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=0∞Bn(x)tnn! (|t|<2π), 2extet+1=∑n=0∞En(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, [1–4].

Some analogues of the classical Bernoulli and Euler polynomials are the Apostol-Bernoulli polynomials ℬn(x;λ) and Apostol-Euler polynomials ℰn(x;μ). They were respectively introduced by Apostol [5] (see also Srivastava [6] for a systematic study) and Luo [7, 8] as follows:
(1.2)textλet-1=∑n=0∞Bn(x;λ)tnn! (|t|<2π if λ=1;|t|<|logλ| otherwise),(1.3)2extλet+1=∑n=0∞En(x;λ)tnn! (|t|<π if λ=1;|t|<|log(-λ)| otherwise).
Moreover, ℬn(λ)=ℬn(0;λ) and ℰn(λ)=2nℰn(1/2;λ) are called the Apostol-Bernoulli numbers and Apostol-Euler numbers, respectively. Obviously ℬn(x;λ) and ℰn(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 [9] 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 [7] presented the close formula for the Apostol-Euler polynomials in a similar technique. After that, Luo [10] obtained some multiplication formulas for the Apostol-Bernoulli and Apostol-Euler polynomials. Further, Luo [11] 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 λ≠1 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)=n∑k=0m(mk)(-1)m-kBm-k(y-x;1λ)Bn+k(y;λμ)n+k+m∑k=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-1⋅1μev-1=(λeuλeu-1+1μev-1)1λμeu+v-1
by uvexu+yv, we obtain
(2.3)uexuλeu-1⋅veyvμev-1=λvue(1+x-y)uλeu-1⋅ey(u+v)λμeu+v-1+uve(y-x)vμev-1⋅ex(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-1⋅veyvμev-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=0∞∂n∂un(exuλμeu-1-δ1,λμu)vnn!.Since ℬ0(x;λ)=1 whenλ=1 and ℬ0(x;λ)=0 when λ≠1 (see e.g., [8]), by (1.2) and (2.5) we get
(2.6)ex(u+v)λμeu+v-1-δ1,λμu+v=∑m=0∞ ∑n=0∞Bn+m+1(x;λμ)n+m+1⋅umm!⋅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) =-λ∑m=0∞ ∑n=0∞[∑k=0m(mk)Bm-k(1+x-y;λ)Bn+k+1(y;λμ)n+k+1]umm!⋅vn+1n! -∑m=0∞ ∑n=0∞[∑k=0n(nk)Bn-k(y-x;μ)Bm+k+1(x;λμ)m+k+1]um+1m!⋅vnn! +∑m=0∞ ∑n=0∞Bm(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=0∞ ∑n=0∞[-λm+1∑k=0m+1(m+1k)Bm+1-k(1+x-y;λ)Bn+k+1(y;λμ)n+k+1 -1n+1∑k=0n+1(n+1k)Bn+1-k(y-x;μ)Bm+k+1(x;λμ)m+k+1 +Bm+1(x;λ)m+1⋅Bn+1(y;μ)n+1]um+1m!⋅vn+1n!.
On the other hand, a simple calculation implies M1=M2=0 when λμ≠1 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=λ∑k=0∞ ∑n=k∞Bn+1(1+x-y;λ)(n+1)!(nk)(u+v)k(-v)n-k=λ∑k=0∞ ∑n=k+1∞Bn+1(1+x-y;λ)(n+1)!(nk+1)(u+v)k+1(-v)n-(k+1)+λ∑n=0∞Bn+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 λℬn(1-x;λ)=(-1)nℬn(x;1/λ) for any nonnegative integer n (see e.g., [8]) that
(2.12)M1+M2=uvδ1,λμ∑k=0∞ ∑n=k+1∞(-1)kBn+1(y-x;1/λ)(n+1)!(nk+1)(u+v)kvn-(k+1)=δ1,λμ∑k=0∞ ∑n=k+1∞(-1)kBn+1(y-x;1/λ)(n+1)!(nk+1)∑m=0k(km)um+1vn-m=δ1,λμ∑m=0∞ ∑k=m∞ ∑n=k+1∞(-1)kBn+1(y-x;1/λ)(n+1)!