tn V q t) = e t 1) 1 ) t 1 + q +1 1 q! = E n;q t n respectively. It is not difficult, from 3.) 3.3) we get the following closed formulas: n 1) q +1)x E n;q x; λ) = 1 q) n 1 + λq +1 3.6) E n;q λ) = 1 q) n n 3.4) 3.5) 1). 1 + λq+1 3.7)

q-apostol EULER POLYNOMIALS AND q-alternating SUMS 1109 Remar 3.1. In the same way, we can also obtain the generating function of q-apostol Bernoulli polynomials as follows: U x;λ;q t) = te t 1) q +1)x ) 1 t 1 λq +1 1 q! = B n;q x; λ) tn. 3.8) 4. q-raabe s multiplication theorem, q-alternating sums their applications. Theorem 4.1 q-apostol-raabe s multiplication theorem). For m, n N, λ C, then we have [m] n m 1 q λ) j E n;q m x + jm ) ; λm, m is odd, E n;q mx; λ) = 4.1) m 1 [m]n q λ) j B n+1;q m x + jm ) ; λm, m is even. Proof. If m is odd, we compute the following sum by 3.): m 1 [m] n q λ) j E n;q m x + jm ) ; λm tn = = e t = e t 1) q +1)mx 1 + λq +1 ) m = 1 ) t 1 q! 1) q +1)mx 1 + λq +1 m 1 1 ) t 1 q! = λq +1) j = E n;q mx; λ) tn. 4.) Comparing the coefficients of tn on the both sides of 4.), we obtain the first formula of Theorem 4.1. If m is even, we calculate the following sum by 3.8) 3.): m 1 [m]n q λ) j B n+1;q m x + jm ) ; λm tn = = e t = e t 1) q +1)mx 1 λq +1 ) m = 1 ) t 1 q! 1) q +1)mx 1 + λq +1 m 1 1 ) t 1 q! = λq +1) j = E n;q mx; λ) tn. 4.3)

1110 Q.-M. LUO Comparing the coefficients of tn on the both sides of 4.3), we obtain the second formula of Theorem 4.1. Theorem 4.1 is proved. Clearly, the above formulas 4.1) of Theorem 4.1 are a q-extensions of the multiplication formulas in [0, p. 386] Eq. 43)). Taing λ = 1 in 4.1), we obtain the following corollary. Corollary 4.1 q-raabe s multiplication theorem). For m, n N, then we have [m] n m 1 q 1) j E n;q m x + j ), m is odd, m E n;q mx) = 4.4) m 1 [m]n q 1) j B n+1;q m x + j ), m is even. m Obviously, the above formulas 4.4) are a q-extensions of the classical Raabe s multiplication theorem of Euler polynomials in [0, p. 386] Eq. 45)). We now define the following alternating sums: m Z ;q m; n; λ) = 1) j+1 λ j q jn ) [j] q = Z ;q m; n) = j=1 = λq n [1] q λ q n ) [] q +... + 1) m+1 λ m q mn ) [m] q, 4.5) m 1) j+1 q jn ) [j] q = q n [1] q q n ) [] q +... + 1) m+1 q mn ) [m] q, j=1 Z m; λ) = 4.6) m 1) j+1 λ j j = λ1 λ +... + 1) m+1 λ m m, 4.7) j=1 Z m) = m 1) j+1 j = 1 +... + 1) m+1 m, 4.8) j=1 m, n, N; n ; λ C, which are called the q-λ-alternating sums, q-alternating sums, λ-alternating sums alternating sums respectively. It is easy to obtain the following generating functions of Z m; λ) Z m) respectively: Z m; λ) x m! = 1) j+1 λ j e jx = λ)m+1 e m+1)x + λe x λe x, 4.9) + 1 j=1 Z m) x m! = 1) j+1 e jx = 1)m+1 e m+1)x + e x e x. 4.10) + 1 j=1

q-apostol EULER POLYNOMIALS AND q-alternating SUMS 1111 Theorem 4.. numbers Let m be odd. For m, n N; λ C, the recursive formula of q-apostol Euler [m] n q E n;q mλ m ) E n;q λ) = n holds true in terms of the q-λ-alternating sums defined by 4.5). Proof. If m is odd, taing x = 0 in 4.1) we obtain = n = [m] qe ;q mλ m )Z n ;q m 1; ; λ) 4.11) m 1 E n;q λ) = [m] n q λ) j E n;q m n ) j m ; λm = ) m 1 n [m] qe ;q mλ m ) λ) j q +1)j [j] q n = [m] qe ;q mλ m )Z n ;q m 1; ; λ) + [m] n q E n;q mλ m ). 4.1) The formula 4.11) follows. Theorem 4. is proved. Clearly, the above formula 4.11) is an q-extension of the formula in [0, p. 389] Eq. 60)). Putting λ = 1 in 4.11), we obtain the following corollary. Corollary 4.. Let m be odd. For m, n N, the recursive formula of Apostol Euler numbers holds [m] n q E n;q m E n;q = n [m] qe ;q mz n ;q m 1; ) m is odd). 4.13) Clearly, the above formula 4.13) is a q-extension of the formula in [0, p. 389] Eq. 61)). Theorem 4.3. Let m be even. For m, n N, λ C, the formula of q-apostol Euler numbers = [m] q E n;q λ) + [m]n+1 q B n+1;q mλ m ) = n+1 holds true in terms of the q-λ-alternating sums defined by 4.5). Proof. If m is even, setting x = 0 in 4.1) we have = ) [m] qb ;q mλ m )Z n+1 ;q m 1; ; λ), 4.14) E n;q λ) = m 1 [m]n q λ) j B n+1;q m n+1 ) [m] 1 q B ;q mλ m ) m 1 ) j m ; λm = 1) j+1 λ j q j [j] n+1 q =

