- Open Access
Hecke operators type and generalized Apostol-Bernoulli polynomials
Fixed Point Theory and Applicationsvolume 2013, Article number: 92 (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.
Our results are new and useful in applied mathematics and computation, analytic number theory and related areas. There are many reasons for being interested by Hecke-type operators. In particular, these operators are linear operators and are closely related to Raabe’s multiplication theorem [9, 10]. We recall the statement of this theorem, for any positive integer we have
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.
In order to state our results, we fix the following notations and definitions. Let a, N be positive integers and and be a primitive root of unity of order N. We consider the functions given by
We define the partial Hecke-type operators associated to and d as follows:
The total Hecke-type operators associated to N and d are defined by
2 Main results
We have the following results.
Theorem 2.1 Let a, N be positive integers and . Assume that . Then we have the following properties for the operators :
The operator is linear and preserves the degree in .
Proof A simple computation gives the linearity of the operator , so we omit it. Since , we can see easily that
From the above equation, we obtain
Let us show how to compute . For , we get
We end the proof by induction. Let , after an elementary manipulation we obtain
and, therefore, (ii) is satisfied. □
We consider the restriction of the partial Hecke operator to the finite dimensional space
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.
Proposition 2.2 For any , let be the canonical ℂ-basis of . Then the matrix corresponding to the operator (restricted to ) in the basis is given by
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 .
Proof We consider the linear operators and and we must show that
for all . This equality is obvious when . For , we have the following equalities:
The linearity of the of the operator implies that
Then we deduce
By setting , we obtain
Finally, we get our desired equality
3 New characterization of Apostol-Bernoulli type polynomials
As an application of our main results, we study the polynomials satisfying the functional equation
where and fixed integer .
Theorem 3.1 Let a, N be positive integers and such that . Then we have the following properties:
There exists an unique sequence of monic polynomials with such that
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).
Suppose that , where and . From (2) and the definition of , we can write
Subtracting (4) from (3), we get
Identifying the coefficients of on both sides, we have , but this contradicts our stipulations that , , and . Hence, the proof of (i) is completed.
We prove (ii). It is easy to see that
and putting , we obtain
Thanks to Theorem 2.4, we find the generating function of polynomials satisfying (2). More precisely, we have the following theorem.
Theorem 3.2 For all , we have the following results:
The difference formula of is given by
Proof We prove (i). From (2) and the definition of the operators , we have
we derive this equation and obtain
Since is monic with degree , from Theorem 3.1(i), we arrive at
We prove (ii). For , by taking and , we have
In the above equation, putting and , respectively, we arrive at
Multiplying each side of (7) by and then substrate it from (6), we have the following relation:
Since , we obtain that for all and .
We prove (iii). We can write
On the other hand, by using Theorem 3.2(i), we get
We multiply the each side of (9) by and then substrate (8) from (9), we arrive to
From Theorem 3.2(ii) and equality , we get
Therefore, we obtain the desired result. □
Using Theorem 2.4 and Theorem 3.1, we can establish the following result.
Theorem 3.3 For , the generating function of is given by
Proof Let integer and write
Using the difference formula in Theorem 3.2, we get
We consider the generating function
We compare the coefficients of in the above equation and we obtain
In particular, if we take , then we have
Therefore, we note that the polynomials and are equal on the infinite set . Then we can write for all ,
Now, by derivation on x we get
We obtain the equality
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
Let d be a positive integer, ψ be a Dirichlet character modulo d. We associate to ψ, d, N the polynomials defined by the generating function
Then we have the interesting relations.
Theorem 4.1 Let d be a positive integer, ψ be a Dirichlet character modulo d. Then we have the identity
which is equivalent to the following equality:
Proof The proof for is trivial, we omit it. For , by using equation (10), we obtain
Taking the coefficients of in the left and right sides of above equation, we have
Now we prove (ii). By equation (10) we have
then we have
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 .
From Theorem 3.1, we have
then for any integer b we have
Summing over all
Therefore, by linearity of the Hecke operator we obtain
We then obtain our formula
Theorem 4.3 For all integer , the difference formula of is given by
Proof From Theorem 3.2, we know that, for all ,
Let . By using Theorem 4.1, we get
Let . By using Theorem 4.1, we get
Serre J-P: Cours D’arithmétique. PUF, Paris; 1970.
Hellegouarch Y: Invitation aux Mathématiques de Fermat-Wiles. Dunod, Paris; 2001.
Kim T: Note on the Euler numbers and polynomials. Adv. Stud. Contemp. Math. 2008, 17(2):131–136.
Kim T: Symmetry of power sum polynomials and multivariate fermionic p -adic invariant integral on . Russ. J. Math. Phys. 2009, 16(1):93–96. 10.1134/S1061920809010063
Ozden H, Simsek Y, Srivastava HM: A unified presentation of the generating functions of the generalized Bernoulli, Euler and Genocchi polynomials. Comput. Math. Appl. 2010, 60: 2779–2787. 10.1016/j.camwa.2010.09.031
Srivastava HM, Ozarslan MA, Kaanoglu C: Some generalized Lagrange-based Apostol-Bernoulli, Apostol-Euler and Apostol-Genocchi polynomials. Russ. J. Math. Phys. 2013, 20: 110–120.
Srivastava HM, Choi J: Zeta and q-Zeta Functions and Associated Series and Integrals. Elsevier, Amsterdam; 2012.
Srivastava HM: Some generalizations and basic (or q -) extensions of the Bernoulli, Euler and Genocchi polynomials. Appl. Math. Inf. Sci. 2011, 5: 390–444.
Bayad A, Aygunes AA, Simsek Y: Hecke operators and generalized Bernoulli-Euler polynomials. J. Algebra Number Theory, Adv. Appl. 2010, 3: 111–122.
Raabe JL: Zurückführung einiger summen and bestimmten integrale auf die Jacob Bernoullische function. J. Reine Angew. Math. 1851, 42: 348–376.
Lehmer DH: A new approach to Bernoulli polynomials. Am. Math. Mon. 1998, 95: 905–911.
Kim T: Symmetry identities for the twisted generalized Euler polynomials. Adv. Stud. Contemp. Math. 2009, 19: 111–118.
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.
The authors declare that they have no competing interests.
All authors completed the paper together. All authors read and approved the final manuscript.