DDNS Discrete Dynamics in Nature and Society 1607-887X 1026-0226 Hindawi Publishing Corporation 927953 10.1155/2012/927953 927953 Research Article Some Formulae of Products of the Apostol-Bernoulli and Apostol-Euler Polynomials He Yuan 1 Wang Chunping 1 Çinar Cengiz 1 Faculty of Science Kunming University of Science and Technology Kunming 650500 China kmust.edu.cn 2012 9 8 2012 2012 17 05 2012 13 07 2012 15 07 2012 2012 Copyright © 2012 Yuan He and Chunping Wang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Some formulae of products of the Apostol-Bernoulli and Apostol-Euler polynomials are established by applying the generating function methods and some summation transform techniques, and various known results are derived as special cases.

1. Introduction

The classical Bernoulli polynomials Bn(x) and Euler polynomials En(x) are usually defined by means of the following generating functions: (1.1)textet-1=n=0Bn(x)tnn!  (|t|<2π),2extet+1=n=0En(x)tnn!  (|t|<π). In particular, Bn=Bn(0) and En=2nEn(1/2) are called the classical Bernoulli numbers and Euler numbers, respectively. These numbers and polynomials play important roles in many branches of mathematics such as combinatorics, number theory, special functions, and analysis. Numerous interesting identities and congruences for them can be found in many papers; see, for example, .

Some analogues of the classical Bernoulli and Euler polynomials are the Apostol-Bernoulli polynomials n(x;λ) and Apostol-Euler polynomials n(x;μ). They were respectively introduced by Apostol  (see also Srivastava  for a systematic study) and Luo [7, 8] as follows: (1.2)textλet-1=n=0Bn(x;λ)tnn!(|t|<2πifλ=1;|t|<|logλ|otherwise),(1.3)2extλet+1=n=0En(x;λ)tnn!(|t|<πifλ=1;|t|<|log(-λ)|otherwise). Moreover, n(λ)=n(0;λ) and n(λ)=2nn(1/2;λ) are called the Apostol-Bernoulli numbers and Apostol-Euler numbers, respectively. Obviously n(x;λ) and n(x;λ) reduce to Bn(x) and En(x) when λ=1. Some arithmetic properties for the Apostol-Bernoulli and Apostol-Euler polynomials and numbers have been well investigated by many authors. For example, in 1998, Srivastava and Todorov  gave the close formula for the Apostol-Bernoulli polynomials in terms of the Gaussian hypergeometric function and the Stirling numbers of the second kind. Following the work of Srivastava and Todorov, Luo  presented the close formula for the Apostol-Euler polynomials in a similar technique. After that, Luo  obtained some multiplication formulas for the Apostol-Bernoulli and Apostol-Euler polynomials. Further, Luo  showed the Fourier expansions for the Apostol-Bernoulli and Apostol-Euler polynomials by applying the Lipschitz summation formula and derived some explicit formulae at rational arguments for these polynomials in terms of the Hurwitz zeta function.

In the present paper, we will further investigate the arithmetic properties of the Apostol-Bernoulli and Apostol-Euler polynomials and establish some formulae of products of the Apostol-Bernoulli and Apostol-Euler polynomials by using the generating function methods and some summation transform techniques. It turns out that various known results are deduced as special cases.

2. The Restatement of the Results

For convenience, in this section we always denote by δ1,λ the Kronecker symbol given by δ1,λ=0 or 1 according to λ1 or λ=1, and we also denote by max(a,b) the maximum number of the real numbers a,b and by [x] the maximum integer less than or equal to the real number x. We now give the formula of products of the Apostol-Bernoulli polynomials in the following way.

Theorem 2.1.

Let m and n be any positive integers. Then, (2.1)Bm(x;λ)Bn(y;μ)=nk=0m(mk)(-1)m-kBm-k(y-x;1λ)Bn+k(y;λμ)n+k+mk=0n(nk)Bn-k(y-x;μ)Bm+k(x;λμ)m+k+(-1)m+1δ1,λμm!n!(m+n)!Bm+n(y-x;1λ).

