CEJM 2(4) 2004 509–515

On the Apostol-Bernoulli Polynomials

Qiu-Ming Luo∗ Department of Mathematics, Jiaozuo University, Jiaozuo City, Henan 454003, The People’s Republic of China

Received 19 July 2001; accepted 23 May 2003

Abstract: In the present paper, we obtain two new formulas of the Apostol-Bernoulli polynomials (see On the Lerch Zeta function. Pacific J. Math., 1 (1951), 161–167.), using the Gaussian hypergeometric functions and Hurwitz Zeta functions respectively, and give certain special cases and applications. °c Central European Science Journals. All rights reserved.

Keywords: Bernoulli numbers, Bernoulli polynomials, Apostol-Bernoulli numbers, Apostol- Bernoulli polynomials, Gaussian hypergeometric functions, Stirling numbers of the second kind, Hurwitz Zeta functions, Lerch functional equation MSC (2000): Primary: 11B68; Secondary: 33C05, 11M35, 30E20

1 Introduction

An analogue of the classical Bernoulli polynomials were defined by T. M. Apostol (see [1]) when he studied the Lipschitz-Lerch Zeta functions. We call this polynomials the Apostol-Bernoulli polynomials. First we rewrite Apostol’s definitions below

Definition 1.1. Apostol-Bernoulli polynomials Bn(x, λ) are defined by means of the generating function (see [1, p.165 (3.1)] or [4, p.83])

∞ zexz zn = B (x, λ) , (|z + ln λ| < 2π) (1) λez − 1 n n! n=0 X setting λ = 1 in (1), Bn(x) = Bn(x, 1) are classical Bernoulli polynomials.

∗ [email protected] 510 Q.-M. Luo / Central European Journal of Mathematics 2(4) 2004 509–515

Definition 1.2. Apostol-Bernoulli numbers Bn(λ) := Bn(0, λ) are defined by means of the generating function

∞ z zn = B (λ) , (|z + ln λ| < 2π) (2) λez − 1 n n! n=0 X setting λ = 1 in (2), Bn = Bn(1) are classical Bernoulli nunbers.

T.M.Apostol not only gave elementary properties of polynomials Bn(x, λ) in [1], but also obtained the recursion formula for the set of numbers Bn(λ) using the Stirling num- bers of the second kind (see [1, p. 166 (3.7)]) as follows

n−1 k k −1−k Bn(λ) = n (−1) k!λ (λ − 1) S(n − 1, k), (n ∈ N0; <(λ) > 0, λ =6 1) (3) k X=1 where S(n, k) denote the Stirling numbers of the second kind which are defined by means of the following expansion (see [3, p.207, Theorem B])

n x xn = k!S(n, k). (4) k k X=0 µ ¶ By applying binomial series expansion and Leibniz’s rule, we first obtain the repre-

sentation of the polynomials Bn(x, λ) involving the Gaussian hypergeometric functions, and thereout deduce Apostol’s formula (3); afterward we prove Theorem 3.1 using Lerch functional equation with related Hurwitz Zeta function. Furthermore we show that the main result in [9, p.1529, Theorem A] is only a special case of Theorem 3.1.

2 Apostol-Bernoulli Polynomials and Gaussian Hypergeometric functions

Theorem 2.1. If n is a positive integer and <(λ) > 0, λ =6 1 are complex numbers, then we have

n− l 1 n − 1 l B (x, λ) = n λl(λ − 1)−l−1 (−1)j jl(x + j)n−l−1 n l j l j=0 X=0 µ ¶ X µ ¶ × F [l − n + 1, l; l + 1; j/(x + j)] (5) where F [a, b; c; z] denotes Gaussian hypergeometric functions defined by (cf. [5, p.44 (4)])

∞ (a) (b) zn F [a, b; c; z] := n n , |z| < 1 (6) (c)n n! n=0 X Γ(λ+n) where (λ)0 = 1, (λ)n = λ(λ + 1) · · · (λ + n − 1) = Γ(λ) , (n ≥ 1). Q.-M. Luo / Central European Journal of Mathematics 2(4) 2004 509–515 511

Proof 2.2. We differentiate both sides of (1) with respect to the variable z. Applying Leibniz’s rule yields

zexz d B (x, λ) =Dn , D = . n z λez − 1 z dz ½ ¾¯z=0 n ¯ −1 (7) −1 ¯n n−k k−1 λ z =(λ − 1) ¯ kx Dz (e − 1) + 1 . k λ − 1 k=0 µ ¶ ½· ¸ ¾¯z=0 X ¯ ¯ Since binomial series expansion ¯

