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The Euler polynomials of order k are defined by the exponential generating function as follows: ∞ k n 2 (k) t (2) ext = etE (x) = E(k)(x) (k ∈ Z = N ∪{0}), et +1 n n! + n=0   X (k) n (k) with the usual convention about replacing (E (x)) by En (x). In the special case, (k) (k) x =0, En (0) := En are called Apostol-Euler numbers of order k (see [13], [15]). In the complex plane, Apostol-Euler polynomials En (x | λ) and Apostol-Bernoulli polynomials Bn (x | λ) are given by [15] ∞ 2 tn (3) ext = E (x | λ) , (|t| < log (−λ)) λet +1 n n! n=0 X∞ t tn (4) ext = B (x | λ) , (|t| < log λ). λet − 1 n n! n=0 X (k) In [15], Apostol-Euler polynomials of higher order En (x | λ) and Apostol-Bernoulli (k) polynomials of higher order Bn (x | λ) are given by the following generating functions: ∞ 2 k tn (5) ext = E(k) (x | λ) , (|t| < log (−λ)) λet +1 n n!   n=0 X∞ tk tn (6) ext = B(k) (x | λ) , (|t| < log λ). t k n n! (λe − 1) n=0 X In the above expressions, we take the principal value of the logarithm log λ, i.e., log λ = log |λ| + i arg λ (−π< arg λ ≤ π) when λ 6= 1; set log1 = 0 when λ = 1. Additionally, (k) (k) in the special case, x = 0 or λ = 1 in (5) and (6), we have En (0 | λ) := En (λ) and (k) (k) (k) (k) (k) (k) Bn (0 | λ) := Bn (λ), En (x | 1) := En (x) and Bn (x | 1) := Bn (x) that stand for Apostol-Euler numbers, Apostol-Bernoulli numbers, the Euler polynomials of order k and the Bernoulli polynomials of order k. Apostol-Euler polynomials of higher order and Apostol-Bernoulli polynomials of higher order can be expressed in terms of their numbers as follows: n n (7) E(k) (x | λ)= xlE(k) (λ) n l n−l l X=0   and n n (8) B(k) (x | λ)= xlB(k) (λ) . n l n−l l X=0   From (1), (2), (3), (4), (5), and (6) we have

(1) (1) En (x | λ) : = En (x | λ) and En (x | 1) := En (x | 1) := En (x) , (1) (1) Bn (x | λ) : = Bn (x | λ) and Bn (x | 1) := Bn (x | 1) := Bn (x) . By (1), we easily get

(0) (0) n (9) En (x | λ)= Bn (x | λ)= x . NEW IDENTITIES ON THE APOSTOL-BERNOULLI POLYNOMIALS 3

Applying derivative operator in the both sides of (8), we have

d (k) (k) (10) B (x | λ)= nB − (x | λ) . dx n n 1 Using (6), we arrive to (k) (k) λB (x +1 | λ) − B (x | λ) − (11) n+1 n+1 = B(k 1)(x | λ), (see [15]) . n +1 n The linear operators Λ and D on the space of real-valued differentiable functions are considered as: For n ∈ N df(x) (12) Λf(x)= λf(x + 1) − f(x) and Df(x)= . dx Notice that ΛD = DΛ. By(12), we have Λ2f (x) = Λ(Λf (x)) = λ2f (x + 2) − 2λf (x +1)+ f (x) 2 2 = (−1)l λlf (x + l) . l l X=0   By continuing this way, we obtain k k Λkf (x)= (−1)l λlf (x + l) . l l X=0   Consequently, we give the following Lemma. Lemma 1. Let f be real valued function and k ∈ N, we have k k Λkf (x)= (−1)l λlf (x + l) . l l X=0   In particular, k k (13) Λkf (0) = (−1)l λlf (l) . l l X=0   Let Pn = {q(x) ∈ Q[x] | deg q(x) ≤ n} be the (n + 1)-dimensional vector space over n Q. Likely, {1, x, ··· , x } is the most natural basis for Pn. (k) (k) (k) Additionally, {B0 (x | λ) ,B1 (x | λ) , ··· ,Bn (x | λ)} is also a good basis for the space Pn for our objective of arithmetical applications of Apostol-Bernoulli polynomials of higher order.