Proof.

Multiplying both sides of the identity (2.2)1λeu-11μev-1=(λeuλeu-1+1μev-1)1λμeu+v-1 by uvexu+yv, we obtain (2.3)uexuλeu-1veyvμev-1=λvue(1+x-y)uλeu-1ey(u+v)λμeu+v-1+uve(y-x)vμev-1ex(u+v)λμeu+v-1. It follows from (2.3) that (2.4)δ1,λμuvu+v(λe(1+x-y)uλeu-1+e(y-x)vμev-1)=uexuλeu-1veyvμev-1-λvue(1+x-y)uλeu-1(ey(u+v)λμeu+v-1-δ1,λμu+v)-uve(y-x)vμev-1(ex(u+v)λμeu+v-1-δ1,λμu+v). By the Taylor theorem we have (2.5)ex(u+v)λμeu+v-1-δ1,λμu+v=n=0nun(exuλμeu-1-δ1,λμu)vnn!.Since 0(x;λ)=1 whenλ=1 and 0(x;λ)=0 when λ1 (see e.g., ), by (1.2) and (2.5) we get (2.6)ex(u+v)λμeu+v-1-δ1,λμu+v=m=0n=0Bn+m+1(x;λμ)n+m+1umm!vnn!. Putting (1.2) and (2.6) in (2.4), with the help of the Cauchy product, we derive (2.7)δ1,λμuvu+v(λe(1+x-y)uλeu-1+e(y-x)vμev-1)=-λm=0n=0[k=0m(mk)Bm-k(1+x-y;λ)Bn+k+1(y;λμ)n+k+1]umm!vn+1n!-m=0n=0[k=0n(nk)Bn-k(y-x;μ)Bm+k+1(x;λμ)m+k+1]um+1m!vnn!+m=0n=0Bm(x;λ)Bn(y;μ)umm!vnn!. If we denote the left-hand side of (2.7) by M1 and (2.8)M2=λδ1,λ(veyvλμev-1-δ1,λμ)+δ1,μ(uexuλμeu-1-δ1,λμ)-δ1,λveyvμev-1-δ1,μ(uexuλeu-1-δ1,λ), then applying (1.2) to (2.8), in light of (2.7), we have (2.9)M1+M2=m=0n=0[-λm+1k=0m+1(m+1k)Bm+1-k(1+x-y;λ)Bn+k+1(y;λμ)n+k+1-1n+1k=0n+1(n+1k)Bn+1-k(y-x;μ)Bm+k+1(x;λμ)m+k+1+Bm+1(x;λ)m+1Bn+1(y;μ)n+1]um+1m!vn+1n!. On the other hand, a simple calculation implies M1=M2=0 when λμ1 and (2.10)M1+M2=δ1,λμuvu+v(λe(1+x-y)uλeu-1-δ1,λu+e(y-x)v(1/λ)ev-1-δ1,1/λv) when λμ=1. Applying un=k=0n(nk)(u+v)k(-v)n-k to (1.2), in view of changing the order of the summation, we obtain (2.11)λe(1+x-y)uλeu-1-δ1,λu=λk=0n=kBn+1(1+x-y;λ)(n+1)!(nk)(u+v)k(-v)n-k=λk=0n=k+1Bn+1(1+x-y;λ)(n+1)!(nk+1)(u+v)k+1(-v)n-(k+1)+λn=0Bn+1(1+x-y;λ)n+1(-v)nn!. It follows from (1.2), (2.10), (2.11), and the symmetric relation for the Apostol-Bernoulli polynomials λn(1-x;λ)=(-1)nn(x;1/λ) for any nonnegative integer n (see e.g., ) that (2.12)M1+M2=uvδ1,λμk=0n=k+1(-1)kBn+1(y-x;1/λ)(n+1)!(nk+1)(u+v)kvn-(k+1)=δ1,λμk=0n=k+1(-1)kBn+1(y-x;1/λ)(n+1)!(nk+1)m=0k(km)um+1vn-m=δ1,λμm=0k=mn=k+1(-1)kBn+1(y-x;1/λ)(n+1)!(nk+1)(km)um+1vn-m=δ1,λμm=0n=m+1(-1)mBn+1(y-x;1/λ)(n+1)!um+1vn-m=δ1,λμm=0n=0(-1)mm!n!Bn+m+2(y-x;1/λ)(n+m+2)!um+1m!vn+1n!. Thus, by equating (2.9) and (2.12) and then comparing the coefficients of um+1vn+1, we complete the proof of Theorem 2.1 after applying the symmetric relation for the Apostol-Bernoulli polynomials.