∞ (1 + w)−1 = (−w)l, |w| < 1 (8) l X=0 λ z setting w = λ−1 (e − 1), we have

n k− n 1 λ l B (x, λ) = (λ − 1)−1 kxn−k Dk−1{(ez − 1)l}| . (9) n k 1 − λ z z=0 k l X=0 µ ¶ X=0 µ ¶ By the definition of Stirling numbers of the second kind (see [5, p.58 (15)])

∞ zr (ez − 1)l = l! S(r, l) (10) r! r l X= yields

n k− n 1 B (x, λ) = k (−1)lλl(λ − 1)−l−1l!S(k − 1, l)xn−k. (11) n k k l X=1 µ ¶ X=0 We change sum order of k and l, and using the formula below (see [5, p.58 (20)])

k 1 k S(n, k) = (−1)k−j jn (12) k! j j=0 X µ ¶ we obtain

n− l n−l− 1 l 1 n − 1 j k B (x, λ) = n λl(λ − 1)−l−1xn−k−l−1 (−1)j jl . (13) n j n − k − 2 x l j=0 k X=0 X µ ¶ X=0 µ ¶µ ¶ Applying (6) to (13) readily yields

n− 1 n − 1 B (x, λ) = n λl(λ − 1)−l−1xn−k−l−1 n l l=0 µ ¶ Xl l × (−1)j jl.F [l − n + 1, 1; l + 1; −j/x] (14) j j=0 X µ ¶ 512 Q.-M. Luo / Central European Journal of Mathematics 2(4) 2004 509–515

Finally, we apply the known transformation [10, 15.3.4]

F [a, b; c; z] = (1 − z)−aF [a, c − b; c; z/(z − 1)], and (2.2) immediately obtain (5).

Remark 2.3. H .M. Srivastava and P. G Todorov considered earlier the formula of the generalized Bernoulli polynomials (see [6, p.510 (3)]), for α = 1, which is a complemen- tarity of our result (5), for λ = 1, as follows

k n k! k B (x) = n (−1)j j2k(x + j)n−k n k=0 k (2k)! j j=0 µ ¶ X µ ¶ P× F [k − n, k − 1; 2k + 1; j/(x + j)]. (15)

Remark 2.4. We will also apply the representation (5) in order to derive an interesting special case considered by T. M. Aspotol in (3). By the well-known formula [10, 15.1.20] Γ(c)Γ(c − a − b) F [a, b; c; 1] = (c =6 0, −1, −2, . . . , <(c − a − b) > 0), Γ(c − a)Γ(c − b) upon setting a = l − n + 1, b = l, and c = l + 1 yields − n − 1 1 F [l − n + 1, l; l + 1; 1] = , (0 ≤ l ≤ n). (16) l µ ¶ In view of (16), the special case of our formula (5) when x = 0 gives Apostol’s represen- tation (3).

Remark 2.5. The formula of classical Bernoulli numbers considered by H. W. Gould in [7, p.49, Eq.(17)] is also a complementarity of the Apostol’s formula (3), for λ = 1

n − n + 1 n + k 1 B = (−1)k S(n + k, k). (17) n n − k k k X=0 µ ¶µ ¶

Remark 2.6. There is a relationship between Apostol-Bernoulli polynomials Bn(x, λ) and Stirling numbers of the second kind:

n k− n 1 B (x, λ) = k (−1)jλj(λ − 1)−j−1j!S(k − 1, j)xn−k, (λ =6 −1). (18) n k k j=0 X=1 µ ¶ X Remark 2.7. Recently, Luo also obtained the relation between the classical Bernoulli polynomials and the Stirling numbers of the second kind [8], which is a complementarity of (18), for λ = 1:

n n−k − n k + s + 1 s + 2k 1 B (x) = S(s + 2k, k)xn−s−k. (19) n s + k s k k s=0 X=0 X µ ¶µ ¶µ ¶ Q.-M. Luo / Central European Journal of Mathematics 2(4) 2004 509–515 513