111 Q.-M. LUO = n+1 ) [m] 1 q B ;q mλ m )Z n +1;q m 1; ; λ) [m]n q B n+1;q mλ m ). 4.15) The formula 4.3) follows. Theorem 4.3 is proved. Clearly, the above formula 4.3) is a q-extension of the formula see [0, p. 390], Eq. 63) for l = 1): me n λ)+ mn+1 B n+1 λ m ) = n+1 ) m B λ m )Z n+1 m 1; λ) m is even), 4.16) where λ-alternating sums defined by 4.7). Putting λ = 1 in 4.3), we have the following corollary. Corollary 4.3. For m be even, m, n N; λ C, the formula of q-euler numbers [m] q E n;q + [m]n+1 q B n+1;q m = n+1 ) [m] qb ;q mz n+1 ;q m 1; n +1), 4.17) holds true in terms of the q-alternating sums defined by 4.6). Clearly, the above formula 4.17) is a q-extension of the formula in [0, p. 390] Eq. 64) for l = 1). Remar 4.1. Setting λ = 1 in 4.16) noting that Z 0 m 1) = 1 for m even, we derive the following interesting sum formula: n ) m B Z n+1 m 1) = m) E n n N; m is even). 4.18) Applying the relation E n = 1 n+1 )B n+1 to 4.18), we find that n ) m B Z n+1 m 1) = m1 n+1 )B n+1 n N; m is even), 4.19) which is just the formula of Howard see [13, p. 167], Eq. 33)). Remar 4.. Separatting the odd even terms in 4.19), noting that B 0 = 1, B 1 = 1, B n+1 = 0, n N, we have n 1 m B Z n m 1) = m1 n )B n + mnz n 1 m 1) Z n m 1) 4.0) =1 n =1 ) m B Z n +1 m 1) = n N; m is even) m) Z n m 1) Z n+1 m 1) 4.1) n N; m is even), where the alternating sums Z m) defined by 4.8).

q-apostol EULER POLYNOMIALS AND q-alternating SUMS 1113 Below we give the evaluations for the alternating sums 4.5), 4.6) given by 4.) 4.4) respectively. Theorem 4.4. For m, n N, λ C, the following formula of q-λ-alternating sums: Z n;q m; ; λ) = m holds true in terms of the q-apostol Euler polynomials. Proof. It is easy to observe that 1) j+1 λ j q j [j] n q = λ)m+1 E n;q m + 1; λ) E n;q λ) λ) m+1 λ) j q m+j+1 e [m+j+1]qt + λ) j q j e [j]qt =, 4.) m 1) j+1 λ j q j e [j]qt. 4.3) By 1.8), 1.9) 4.3), via simple computation, we arrive at the desire 4.) immediately. Theorem 4.4 is proved. Clearly, the above formula 4.) is a q-extension of the formula in [0, p. 388] Eq. 55)). Setting λ = 1 in 4.), then we have Z n;q m; ) = m 1) j+1 q j [j] n q = 1)m E n;q m + 1) + E n;q, 4.4) which is a q-extension of the well-nown formula in [0, p. 388] Eq. 56)) [1, p. 804], 3.1.4). 5. Some relationships between the q-apostol Euler polynomials q-hurwitz Lerch zeta-function. The Hurwitz Lerch zeta-function Φz, s, a) defined by cf., e.g., [9, p. 11]). Φz, s, a) := z n n + a) s a C \ Z 0 ; s C when z < 1; Rs) > 1 when z = 1 ) contains, as its special cases, not only the Riemann Hurwitz or generalized) zeta-functions ζs) := Φ1, s, 1) = ζ s, 1) = 1 s 1 ζ s, 1 ), the Lerch zeta-function: l s ξ) := ζs, a) := Φ1, s, a) = n=1 e nπiξ n s 1 n + a) s, R s) > 1, a / Z 0, ) = e πiξ Φ e πiξ, s, 1, ξ R, Rs) > 1, but also such other functions as the polylogarithmic function: Li s z) := n=1 z n = z Φz, s, 1) ns

