Some Properties of Bernoulli Polynomials and Their Generalizations

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Applied Mathematics Letters 24 (2011) 746–751

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Applied Mathematics Letters

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Some properties of Bernoulli polynomials and their generalizations Da-Qian Lu East China Normal University, Department of Mathematics, Shanghai 200241, PR China article info a b s t r a c t

Article history: In this work, we investigate some well-known and new properties of the Bernoulli Received 18 July 2010 polynomials and their generalizations by using quasi-monomial, lowering operator and Accepted 21 December 2010 operational methods. Some of these general results can indeed be suitably specialized in order to deduce the corresponding properties and relationships involving the (generalized) Keywords: Bernoulli polynomials. Quasi-monomial © 2010 Elsevier Ltd. All rights reserved. Generating functions Recursion formulas Connection problems Bernoulli polynomials and numbers Apostol–Bernoulli polynomials and numbers

1. Introduction ˆ ˆ A polynomial pn(x)(n ∈ N, x ∈ C) is said to be a quasi-monomial [1] whenever two operators M, P, called the multiplicative and derivative (or lowering) operators respectively, can be defined in such a way that ˆ Ppn(x) = npn−1(x), (1.1) ˆ Mpn(x) = pn+1(x), (1.2) which can be combined to get the identity ˆ ˆ MPpn(x) = npn(x). (1.3)

The classical Bernoulli polynomials Bn(x) are defined by [2] ∞ zexz − zn = Bn(x) (|z| < 2π), (1.4) ez − 1 n! n=0 and, consequently, the classical Bernoulli numbers Bn := Bn(0) can be obtained by using the generating function ∞ z − zn = Bn . (1.5) ez − 1 n! n=0 Moreover, we have [3]   n 1 1 Bn(0) = (−1) Bn(1) = Bn (n ∈ N0). (1.6) 21−n − 1 2

E-mail address: [email protected].

(α) The generalized Bernoulli polynomials Bn (x) are defined by [4]

 α ∞ z − zn exz = B(α)(x) (|z| < 2π). (1.7) ez − 1 n n! n=0 (α) Clearly, the generalized Bernoulli numbers Bn are given by (α) := (α) Bn Bn (0) and := (1) ∈ Bn(x) Bn (x)(n N0).

In 2005, Luo defined the Apostol–Bernoulli numbers Bn(λ) and polynomials Bn(x; λ) as ∞ z − zn = Bn(λ) , (1.8) ez − 1 n! λ n=0 ∞ zexz − zn = Bn(x; λ) , (1.9) ez − 1 n! λ n=0 (|z| < 2π when λ = 1; |z| < | log λ| when λ ̸= 1). (α) (α) The generalized Apostol–Bernoulli numbers Bn (λ) and polynomials Bn (x; λ) are defined by [5]

 α ∞ z − zn = B(α)(λ) , (1.10) ez − 1 n n! λ n=0

 α ∞ z − zn exz = B(α)(x; λ) , (1.11) ez − 1 n n! λ n=0 (|z| < 2π when λ = 1; |z| < | log λ| when λ ̸= 1). Clearly, we have (α) = (α) ; (α) := (α) ; Bn (x) Bn (x 1) and Bn (λ) En (0 λ), (1.12) ; := (1) ; := (1) Bn(x λ) Bn (x λ) and Bn(λ) Bn (λ). (1.13) The Appell polynomials [6] can be defined by considering the following generating function: ∞ − Rn(x) A(t)ext = tn, (1.14) n! n=0 where ∞ − Rk A(t) = tk,(A(0) ̸= 0) (1.15) k! k=0 is an analytic function at t = 0.  α (α) It is easy to see that if A t = t , then R t = x . ( ) λet −1 n( ) Bn ( )

From [7], we know that the multiplicative and derivative operators of Rn(x) are

n−1 ˆ − αn−k n−k M = (x + α0) + D , (1.16) n − k ! x k=0 ( ) ˆ P = Dx, (1.17) where ′ ∞ A (t) − tn = αn . (1.18) A t n! ( ) n=0 By using (1.3), we have the following lemma. 748 D.-Q. Lu / Applied Mathematics Letters 24 (2011) 746–751

Lemma 1.1 ([7]). The Appell polynomials Rn(x) defined by (1.14) satisfy the differential equation

