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On the Apostol-Bernoulli Polynomials 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. Paci¯c 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 de¯ned 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 de¯nitions below De¯nition 1.1. Apostol-Bernoulli polynomials Bn(x; ¸) are de¯ned by means of the generating function (see [1, p.165 (3.1)] or [4, p.83]) 1 zexz zn = B (x; ¸) ; (jz + ln ¸j < 2¼) (1) ¸ez ¡ 1 n n! n=0 X setting ¸ = 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 De¯nition 1.2. Apostol-Bernoulli numbers Bn(¸) := Bn(0; ¸) are de¯ned by means of the generating function 1 z zn = B (¸) ; (jz + ln ¸j < 2¼) (2) ¸ez ¡ 1 n n! n=0 X setting ¸ = 1 in (2), Bn = Bn(1) are classical Bernoulli nunbers. T.M.Apostol not only gave elementary properties of polynomials Bn(x; ¸) in [1], but also obtained the recursion formula for the set of numbers Bn(¸) using the Stirling num- bers of the second kind (see [1, p. 166 (3.7)]) as follows n¡1 k k ¡1¡k Bn(¸) = n (¡1) k!¸ (¸ ¡ 1) S(n ¡ 1; k); (n 2 N0; <(¸) > 0; ¸ =6 1) (3) k X=1 where S(n; k) denote the Stirling numbers of the second kind which are de¯ned 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 ¯rst obtain the repre- sentation of the polynomials Bn(x; ¸) 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 <(¸) > 0; ¸ =6 1 are complex numbers, then we have n¡ l 1 n ¡ 1 l B (x; ¸) = n ¸l(¸ ¡ 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 de¯ned by (cf. [5, p.44 (4)]) 1 (a) (b) zn F [a; b; c; z] := n n ; jzj < 1 (6) (c)n n! n=0 X ¡(¸+n) where (¸)0 = 1; (¸)n = ¸(¸ + 1) ¢ ¢ ¢ (¸ + n ¡ 1) = ¡(¸) ; (n ¸ 1). Q.-M. Luo / Central European Journal of Mathematics 2(4) 2004 509{515 511 Proof 2.2. We di®erentiate both sides of (1) with respect to the variable z. Applying Leibniz's rule yields zexz d B (x; ¸) =Dn ; D = : n z ¸ez ¡ 1 z dz ½ ¾¯z=0 n ¯ ¡1 (7) ¡1 ¯n n¡k k¡1 ¸ z =(¸ ¡ 1) ¯ kx Dz (e ¡ 1) + 1 : k ¸ ¡ 1 k=0 µ ¶ ½· ¸ ¾¯z=0 X ¯ ¯ Since binomial series expansion ¯ 1 (1 + w)¡1 = (¡w)l; jwj < 1 (8) l X=0 ¸ z setting w = ¸¡1 (e ¡ 1), we have n k¡ n 1 ¸ l B (x; ¸) = (¸ ¡ 1)¡1 kxn¡k Dk¡1f(ez ¡ 1)lgj : (9) n k 1 ¡ ¸ z z=0 k l X=0 µ ¶ X=0 µ ¶ By the de¯nition of Stirling numbers of the second kind (see [5, p.58 (15)]) 1 zr (ez ¡ 1)l = l! S(r; l) (10) r! r l X= yields n k¡ n 1 B (x; ¸) = k (¡1)l¸l(¸ ¡ 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; ¸) = n ¸l(¸ ¡ 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; ¸) = n ¸l(¸ ¡ 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 ® = 1, which is a complemen- tarity of our result (5), for ¸ = 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] ¡(c)¡(c ¡ a ¡ b) F [a; b; c; 1] = (c =6 0; ¡1; ¡2; : : : ; <(c ¡ a ¡ b) > 0); ¡(c ¡ a)¡(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 ¸ = 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; ¸) and Stirling numbers of the second kind: n k¡ n 1 B (x; ¸) = k (¡1)j¸j(¸ ¡ 1)¡j¡1j!S(k ¡ 1; j)xn¡k; (¸ =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 ¸ = 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 de¯ned by the in¯nite series (see [5, p.121, (1)]) 1 zk ©(z; a; s) = ; (20) (a + k)s k=0 C Z¡ C X (a 2 n 0 ; s 2 ; when jzj < 1; <(s) > 1; when jzj = 1): Setting z = e2¼iz gives the Lipschitz-Lerch Zeta function Á(z; a; s) (see [1, p.161]) 1 e2k¼iz Á(z; a; s) = ; (21) (a + k)s k=0 C Z¡ R Z X Z (a 2 n 0 ; z 2 n ; <(s) > 0; z 2 ; <(s) > 1): When s = ¡n is negative integer, the Lipschitz-Lerch Zeta function Á(z; a; s) was evalu- ated by T. M. Apostol using polynomials Bn(x; ¸) (see [1, p.164]): B (a; e2¼iz) Á(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) ³(a; ¡n) = ¡ n+1 (23) n + 1 1 1 where ³(a; s) are Hurwitz Zeta functions de¯ned by ³(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 ³(¡n) = ¡ n+1 (24) n + 1 1 1 where ³(s) are Riemann Zeta functions de¯ned by ³(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; ¸) at rational points x = .
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