4.1 Apostol-Euler-Nörlund polynomials
The Apostol-Euler-Nörlund polynomials are defined, for , by the generating function
(25)
We consider complex numbers α, λ and x such that and . We first prove the analytic continuation of , and we compute special values of .
Theorem 4.1 Let λ be a complex number with . The function has analytic continuation to the whole complex plane, except possible simple poles at non-positive integers with residue
(26)
Moreover, the function has analytic continuation as an entire function to the whole complex plane, and for all non-negative integer n, we have
(27)
Proof Let λ be a complex number other than 1. We choose a real number so that . We split the integral from zeta’s definition as
As and , for all complex number α, the integral
defines an entire function of .
We substitute in the first integral, for , the generating expansion of the Apostol-Euler-Nörlund polynomials
It hence follows that
For a given non-negative integer m, we know that
and . This proves the analytic continuation of the function as an entire function to the whole complex plane, and
(28)
□
Remark 4.2 Equality (28) has been proved, using different method, by Luo [[11], Theorem 2.1].
Now, by applying our Theorem 2.4, Theorem 2.5 and Theorem 4.1, we deduce the reduction and duality formulas for the Apostol-Euler-Nörlund polynomials .
Theorem 4.3 For any non-negative integers n, N and complex numbers α, with , we have
and
(29)
4.2 Explicit formula for the Apostol-Euler-Nörlund polynomials
In particular for , and by using formula (29), we get this explicit formula for the Apostol-Euler-Nörlund polynomials.
Proposition 4.4 Let α, λ be complex numbers with , . For any positive integer N, we have
(30)
The above formula, when combined with the well-known equality
gives this other explicit expression
(31)
4.3 Differential formula for the Apostol-Euler-Nörlund polynomials
We consider the differential operator . From the series representation (2.2) of the Hurwitz-Lerch zeta functions, we have
(32)
Using Theorem 4.1 at , we obtain the differential formula
(33)
4.4 Bernoulli-Nörlund polynomials
In this subsection we investigate convolution formulas for the Bernoulli-Nörlund polynomials . We define them by the generating function
The Nörlund polynomials are , see [12].
We set , and we introduce the modified Hurwitz-Lerch zeta function defined by the integral representation
(34)
Theorem 4.5 ()
Let α, x be complex numbers with . The function has analytic continuation to the whole complex plane, except simple poles at with residue , for any non-negative integer (when ). Moreover, for all non-negative integer , we have
(35)
Proof
The proof is similar to that of Theorem 4.1. We split the integral
As , for all complex number α, the integral
defines an entire function of .
We use in the first integral, for , the generating function of the Bernoulli-Nörlund polynomials
It hence follows that
For a given non-negative integer (if ), we have a simple pole at with residue .
For a given integer , it is known that
and . Then we obtain
This proves the analytic continuation of the function as an entire function to the whole complex plane, except simple poles at , if .
This completes the proof of the theorem. □
Remark 4.6 For any positive integer α, the relation (35) recovers the results in the paper [13].
Observe that
Hence, from Theorem 2.4 and Theorem 4.1, we deduce the following reduction and duality formulas.
Theorem 4.7 Under the hypothesis of Theorem 2.4, we have
By use of Theorem 4.1 and equalities (36), (37), we get the convolution identities on the Bernoulli-Nörlund polynomials.
Theorem 4.8 For any non-negative integer n and any positive integer N, we have the convolution identity and its dual version
Note that from Theorem 4.8 we have the following corollaries.
Corollary 4.9 Under the hypothesis of Theorem 4.8, for , and indeterminate α, we obtain, among the so-called Nörlund polynomials , the formula
Corollary 4.10 Under the hypothesis of Theorem 4.8, for , we obtain the following formula:
Corollary 4.11 Under the hypothesis of Theorem 4.8, and if , , we obtain Euler’s identity type on the Bernoulli numbers of order N
This formula is an analogue of the nice identity for Bernoulli numbers obtained by Euler and given by
and its generalization to Bernoulli numbers of arbitrary level N, cf. [14, 15].
Corollary 4.12 Under the hypothesis of Theorem 4.8, taking in the dual formula (39), we obtain