It follows that we show some special cases of Theorem 2.1. By setting x=y in Theorem 2.1, we have the following.

Corollary 2.2.

Let m and n be any positive integers. Then, (2.13)Bm(x;λ)Bn(x;μ)=nk=0m(mk)(-1)m-kBm-k(1λ)Bn+k(x;λμ)n+k+mk=0n(nk)Bn-k(μ)Bm+k(x;λμ)m+k+(-1)m+1δ1,λμm!n!(m+n)!Bm+n(1λ).

It is well known that the classical Bernoulli numbers with odd subscripts obey 1=-1/2 and 2n+1=0 for any positive integer n (see, e.g., ). Setting λ=μ=1 in Corollary 2.2, we immediately obtain the familiar formula of products of the classical Bernoulli polynomials due to Carlitz  and Nielsen  as follows.

Corollary 2.3.

Let m and n be any positive integers. Then, (2.14)Bm(x)Bn(x)=k=0max([m/2],[n/2]){n(m2k)+m(n2k)}B2kBm+n-2k(x)m+n-2k+(-1)m+1m!n!(m+n)!Bm+n.

Since the Apostol-Bernoulli polynomials n(x;λ) satisfy the difference equation (/x)n(x;λ) = nn-1(x;λ) for any positive integer n (see, e.g., ), by substituting x+y for x in Theorem 2.1 and then taking differences with respect to y, we get the following result after replacing x by x-y.

Corollary 2.4.

Let m and n be any positive integers. Then, (2.15)1mk=0m(mk)(-1)m-kBm-k(y-x;1λ)Bn-1+k(y;λμ)-1mBm(x;λ)Bn-1(y;μ)=-1nk=0n(nk)Bn-k(y-x;μ)Bm-1+k(x;λμ)+1nBn(y;μ)Bm-1(x;λ).

Setting x=t and y=1-t in Corollary 2.4, by λn(1-x;λ)=(-1)nn(x;1/λ) for any nonnegative integer n, we get the following.

Corollary 2.5.

Let m and n be any positive integers. Then, (2.16)1mk=0m(mk)(-1)kBm-k(2t;λ)Bn-1+k(t;1λμ)-1mBm(t;λ)Bn-1(t;1μ)=1nk=0n(nk)(-1)kBn-k(2t;1μ)Bm-1+k(t;λμ)-1nB(t;1μ)Bm-1(t;λ).

In particular, the case λ=μ=1 in Corollary 2.5 gives the following generalization for Woodcock’s identity on the classical Bernoulli numbers, see [15, 16],

Corollary 2.6.

Let m and n be any positive integers. Then, (2.17)1mk=0m(mk)(-1)kBm-k(2t)Bn-1+k(t)-1mBm(t)Bn-1(t)=1nk=0n(nk)(-1)kBn-k(2t)Bm-1+k(t)-1nBn(t)Bm-1(t).

We next present some mixed formulae of products of the Apostol-Bernoulli and Apostol-Euler polynomials and numbers.

Theorem 2.7.