αn−1 (n) αn−2 (n−1) α1 ′′ ′ y + y + · · · + y + (x + α0)y − ny = 0, (1.19) (n − 1)! (n − 2)! 1! where the numerical coefficients αk, k = 1, 2,..., n − 1, are defined in (1.19), and are linked to the values Rk by the following relations: −k  k  Rk+1 = Rhαk−h. h h=0

Let P be the vector space of polynomials with coefficients in C, and the polynomial sequence {Pn}n≥0 be a polynomial set. {Pn}n≥0 is called a σ -Appell polynomial set of the transfer power series A generated by ∞ − Pn(x) n G(x, t) = A(t)G0(x, t) = t , (1.20) n! n=0 where G0(x, t) is a solution of the system

σ G0(x, t) = tG0(x, t),

G0(x, 0) = 1. In [8], the authors investigated the coefficients of connection between two polynomials. And there is a result concerning coefficients of connection between two σ -Appell polynomial sets.

(−1) Lemma 1.2 ([8]). Let σ ∈ Λ . Let {Pn}n≥0 and {Qn}n≥0 be two σ -Appell polynomial sets of transfer power series A1 and A2, respectively. Then

−n n! Qn(x) = αn−mPm(x), (1.21) m! m=0 where ∞ A2(t) − k = αkt . A t 1( ) k=0

Recently, several interesting properties and relationships involving the classical as well as the generalized Bernoulli polynomials and the Apostol–Bernoulli polynomials were investigated [3,9,5,10–15]. In this work, we want to investigate some well-known and new properties of these polynomials from different angles. In Section 2, we propose to prove some relationships and differential equations involving the generalized Bernoulli polynomials. In Section 3, we want to consider the problem of connection of the generalized Apostol–Bernoulli polynomials with some other polynomials. In Section 4, we shall extend the generalized Apostol–Bernoulli polynomials to Hermite-based generalized Apostol– Bernoulli polynomials by an operational method. And some of their properties are given.

2. Recursion formulas and differential equations

(α) α From Section 1, we know that the generalized Bernoulli polynomials B x are Appell polynomials with A t =  t  . n ( ) ( ) λet −1 And by using (1.4), we obtain ′ ∞ n A (t) − Bn+1(1) t = −α . A t n + 1 n! ( ) n=0 By using (1.16)–(1.18), we can obtain the multiplicative and derivative operators of the generalized Bernoulli polynomials: −  1  n 1 B 1 ˆ − n−k+1( ) n−k ˆ M = x − α − α D , P = Dx. (2.1) 2 n − k + 1 ! x k=0 ( ) From the generating function (1.8), we can easily obtain p ∂ (α) n! (α) B (x) = B − (x). (2.2) ∂xp n (n − p)! n p By using (1.2), (2.1) and (2.2), we obtain the following result. D.-Q. Lu / Applied Mathematics Letters 24 (2011) 746–751 749

(α) Theorem 2.1. For any integral n ≥ 1, the following linear recurrence relation for the generalized Bernoulli polynomials Bn (x) holds true:   n−1 (α) 1 (α) −  n  Bn−k+1(1) (α) B + (x) = x − α B (x) − α B (x). (2.3) n 1 2 n k n − k + 1 k k=0

(α) Furthermore, applying Lemma 1.1 to Bn (x), we obtain (α) Theorem 2.2. The generalized Bernoulli polynomials Bn (x) satisfy the differential equation   Bn(1) Bn−1(1) − B2(1) ′′ x 1 ′ n y(n) + y(n 1) + · · · + y − − y + y = 0. (2.4) (n)! (n − 1)! 2! α 2 α

In the special case of (2.3) and (2.4) when α = 1, we obtain the following results.

Corollary 2.1. For any integral n ≥ 1, the following linear recurrence relation for the Bernoulli polynomials Bn(x) holds true:

  n−1 1 −  n  Bn−k+1(1) Bn+1(x) = x − Bn(x) − Bk(x). (2.5) 2 k n − k + 1 k=0

Corollary 2.2. The Bernoulli polynomials Bn(x) satisfy the differential equation   Bn(1) Bn−1(1) − B2(1) ′′ 1 ′ y(n) + y(n 1) + · · · + y − x − y + ny = 0. (2.6) (n)! (n − 1)! 2! 2

