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Hecke operators type and generalized ApostolBernoulli polynomials
Fixed Point Theory and Applications volume 2013, Article number: 92 (2013)
Abstract
In this paper, we construct some Hecketype 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 ApostolBernoulli type polynomials which are eigenfunctions of these Hecketype operators. Moreover, we derive many useful identities related to these operators and polynomials.
MSC:11M35, 30B40, 30B50.
1 Introduction
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 Heckeeigenforms, which satisfy many interesting arithmetic relations. For more details on Hecke operators, see [1, 2]. Recently, the Hurwitz zeta functions and the ApostolBernoulli 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 Hecketype operators on the ring of \mathbb{C}[x]. We study fundamental properties of these operators. We derive relations between these operators, the Hurwitz zeta functions and ApostolBernoulli 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 Hecketype 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 m\ge 1 we have
where {B}_{n}(x) are the wellknown Bernoulli polynomials. Conversely, from the paper [11] 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 [11] approach will be generalized to the ApostolBernoulli 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 d\in \mathbb{C}\mathrm{\setminus}\{0\} and {\xi}_{N} be a primitive root of unity of order N. We consider the functions {\chi}_{a,N}:\mathbb{N}\to \mathbb{C} given by
We define the partial Hecketype operators associated to {\chi}_{a,N} and d as follows:
The total Hecketype 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 d\in \mathbb{C}\mathrm{\setminus}\{0\}. Assume that a\not\equiv 0(modN). Then we have the following properties for the operators {T}_{a,d,N}:

(i)
The operator {T}_{a,d,N} is linear and preserves the degree in \mathbb{C}[x].

(ii)
\mathrm{\forall}m\ge 1,
{T}_{a,d,N}\left({x}^{m}\right)=\{\begin{array}{cc}{S}_{d,0},\hfill & m=0;\hfill \\ {a}^{m}{x}^{m}+{a}^{m}{\sum}_{v=0}^{m1}\left(\genfrac{}{}{0ex}{}{m}{v}\right){S}_{d,mv}({\chi}_{a,N})\cdot {x}^{v},\hfill & m\ge 1,\hfill \end{array}
where
and {S}_{d,0}={S}_{d,0}({\chi}_{a,N}).
Proof A simple computation gives the linearity of the operator {T}_{a,d,N}, so we omit it. Since a\not\equiv 0(modN), we can see easily that
From the above equation, we obtain
Let us show how to compute {T}_{a,d,N}({x}^{m}). For m=0, we get
We end the proof by induction. Let m\ge 1, 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 {T}_{a,d,N} in the canonical basis {\beta}_{m}=(1,x,{x}^{2},\dots ,{x}^{m}), 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 m\in \mathbb{N}, let {\beta}_{m}=(1,x,{x}^{2},\dots ,{x}^{m}) be the canonical ℂbasis of {\mathbb{C}}_{m}[x]. Then the matrix corresponding to the operator (restricted to {\mathbb{C}}_{m}[x]) in the basis {\beta}_{m} is given by
Remark 2.3 For any positive integer a\ge 2, the eigenvalues {S}_{d,0},{a}^{1}{S}_{d,0},{a}^{2}{S}_{d,0},\dots ,{a}^{m}{S}_{d,0} 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 {({P}_{n,d,N})}_{n\in \mathbb{N}}, which is a sequence of eigenpolynomials of (1). For more details, see the next section.
Theorem 2.4 The operators {T}_{a,d,N} and {T}_{b,d,N} commute if a\equiv b\equiv 1(modN).
Proof We consider the linear operators {T}_{a,d,N} and {T}_{b,d,N} and we must show that
for all a\equiv 1(modN). This equality is obvious when N=1. For N\ge 2, we have the following equalities:
The linearity of the of the operator {T}_{a,d,N} implies that
Then we deduce
By setting k={k}_{1}+b{k}_{2}, we obtain
Finally, we get our desired equality
□
3 New characterization of ApostolBernoulli type polynomials
As an application of our main results, we study the polynomials P\in \mathbb{C}[x] satisfying the functional equation
where a\equiv 1(N) and fixed integer n\ge 1.
Theorem 3.1 Let a, N be positive integers and d\in \mathbb{C}\mathrm{\setminus}\{0\} such that a\equiv 1(N). Then we have the following properties:

(i)
There exists an unique sequence of monic polynomials {P}_{n,d,N}\in \mathbb{C}[x] with deg{P}_{n,d,N}=n such that
{T}_{a,d,N}({P}_{n,d,N})={a}^{n}{P}_{n,d,N}. 
(ii)
Polynomials {P}_{n,d,N}(x) are eigenfunctions for the operators {T}_{n,d,N} with eigenvalues {N}^{n}\zeta (n,\frac{1}{N}), that is
{T}_{d,N}({P}_{n,d,N})(x)={N}^{n}\zeta (n,\frac{1}{N}){P}_{n,d,N}(x),
where \zeta (s,x)={\sum}_{k\ge 0}\frac{1}{{(x+k)}^{s}} 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 {({P}_{n,d,N})}_{n\in \mathbb{N}}. For this end, we take two different monic polynomials {P}_{n,d,N} and {R}_{n,d,N} of degree n satisfying (2).
Suppose that {P}_{n,d,N}(x){R}_{n,d,N}(x)={\mathrm{\Delta}}_{m}(x)={A}_{0}{x}^{m}+{A}_{1}{x}^{m1}+\cdots , where 1\le m<n and {A}_{0}\ne 0. From (2) and the definition of {T}_{a,d,N}, we can write
and
Subtracting (4) from (3), we get
Identifying the coefficients of {x}^{m} on both sides, we have {A}_{0}={a}^{n}{A}_{0}, but this contradicts our stipulations that {A}_{0}\ne 0, m<n, and a\ge 2. Hence, the proof of (i) is completed.
We prove (ii). It is easy to see that
and putting a=1+kN, we obtain
□
Thanks to Theorem 2.4, we find the generating function of polynomials {({P}_{n,d,N})}_{n\in \mathbb{N}} satisfying (2). More precisely, we have the following theorem.
Theorem 3.2 For all n\ge 1, we have the following results:

(i)
\frac{d}{dx}{P}_{n,d,N}(x)=n{P}_{n1,d,N}(x).

(ii)
{P}_{n,d,N}(d)={\xi}_{N}^{1}{P}_{n,d,N}(0).

(iii)
The difference formula of {({P}_{n,d,N})}_{n\in \mathbb{N}} is given by
{P}_{n,d,N}(x+d){\xi}_{N}^{1}{P}_{n,d,N}(x)=\{\begin{array}{cc}n{x}^{n1},\hfill & N=1;\hfill \\ (1{\xi}_{N}^{1}){x}^{n},\hfill & N\ge 2.\hfill \end{array}
Proof We prove (i). From (2) and the definition of the operators {T}_{a,d,N}, we have
we derive this equation and obtain
Since \frac{{P}_{n,d,N}^{\mathrm{\prime}}(x)}{n} is monic with degree n1, from Theorem 3.1(i), we arrive at
We prove (ii). For N\ge 2, by taking {\chi}_{a,N}(k)={\xi}_{N}^{k} and a=N+1, we have
In the above equation, putting x=0 and x=d, respectively, we arrive at
and
Multiplying each side of (7) by {\xi}_{N} and then substrate it from (6), we have the following relation:
Since {\xi}_{N}^{N+1}={\xi}_{N}, we obtain that {P}_{n,d,N}(d)={\xi}_{N}^{1}{P}_{n,d,N}(0) for all n\ge 1 and N\ge 2.
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 {\xi}_{N} and then substrate (8) from (9), we arrive to
From Theorem 3.2(ii) and equality {P}_{0,d,N}(d)={P}_{0,d,N}(d)=1, 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 a\equiv 1(N), the generating function of {({P}_{n,d,N})}_{n\in \mathbb{N}} is given by
Proof Let N\ge 2 integer and write
Using the difference formula in Theorem 3.2, we get
We consider the generating function
thus,
We compare the coefficients of {x}^{n} in the above equation and we obtain
In particular, if we take m\equiv 0(N), then we have
Therefore, we note that the polynomials {P}_{n,d,N}(x){P}_{n,d,N}(0,{\xi}_{N}) and {Q}_{n,d}(x,{\xi}_{N}){Q}_{n,d}(0,{\xi}_{N}) are equal on the infinite set \{x=md:\phantom{\rule{0.25em}{0ex}}\text{with}m\equiv 0(modN)\}. Then we can write for all x\in \mathbb{C},
Now, by derivation on x we get
We obtain the equality
Hence, we obtain the generating function of {P}_{n,d,N}. □
Remark 3.4 The case d=1 of Theorem 3.3 recovers the socalled generalized Bernoulli and Euler polynomials, which are studied in [9].
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 {P}_{n,\psi ,d}(x,{\xi}_{N}) 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 N=1 is trivial, we omit it. For N\ge 2, by using equation (10), we obtain
Taking the coefficients of \frac{{t}^{n}}{n!} 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 \frac{{t}^{n}}{n!} in both sides in the last equality, we obtain the desired result. □
Theorem 4.2 For any positive integers N and a such that a\equiv 1(N). Then the polynomials {P}_{n,\psi ,d} are eigenpolynomials for Hecke type operators {T}_{a,d,N}.
Proof
From Theorem 3.1, we have
then for any integer b we have
Summing over all 1\le b\le d
Therefore, by linearity of the Hecke operator we obtain
We then obtain our formula
□
Theorem 4.3 For all integer n\ge 1, the difference formula of ({P}_{n,\psi ,d})(x,{\xi}_{N}) is given by
Proof From Theorem 3.2, we know that, for all n\ge 1,
Let N\ge 2. By using Theorem 4.1, we get
and
Therefore,
Let N=1. By using Theorem 4.1, we get
and
Therefore,
□
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Acknowledgements
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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Aygunes, A.A., Bayad, A. & Simsek, Y. Hecke operators type and generalized ApostolBernoulli polynomials. Fixed Point Theory Appl 2013, 92 (2013). https://doi.org/10.1186/16871812201392
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DOI: https://doi.org/10.1186/16871812201392
Keywords
 Hecke operators
 Hurwitz zeta function
 generalized ApostolBernoulli type polynomials
 Bernoulli polynomials
 Euler polynomials