3 Apostol-Bernoulli polynomials and the Hurwitz Zeta Function

It is well-known that Hurwitz-Lerch Zeta functions are defined by the infinite series (see [5, p.121, (1)]) ∞ zk Φ(z, a, s) = , (20) (a + k)s k=0 C Z− C X (a ∈ \ 0 ; s ∈ , when |z| < 1; <(s) > 1, when |z| = 1). Setting z = e2πiz gives the Lipschitz-Lerch Zeta function φ(z, a, s) (see [1, p.161]) ∞ e2kπiz φ(z, a, s) = , (21) (a + k)s k=0 C Z− R Z X Z (a ∈ \ 0 ; z ∈ \ , <(s) > 0; z ∈ , <(s) > 1). When s = −n is negative integer, the Lipschitz-Lerch Zeta function φ(z, a, s) was evalu- ated by T. M. Apostol using polynomials Bn(x, λ) (see [1, p.164]): B (a, e2πiz) φ(z, a, −n) = − n+1 . (22) n + 1 If we set z = 0 in (22), then we have the formula (see [2, p.264, Theorem 12.13]) B (a) ζ(a, −n) = − n+1 (23) n + 1

∞ 1 where ζ(a, s) are Hurwitz Zeta functions defined by ζ(a, s) := (see [2, k=0 (a + k)s p.249]) and Bn(x) are classical Bernoulli polynomials. P Further, if we set a = 0 in (23), then we have the known formula (see [2, p.266, Theorem 12.16]): B ζ(−n) = − n+1 (24) n + 1 ∞ 1 where ζ(s) are Riemann Zeta functions defined by ζ(s) := (see [2, p.249]) and k=0 ks Bn := Bn(0) are classical Bernoulli numbers. P In this section, we will apply the Lerch functional equation to obtain the representa- p tion of Apostol-Bernoulli polynomials B (x, λ) at rational points x = . Clearly, it is a n q generalized form of [9, p.1529, Theorem A].

Theorem 3.1. For n ∈ N \ {1}; p ∈ Z, q ∈ N, z ∈ C we have

q B p 2πiz n! z + j − 1 n 2(z + j − 1)p n , e = − n ζ , n exp − πi q (2qπ) ( q 2 q ³ ´ j=1 ³ ´ h³ ´ i q X j − z n 2(j − z)p + ζ , n exp − + πi . (25) q 2 q j=1 ) X ³ ´ h³ ´ i 514 Q.-M. Luo / Central European Journal of Mathematics 2(4) 2004 509–515

Proof 3.2. In view of the Lipschitz-Lerch Zeta functions φ(z, a, s) (21), and using the known series identity ∞ q ∞ f(k) = f(qk + j), (q ∈ N) (26) k j=1 k X=1 X X=0 we obtain that q a + j − 1 φ(z, a, s) = q−s ζ , s exp(2(kq + j − 1)πiz), (k ∈ N ). (27) q 0 j=1 X µ ¶ p Setting z = yields q q p a + j − 1 2(j − 1)pπi φ , a, s = q−s ζ , s exp , (p ∈ Z, q ∈ N). (28) q q q j=1 µ ¶ X µ ¶ µ ¶ On the other hand, by the Lerch functional equation (see [5, p.125, (29)]), for 0 < z < 1, 0 < a < 1, s ∈ C Γ(s) 1 φ(z, a, 1 − s) = exp πi s − 2az φ(−a, z, s) (2π)s 2 n h1 ³ ´i + exp πi − s + 2a(1 − z) φ(a, 1 − z, s) . (29) 2 p h ³ ´i o Setting a = above and applying (28) yields q q p Γ(s) z + j − 1 s 2(z + j − 1)p φ(z, , 1 − s) = s ζ , s exp − πi q (2qπ) ( q 2 q j=1 ³ ´ h³ ´ i q X j − z s 2(j − z)p + ζ , s exp − + πi , (p ∈ Z, q ∈ N).(30) q 2 q j=1 ) X ³ ´ h³ ´ i Finally, set s = n in (30), and Apostol’s formula (22) leads immediately to formula (25).

Remark 3.3. If z in Theorem 3.1 is an integer, then we deduce readily the Cvijovic and Klinowski’s result (see [9, p.1529, Theorem A]) q p 2 · n! j 2jpπ nπ B = − ζ , n cos − , (31) n q (2qπ)n q q 2 j=1 ³ ´ X ³ ´ ³ ´ (n ∈ N \ {1}; p ∈ N0, q ∈ N; 0 ≤ p ≤ q).

Acknowledgements

The author appreciate the anonymous referee and the editor, Professor Monika Sperling, for their valuable comments and addition to references. The author was supported in part by NNSF (#10001016) of China, SF for the Promi- nent Youth of Henan Province China (#0112000200). Q.-M. Luo / Central European Journal of Mathematics 2(4) 2004 509–515 515

References

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