3. Connection problems

From the generating function (1.11), we can easily obtain

∂ (α) (α) B (x; λ) = nB − (x; λ). ∂x n n 1 So by the definition of the σ -Appell polynomial (1.20), we know that the generalized Apostol–Bernoulli polynomials are a D-Appell polynomial set, D being the derivative operator. From Table 1 in [8], we know that the lowering operators of monomials xn and the Gould–Hopper polynomials [16] m htm gn (x, h) are all D. And their transfer power series A(t) are 1 and e respectively. n (α) Applying Lemma 1.2 to Pn(x) = x , Qn(x) = Bn (x; λ), we get the well-known identity [5] n (α) −  n  (α) m B (x; λ) = B − (λ)x . (3.1) n m n m m=0 (α) Applying Lemma 1.2 to Pn(x) = Bn(x; λ), Qn(x) = Bn (x; λ), we get the well-known identity [5] n (α) −  n  (α−1) B (x; λ) = B − (λ)Bm(x; λ). (3.2) n m n m m=0 = m = (α) ; Applying Lemma 1.2 to Pn(x) gn (x, h), Qn(x) Bn (x λ), we get a new identity: [ − ]  n ! n r/m k (α) − n − k h (α) m B (x; λ) = (−1) B − − (λ) g (x, h). (3.3) n r! k! n − r − mk ! n r mk r r=0 k=0 ( ) = 2 − In particular, for Hermite polynomials, since Hn(x) gn (2x, 1), we have

n [n−r/2]  (α) − n! − 1 (α) B (2x; λ) = B − − (λ) Hr (x). (3.4) n r! k! n − r − 2k ! n r 2k r=0 k=0 ( ) In the special case of (3.1) when α = 1 and λ = 1, we have n −  n  m Bn(x) = Bn−mx . (3.5) m m=0 n And if we apply Lemma 1.2 to Pn(x) = Bn(x), Qn(x) = x , we can obtain the following familiar expansion [4]: n  +  n 1 − n 1 x = Bm(x). (3.6) n + 1 m m=0 750 D.-Q. Lu / Applied Mathematics Letters 24 (2011) 746–751

4. Hermite-based generalized Apostol–Bernoulli polynomials

The two-variable Hermite–Kampé de Fériet polynomials (2VHKdFP) Hn(x, y) are defined by the series [17] [n/2] − − xn 2r yr Hn(x, y) = n! (4.1) r! n − 2r ! r=0 ( ) with the following generating function:

∞ n 2 − t exp(xt + yt ) = Hn(x, y). (4.2) n! n=0

And the 2VHKdFP Hn(x, y) are also defined through the operational identity  2  ∂ n exp y {x } = Hn(x, y). (4.3) ∂x2  2  Acting with the operator exp y ∂ on (1.11), and by the identity [18] ∂x2    ∂ 2  1 ax2 − bx − b2y exp y {exp(−ax2 + bx)} = √ exp − , (4.4) ∂x2 1 + 4ay 1 + 4ay

(α) we define the Hermite-based generalized Apostol–Bernoulli polynomials H Bn (x, y; λ) through the generating function α   ∞ n z xz+yz2 − (α) z e = H B (x, y; λ) . (4.5) ez − 1 n n! λ n=0 Clearly, we have := (1) = (1) H Bn(x, y, λ) H Bn (x, y, λ), H Bn(x, y) H Bn (x, y, 1). From the generating function (4.5), we easily obtain

∂ (α) (α) H B (x, y; λ) = nH B − (x, y; λ) (4.6) ∂x n n 1 and

∂ (α) (α) H B (x, y; λ) = n(n − 1)H B − (x, y; λ), (4.7) ∂y n n 2 which can be combined to get the identity

2 ∂ (α) ∂ (α) H B (x, y; λ) = H B (x, y; λ). (4.8) ∂x2 n ∂y n  2  Acting with the operator exp y ∂ on both sides of (3.1), (3.2) and (3.6), and by using (4.3), we obtain ∂x2

n (α) −  n  (α) H B (x, y; λ) = B − (λ)Hm(x, y), (4.9) n m n m m=0 n (α) −  n  (α−1) H B (x, y; λ) = B − (λ)H Bm(x, y, λ), (4.10) n m n m m=0 1 −n  n + 1  Hn(x, y) = H Bn(x, y). (4.11) n + 1 m m=0

Acknowledgements

The author would like to thank the referees and editors for their many valuable comments and suggestions. This work was supported by the Shanghai Natural Science Foundation, grant number 10ZR1409100.

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