If q(x) ∈ Pn, then q(x) can be written as n (k) (14) q(x)= bjBj (x | λ). j=0 X Recently, many mathematicians have studied on the applications of polynomials and q-polynomials for their finite evaluation schemes, closure under addition, multiplication, differentiation, integration, and composition, they are also richly utilized in construction of their generating functions for finding many identities and formulas (see [1]–[22]). NEW IDENTITIES ON THE APOSTOL-BERNOULLI POLYNOMIALS 4

In this paper, we discover methods for determining bj from the expression of q(x) in (14), and apply those results to arithmetically and combinatorially interesting identities (k) (k) (k) involving B0 (x | λ), B1 (x | λ), ..., Bn (x | λ).

2. Identities on the Apostol-Bernoulli polynomials of higher order By (11) and (12), we see that

(k) (k) (k) (k−1) (15) ΛBn (x | λ)= λBn (x +1 | λ) − Bn (x | λ)= nBn−1 (x | λ) , and (k) (k) (16) DBn (x | λ)= nBn−1(x | λ). (k) Let us assume that q(x) ∈ Pn. Then q(x) can be generated by means of B0 (x | λ), (k) (k) B1 (x | λ),. . . , Bn (x | λ) as follows: n (k) (17) q(x)= bj Bj (x | λ). j=0 X Thus, by (17), we get n n (k) (k−1) Λq (x)= bjΛBj (x | λ)= bljBj−1 (x | λ), j=0 j=1 X X and n 2 (k−2) Λ q(x) = Λ [Λq (x)] = bjj (j − 1) Bj−2 (x | λ). j=2 X By continuing this way, we have n k (0) (18) Λ q(x)= bjj (j − 1) ··· (j − k + 1) Bj−k(x | λ). j k X= By (9) and (18), we see that n j! − − (19) DsΛkq(x)= b xj k s j (j − k − s)! j k s =X+ Let us take x = 0 in (19). Then we derive the following: 1 (20) DsΛkq(0) = b . (k + s)! k+s From (13) and (20), we have 1 1 (21) b = DsΛkq(0) = ΛkDsq(0) k+s (k + s)! (k + s)! k 1 k = (−1)a λaDsq(a). (k + s)! a a=0 X   Therefore, by (17) and (21), we have the following theorem. NEW IDENTITIES ON THE APOSTOL-BERNOULLI POLYNOMIALS 5

Theorem 1. For k ∈ Z+ and q (x) ∈ Pn, we have n k 1 k − k q(x)= (−1)a λaDj kq(a) B( )(x | λ). j! a j j k a=0 ! X= X   n j−k n n! n−j+k Let us take q(x)= x ∈ Pn. Then we derive that D x = (n−j+k)! x .

Thus, by Theorem 1, we get n k 1 k n! − k (22) xn = (−1)a λa an j+k B( )(x | λ). j! a (n − j + k)! j j k a=0 ! X= X   Therefore, by (22), we arrive at the following corollary.

Corollary 1. For k,n ∈ Z+, we have n k 1 k n! − k xn = (−1)a λa an j+k B( )(x | λ). j! a (n − j + k)! j j k a=0 ! X= X   (k) Let q (x)= En (x) ∈ Pn. Also, it is well known in [12] that

− n! (23) Dj kE(k) (x)= E(k) (x). n (n − j + k)! n−j+k By Theorem 1 and (23), we get the following theorem.

Theorem 2. For k,n ∈ Z+, we have n−j k − n k + k n j+k al (−λ)a n! E(k) (x)= a l E(k) B(k)(x | λ). n j! (n − j + k)! n−j+k−l j j=k a=0 l=0

X X X (k) Let us consider q (x)= Bn (x) ∈ Pn. Then we see that

− n! (24) Dj kB(k) (x)= B(k) (x). n (n − j + k)! n−j+k Thanks to Theorem 1 and (24), we obtain the following theorem.