1114 Q.-M. LUO s C when z < 1; Rs) > 1 when z = 1 ) the Lipschitz Lerch zeta-function cf. [9, p. 1], Eq..5 11)): e nπiξ ) φξ, a, s) := n + a) s = Φ e πiξ, s, a =: L ξ, s, a) a C \ Z 0 ; Rs) > 0 when ξ R \ Z; Rs) > 1 when ξ Z ). We define the q-hurwitz Lerch zeta-functions as follows: Definition 5.1. For Ra) > 0, q-hurwitz Lerch zeta-function is defined by Theorem 5.1. Φ q z, s, a) := The following relationship: z n q [ n+a ] s, R a) > 0, a / Z n + a 0. q E n;q a; λ) = Φ q λ, n, a), n N, λ 1, a C \ Z 0, 5.1) holds true between the q-apostol Euler polynomials the q-hurwitz Lerch zeta-function. Proof. We differentiate the both sides of 1.9) with respect to the variable t yields that E n;q a; λ) = dn dt n V { a;λ;qt) = λ) +a dn q t=0 dt n e [+a]qt} t=0 = = λ) q +a [ + a] q = Theorem 5.1 is proved. Letting q 1 in 5.1), we have the following corollary. Corollary 5.1. The following relationship: λ) q +a [ ] n. + a q E n a; λ) = Φ λ, n, a), n N, λ 1, a C \ Z 0, holds true between the Apostol Euler polynomials Hurwitz Lerch zeta-function. On the other h, we define an analogue of the Hurwitz zeta-function as follows: Definition 5.. For Rs) > 1, a / Z 0, L-function is defined by 1) n Ls, a) := n + a) s. ) ) 1 1 Clearly, Ls, a) = φ, a, s = Φe πi, s, a) = L, s, a. Next we define an q-extension of the L-function. Definition 5.3. For Rs) > 1, a / Z 0, the q-l-function is defined by 1) n q n+a L q s, a) := [n + a] s. q

q-apostol EULER POLYNOMIALS AND q-alternating SUMS 1115 In the same way, we can obtain the following relationships. Theorem 5.. The following relationship: E n,q a) = L q n, a), n N, a C \ Z 0, holds true between the q-euler polynomials q-l-function. Corollary 5.. The following relationship: E n a) = L n, a), n N, a C \ Z 0, holds true between the Euler polynomials the L-function. We define an analogue of the Riemann zeta-function: Definition 5.4. The l-function is defined by ls) := 1) n n=1 n s, R s) > 1. ) 1 Obviously, ls) = l s = Li s 1). It follows that we define an q-extension of the l-function: Definition 5.5. The q-l-function is defined by l q s) := 1) n q n [n] s, R s) > 1. q n=1 Similarly, we can obtain the following explicit relationship: Theorem 5.3. The following relationship: E n,q = l q n), n N, holds true between the q-euler numbers q-l-function. Corollary 5.3. The following relationship: E n = l n), n N, holds true between the Euler numbers l-function. 6. Some explicit relationships between the q-apostol Bernoulli q-apostol Euler polynomials. In this section, we will investigate some relationships between the q-apostol Bernoulli q-apostol Euler polynomials. We also obtain an q-extension of Howard s result. It is easy to observe that λ n q n+x e [n+x]qt λ n q n+x e [n+x]qt = By 1.7) 1.9), via the simple computation, we obtain E n;q x; λ) = λ) n q n+x e [n+x]qt. [ B n+1;q x; λ) [] n q B n+1;q x ; λ)], 6.1) which is just an q-extension of the formula of Luo Srivastava see [19, p. 636], Eq. 38))

1116 Q.-M. LUO Setting x = 0 in 6.1), we get Putting λ = 1 in 6.1), we have E n x; λ) = [ x B n+1 x; λ) n+1 B n+1 ; λ)]. E n;q λ) = [ Bn+1;q λ) [] n q B n+1;q λ )]. 6.) E n;q x) = which is an q-extension of the well-nown formula see [1]) Remar 6.1. E n x) = If taing x = 0 in 6.3), we obtain [ x )] B n+1;q x) [] n q B n+1;q, 6.3) [ B n+1 x) n+1 B n+1 x E n;q = Bn+1;q [] n q B n+1;q ) )]. is an q-extension of the formula see [1, p. 805] Entry 3.1.0)) [5, p. 9]) E n = 1 n+1 )B n+1. Remar 6.. By 4.3) 6.), we easily obtain the following explicit recursive formula for the q-apostol Bernoulli numbers: [m] q Bn;q λ) [] n 1 q B n;q λ ) ) n = [m] qb ;q mλ m )Z n ;q m 1; n; λ) [m] n q B n;q mλ m ). 6.4) Remar 6.3. Setting λ = 1 in 6.4), we have [m] q Bn;q [] n 1 q B n;q ) = n [m] qb ;q mz n ;q m 1; n) [m] n q B n;q m is an q-extension of Howard s formula [13, p. 167] Eq. 33)) n 1 m1 n )B n = B m Z n m 1) m is even). m is even) Remar 6.4. Letting q 1 in 6.4), we obtain a new formula for the Apostol Bernoulli numbers as follows: m B n λ) n B n λ ) ) n = m B λ m )Z n m 1; λ) m n B n λ m ) m is even). 6.5) Obviously, by setting λ = 1 in 6.5) we get a new recurrence formula for the Bernoulli numbers: n m 1 Z n m 1) B n = 1 n + m n 1 B m is even).

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