(nk+1)(km)um+1vn-m=δ1,λμ∑m=0∞∑n=m+1∞(-1)mBn+1(y-x;1/λ)(n+1)!um+1vn-m=δ1,λμ∑m=0∞ ∑n=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)=n∑k=0m(mk)(-1)m-kBm-k(1λ)Bn+k(x;λμ)n+k+m∑k=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 ℬ1=-1/2 and ℬ2n+1=0 for any positive integer n (see, e.g., [12]). Setting λ=μ=1 in Corollary 2.2, we immediately obtain the familiar formula of products of the classical Bernoulli polynomials due to Carlitz [13] and Nielsen [14] 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 ℬn(x;λ) satisfy the difference equation (∂/∂x)ℬn(x;λ) = nℬn-1(x;λ) for any positive integer n (see, e.g., [8]), 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)1m∑k=0m(mk)(-1)m-kBm-k(y-x;1λ)Bn-1+k(y;λμ)-1mBm(x;λ)Bn-1(y;μ) =-1n∑k=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 λℬn(1-x;λ)=(-1)nℬn(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)1m∑k=0m(mk)(-1)kBm-k(2t;λ)Bn-1+k(t;1λμ)-1mBm(t;λ)Bn-1(t;1μ) =1n∑k=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)1m∑k=0m(mk)(-1)kBm-k(2t)Bn-1+k(t)-1mBm(t)Bn-1(t) =1n∑k=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)=2∑k=0m(mk)(-1)m-kEm-k(y-x;1λ)Bn+k+1(y;λμ)n+k+1-2∑k=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+1⋅1μev+1=(λeuλeu+1-1μev+1)1λμeu+v-1
by 2exu+yv, we obtain
(2.20)12⋅2exuλeu+1⋅2eyvμev+1=λ2e(1+x-y)uλeu+1⋅ey(u+v)λμeu+v-1-2e(y-x)vμev+1⋅ex(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) =12⋅2exuλeu+1⋅2eyvμev+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=0∞ ∑n=0∞[12Em(x;λ)En(y;μ)-λ∑k=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 λμ≠1, 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=λ∑k=0∞ ∑n=k∞En(1+x-y;λ)n!(nk)(u+v)k(-v)n-k=λ∑k=0∞ ∑n=k+1∞En(1+x-y;λ)n!(nk+1)(u+v)k+1(-v)n-(k+1) +λ∑n=0∞En(1+x-y;λ)(-v)nn!.
It follows from (1.3), (2.23), and the symmetric relation for the Apostol-Euler polynomials λℰn(1-x;λ)=(-1)nℰn(x;1/λ) for any non-negative integer n (see, e.g., [7]) that
(2.24)δ1,λμu+v(λ2e(1+x-y)uλeu+1-2e(y-x)vμev+1) =δ1,λμ∑k=0∞ ∑n=k+1∞(-1)k+1En(y-x;1/λ)n!(nk+1)(u+v)kvn-(k+1) =δ1,λμ∑k=0∞ ∑n=k+1∞(-1)k+1En(y-x;1/λ)n!(nk+1)∑m=0k(km)umvn-(m+1) =δ1,λμ∑m=0∞∑k=m∞∑n=k+1∞(-1)k+1En(y-x;1/λ)n!(nk+1)(km)umvn-(m+1) =δ1,λμ∑m=0∞∑n=m+1∞(-1)m+1En(y-x;1/λ)n!umvn-(m+1) =δ1,λμ∑m=0∞∑n=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)=2∑k=0m(mk)(-1)m-kEm-k(0;1λ)Bn+k+1(x;λμ)n+k+1-2∑k=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., [12]), 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)=-2∑k=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+n≡0 (mod 2) 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)=n2∑k=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+1⋅1μev-1=(λeuλeu+1+1μev-1)1λμeu+v+1
by 2vexu+yv, we obtain
(2.29)2exuλeu+1⋅veyvμev-1=λv2⋅2e(1+x-y)uλeu+1⋅2ey(u+v)λμeu+v+1+ve(y-x)vμev-1⋅2ex(u+v)λμeu+v+1.
By (1.3) and the Taylor theorem, we have
(2.30)2ex(u+v)λμeu+v+1=∑m=0∞∑n=0∞En+m(x;λμ)umm!⋅vnn!.
Applying (1.2), (1.3), and (2.30) to (2.29), we get
(2.31)∑m=0∞∑n=0∞Em(x;λ)Bn(y;μ)umm!⋅vnn! =λ2∑m=0∞∑n=0∞[∑k=0m(mk)Em-k(1+x-y;λ)En+k(y;λμ)]umm!⋅vn+1n! +∑m=0∞∑n=0∞[∑k=0n(nk)Bn-k(y-x;μ)Em+k(x;λμ)]umm!⋅vnn!,
which means
(2.32)∑m=0∞∑n=0∞Em(x;λ)Bn+1(y;μ)n+1⋅umm!⋅vn+1n! =∑m=0∞∑n=0∞[λ2∑k=0m(mk)Em-k(1+x-y;λ)En+k(y;λμ) +1n+1∑k=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)=n2∑k=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 [14].