Let m and n be non-negative integers. Then, (2.18)Em(x;λ)En(y;μ)=2k=0m(mk)(-1)m-kEm-k(y-x;1λ)Bn+k+1(y;λμ)n+k+1-2k=0n(nk)En-k(y-x;μ)Bm+k+1(x;λμ)m+k+1+(-1)m+12δ1,λμm!n!(m+n+1)!Em+n+1(y-x;1λ).

Proof.

Multiplying both sides of the identity (2.19)1λeu+11μev+1=(λeuλeu+1-1μev+1)1λμeu+v-1 by 2exu+yv, we obtain (2.20)122exuλeu+12eyvμev+1=λ2e(1+x-y)uλeu+1ey(u+v)λμeu+v-1-2e(y-x)vμev+1ex(u+v)λμeu+v-1. It follows from (2.20) that (2.21)δ1,λμu+v(λ2e(1+x-y)uλeu+1-2e(y-x)vμev+1)=122exuλeu+12eyvμev+1-λ2e(1+x-y)uλeu+1(ey(u+v)λμeu+v-1-δ1,λμu+v)+2e(y-x)vμev+1(ex(u+v)λμeu+v-1-δ1,λμu+v). Applying (1.3) and (2.6) to (2.21), in view of the Cauchy product, we get (2.22)δ1,λμu+v(λ2e(1+x-y)uλeu+1-2e(y-x)vμev+1)=m=0n=0[12Em(x;λ)En(y;μ)-λk=0m(mk)Em-k(1+x-y;λ)Bn+k+1(y;λμ)n+k+1+k=0n(nk)En-k(y-x;μ)Bm+k+1(x;λμ)m+k+1]umm!vnn!. On the other hand, since the left-hand side of (2.22) vanishes when λμ1, it suffices to consider the case λμ=1. Applying un=k=0n(nk)(u+v)k(-v)n-k to (1.3), in view of changing the order of the summation, we have (2.23)λ2e(1+x-y)uλeu+1=λk=0n=kEn(1+x-y;λ)n!(nk)(u+v)k(-v)n-k=λk=0n=k+1En(1+x-y;λ)n!(nk+1)(u+v)k+1(-v)n-(k+1)+λn=0En(1+x-y;λ)(-v)nn!. It follows from (1.3), (2.23), and the symmetric relation for the Apostol-Euler polynomials λn(1-x;λ)=(-1)nn(x;1/λ) for any non-negative integer n (see, e.g., ) that (2.24)δ1,λμu+v(λ2e(1+x-y)uλeu+1-2e(y-x)vμev+1)=δ1,λμk=0n=k+1(-1)k+1En(y-x;1/λ)n!(nk+1)(u+v)kvn-(k+1)=δ1,λμk=0n=k+1(-1)k+1En(y-x;1/λ)n!(nk+1)m=0k(km)umvn-(m+1)=δ1,λμm=0k=mn=k+1(-1)k+1En(y-x;1/λ)n!(nk+1)(km)umvn-(m+1)=δ1,λμm=0n=m+1(-1)m+1En(y-x;1/λ)n!umvn-(m+1)=δ1,λμm=0n=0(-1)m+1m!n!Em+n+1(y-x;1/λ)(m+n+1)!umm!vnn!. Thus, by equating (2.22) and (2.24) and then comparing the coefficients of umvn, we complete the proof of Theorem 2.7 after applying the symmetric relation for the Apostol-Euler polynomials.

Next, we give some special cases of Theorem 2.7. By setting x=y in Theorem 2.7, we have the following.

Corollary 2.8.

Let m and n be non-negative integers. Then, (2.25)Em(x;λ)En(x;μ)=2k=0m(mk)(-1)m-kEm-k(0;1λ)Bn+k+1(x;λμ)n+k+1-2k=0n(nk)En-k(0;μ)Bm+k+1(x;λμ)m+k+1+(-1)m+12δ1,λμm!n!(m+n+1)!Em+n+1(0;1λ).