Theorem 3. For k,n ∈ Z+, we have n−j k − n k + (−λ)a k n j+k aln! B(k) (x)= a l B(k) B(k)(x | λ). n j! (n − j + k)! n−j+k−l j j=k a=0 l=0

X X X Hansen [6] derived the following convolution formula: m m (25) B (x) B − (y)= k k m k k X=0   (1 − m) Bm (x + y) + (x + y − 1) mBm−1 (x + y) . We note that the special case x = y = 0 of the last identity m−2 m BkBm−k B = − k=2 k m m +1 P
is originally constructed by Euler and Ramanujan (cf. [4]). NEW IDENTITIES ON THE APOSTOL-BERNOULLI POLYNOMIALS 6

Let us now write the following

n n (26) q (x)= B (x) B − (y) ∈ P . k k n k n k X=0   By using derivative operator Ds in the both sides of (25), we derive

j−k n! (27) D q (x)=(1 − n) B − (x + y) (n − j + k)! n j+k n! + (x + y − 1) B − − (x + y) (n − j + k − 1)! n j+k 1 n! + (j − k) B − (x + y) (n − j + k)! n j+k By Theorem 1, (26) and (27), we arrive at the following theorem.

Theorem 4. For k,n ∈ Z+, we have

n n B (x) B − (y) k k n k k X=0   n k 1 a k a n! = (−1) λ {(1 − n) B − (a + y) j! a (n − j + k)! n j+k j k a=0 X= X   n! + (a + y − 1) B − − (a + y) (n − j + k − 1)! n j+k 1

n! (k) + (j − k) B − (a + y)}B (x | λ). (n − j + k)! n j+k j Dilcher [4] introduced the following interesting identity:

n n E (x) E − (y)=2(1 − x − y) E (x + y)+2E (x + y). k k n k n n+1 k X=0   n n Let k=0 k Ek (x) En−k (y) ∈ Pn, then we write that n P
n (28) q (x)= E (x) E − (y) . k k n k k X=0   By (28), we have

j−k n! D q (x)=2{ (1 − x − y) E − (x + y) (n − j + k)! n j+k n! − (j − k) E − (x + y) (n − j + k + 1)! n j+k+1 (n + 1)! + E − (x + y)}. (n +1 − j + k)! n+1 j+k As a result of the last identity and Theorem 1, we derive the following. NEW IDENTITIES ON THE APOSTOL-BERNOULLI POLYNOMIALS 7

Theorem 5. The following equality holds: n n E (x) E − (y) k k n k k X=0   n k 1 a k a n! =2 (−1) λ { (1 − x − y) E − (x + y) j! a (n − j + k)! n j+k j k a=0 X= X   n! − (j − k) E − (x + y) (n − j + k + 1)! n j+k+1

(n + 1)! (k) + E − (x + y)}B (x | λ). (n +1 − j + k)! n+1 j+k j Remark 1. Throughout this paper when we take λ = 1, our results can easily be related to Bernoulli polynomials of higher order. Remark 2. Theorem 1 seems to be plenty large enough for obtaining interesting identities related to special functions in connection with Apostol-Bernoulli polynomials of higher order.

Acknowledgements The authors wish to thank Bernd Kellner for his valuable suggestions for the present paper.

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Department of Mathematics and Statistics, American University of the Middle East, Kuwait City, 15453 Egaila, Kuwait; and Russian Academy of Sciences, Institute for Control Sciences, 65 Profsoyuznaya, 117997 Moscow, Russia; E-mail address: [email protected]

Department of Economics, Faculty of Economics, Administrative and Social Sciences, Hasan Kalyoncu University, TR-27410 Gaziantep, Turkey E-mail address: [email protected]

Department of Mathematics, Faculty of Arts and Sciences, University of Gaziantep TR- 27310 Gaziantep, Turkey E-mail address: [email protected]

Faculty of Science, Kunming University of Science and Technology, Kunming, Yunnan 650500, People’s Republic of China E-mail address: [email protected]