Hecke operators type and generalized Apostol-Bernoulli polynomials
© Aygunes et al.; licensee Springer 2013
Received: 24 January 2013
Accepted: 26 March 2013
Published: 10 April 2013
In this paper, we construct some Hecke-type operators acting on the complex polynomials space, and we prove their commutativity. By means of this commutativity, we find a new approach to establish the generating function of the Apostol-Bernoulli type polynomials which are eigenfunctions of these Hecke-type operators. Moreover, we derive many useful identities related to these operators and polynomials.
MSC:11M35, 30B40, 30B50.
The Hecke operators have many applications in various spaces like the space of elliptic modular forms, the space of polynomials and others. Many mathematicians applied them to obtain applications in analytic number theory, harmonic analysis, theoretical physics, equidistribution of Hecke points on a family of homogeneous varieties, and cohomology. For instance, Hecke operators are used to investigate and study Fourier coefficients of modular forms, to explore other properties of the Hecke-eigenforms, which satisfy many interesting arithmetic relations. For more details on Hecke operators, see [1, 2]. Recently, the Hurwitz zeta functions and the Apostol-Bernoulli polynomials have been studied by many authors, for example, see (cf. [3–12], the others).
The main motivation of this paper is to introduce and study new Hecke-type operators on the ring of . We study fundamental properties of these operators. We derive relations between these operators, the Hurwitz zeta functions and Apostol-Bernoulli type polynomials.
where are the well-known Bernoulli polynomials. Conversely, from the paper  of Lehmer, it is well known that the Raabe’s theorem gives a characterization of the Bernoulli polynomials. As an application, of the main result of this paper, the Lehmer’s  approach will be generalized to the Apostol-Bernoulli type polynomials. These polynomials plays a central role in the computational number theory.
2 Main results
We have the following results.
The operator is linear and preserves the degree in .
and, therefore, (ii) is satisfied. □
By writing the operator in the canonical basis , and from (ii), we get the corresponding matrix. Using linear algebra, we will see that this matrix representation is useful and gives interesting results.
Remark 2.3 For any positive integer , the eigenvalues of the matrix (1) are distinct. Then from the theory of linear algebra we deduce that the matrix (1) is a diagonalizable. Again, thanks to linear algebra, we know that there exists a sequence of polynomials , which is a sequence of eigenpolynomials of (1). For more details, see the next section.
Theorem 2.4 The operators and commute if .
3 New characterization of Apostol-Bernoulli type polynomials
where and fixed integer .
- (i)There exists an unique sequence of monic polynomials with such that
- (ii)Polynomials are eigenfunctions for the operators with eigenvalues , that is
where is the Hurwitz zeta function.
Proof The existence of a sequence of monic polynomials P is satisfied from Theorem 2.1 and Theorem 2.4.
Now we must observe the uniqueness of . For this end, we take two different monic polynomials and of degree n satisfying (2).
Identifying the coefficients of on both sides, we have , but this contradicts our stipulations that , , and . Hence, the proof of (i) is completed.
Thanks to Theorem 2.4, we find the generating function of polynomials satisfying (2). More precisely, we have the following theorem.
- (iii)The difference formula of is given by
Since , we obtain that for all and .
Therefore, we obtain the desired result. □
Using Theorem 2.4 and Theorem 3.1, we can establish the following result.
Hence, we obtain the generating function of . □
Remark 3.4 The case of Theorem 3.3 recovers the so-called generalized Bernoulli and Euler polynomials, which are studied in .
4 Eigenpolynomials attached to Dirichlet characters
Then we have the interesting relations.
Comparing the coefficients of in both sides in the last equality, we obtain the desired result. □
Theorem 4.2 For any positive integers N and a such that . Then the polynomials are eigenpolynomials for Hecke type operators .
Dedicated to Professor Hari M Srivastava.
All authors are partially supported by Research Project Offices Akdeniz Universities. We would like to thank for referees for their valuable comments.
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