Since the classical Euler polynomials En(x) at zero arguments satisfy E0(0)=1, E2n(0)=0, and E2n-1(0)=(1-22n)B2n/n for any positive integer n (see, e.g., ), by setting λ=μ=1 in Corollary 2.8, we obtain the following.

Corollary 2.9.

Let m and n be non-negative integers. Then, (2.26)Em(x)En(x)=-2k=1max([(m+1)/2],[(n+1)/2]){(m2k-1)+(n2k-1)}(1-22k)B2kk×Bn+n+2-2k(x)m+n+2-2k+(-1)m+12m!n!(m+n+1)!Km,n, where Km,n=2(1-2m+n+2)Bm+n+2/(m+n+2) when m+n0 (mod2) and Km,n=0 otherwise.

Theorem 2.10.

Let m be non-negative integer and n positive integer. Then, (2.27)Em(x;λ)Bn(y;μ)=n2k=0m(mk)(-1)m-kEm-k(y-x;1λ)En+k-1(y;λμ)+k=0n(nk)Bn-k(y-x;μ)Em+k(x;λμ).

Proof.

Multiplying both sides of the identity (2.28)1λeu+11μev-1=(λeuλeu+1+1μev-1)1λμeu+v+1 by 2vexu+yv, we obtain (2.29)2exuλeu+1veyvμev-1=λv22e(1+x-y)uλeu+12ey(u+v)λμeu+v+1+ve(y-x)vμev-12ex(u+v)λμeu+v+1. By (1.3) and the Taylor theorem, we have (2.30)2ex(u+v)λμeu+v+1=m=0n=0En+m(x;λμ)umm!vnn!. Applying (1.2), (1.3), and (2.30) to (2.29), we get (2.31)m=0n=0Em(x;λ)Bn(y;μ)umm!vnn!=λ2m=0n=0[k=0m(mk)Em-k(1+x-y;λ)En+k(y;λμ)]umm!vn+1n!+m=0n=0[k=0n(nk)Bn-k(y-x;μ)Em+k(x;λμ)]umm!vnn!, which means (2.32)m=0n=0Em(x;λ)Bn+1(y;μ)n+1umm!vn+1n!=m=0n=0[λ2k=0m(mk)Em-k(1+x-y;λ)En+k(y;λμ)+1n+1k=0n+1(n+1k)Bn+1-k(y-x;μ)Em+k(x;λμ)]umm!vn+1n!. Thus, by comparing the coefficients of umvn+1 in (2.32), we conclude the proof of Theorem 2.10 after applying the symmetric relation for the Apostol-Euler polynomials.

Obviously, by setting x=y in Theorem 2.10, we have the following.

Corollary 2.11.

Let m be non-negative integer and n positive integer. Then, (2.33)Em(x;λ)Bn(x;μ)=n2k=0m(mk)(-1)m-kEm-k(0;1λ)En+k-1(x;λμ)+k=0n(nk)Bn-k(μ)Em+k(x;λμ).

Since B1=-1/2, E0(0)=1, E2n(0)=0, and E2n-1(0)=(1-22n)B2n/n for any positive integer n, by setting λ=μ=1 in Corollary 2.11, we obtain the following.

Corollary 2.12.

Let m be non-negative integer and n positive integer. Then, (2.34)Em(x)Bn(x)=k=1max([(m+1)/2],[n/2]){n(22k-1)2k(m2k-1)+(n2k)}B2kEm+n-2k(x)+Em+n(x).

Remark 2.13.

For the equivalent forms of Corollaries 2.9 and 2.12, the interested readers may consult .

Acknowledgments

The authors are very grateful to anonymous referees for helpful comments on the previous version of this work. They express their gratitude to Professor Wenpeng Zhang who provided them with some suggestions. This paper is supported by the National Natural Science Foundation of China (Grant no. 10671194).

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