Abstract
The aim of this paper is to give not only the matrix representation of partial Hecketype operators by means of Bernoulli polynomials and Euler polynomials, but also functional equations and differential equations related to partial Hecketype operators and special polynomials. By using these functional equations and differential equations, we derive some identities associated with special polynomials and partial Hecketype operators. Moreover, we find several useful identities and relations using the partial Hecke operators.
MSC: 05B20, 11B68, 11F25.
Keywords:
Bernoulli polynomial; Euler polynomial; partial Hecketype operator; total Hecketype operator; Hurwitz zeta function; partial zeta function1 Introduction
Recently, there have been many applications of Bernoulli polynomials and Euler polynomials in differential equations, in analytic number theory and in engineering. Highorder linear differentialdifference equations have also been solved in terms of Bernoulli polynomials. These polynomials are also related to several linear operators. In this paper, we investigate and derive several new identities related to the Hecketype operators and generating functions for special polynomials.
Recently, many authors introduced and investigated the following generating functions which give us the Bernoulli polynomials and the Euler polynomials , respectively:
and
For , (1) and (2) are reduced to the generating functions for the Bernoulli numbers and the Euler numbers , respectively (cf.[14]), and see also the references cited in each of these earlier works.
The multiplication formulas for the Bernoulli and Euler polynomials are given as follows:
and for odd m,
(cf.[58]), and see also the references cited in each of these earlier works.
The Bernoulli polynomials satisfy the following wellknown identity:
where m and n are positive integers (cf.[5,9,10]).
Bayad et al. [11] introduced and systematically studied the following family of partial Hecketype operators on .
Throughout this paper, we use the following notations: . Let and .
where
Lemma 1.1 [[11], p.114, Lemma 1]
For anysuch that, we have the following properties:
(ii) By induction,
where
(iii) For any, letbe the canonical ℂbasis of
Then the matrixcorresponding to the operator (restricted to) in the basisis represented by:
Consequently, for a given integern, there is only one monic polynomialwith degreeninxsatisfying the functional equation (6).
The operator satisfies the following equation:
For , from (6) and Lemma 1.1, we know that is a monic polynomial (cf.[11]).
Remark 1.2 Equations (3) and (4) are closely related to the functional equation of (6). For and , equation (6) is reduced to and , respectively. For fixed , we know that there is only one monic polynomial satisfying (6) by Lemma 1.1, and there already exist the functional equations as (3) and (4).
The total Hecketype operators, associated with partial Hecketype operators, are defined by Bayad et al. [[11], p.112, Eq. (1.6)] as follows:
Theorem 1.3[11]
Polynomialsare eigenfunctions for the operatorswith eigenvalues, that is,
whereis the Hurwitz zeta function defined by
2 Differential equations related to the partial Hecketype operators and special polynomials
In this section, we derive some ordinary and partial differential equations not only for a generating function, but also for partial Hecketype operators. We also give a functional equation for the generating function. We set
We now give an explicit formula of the generating function as follows.
Theorem 2.1[11]
Generating functions for the polynomialsare given by
The polynomials are the socalled BernoulliEulertype polynomials.
We derive the following partial differential equation for as follows:
Proof By using (7), for , we obtain
Therefore, by comparing the coefficients of on both sides of equation (8), we have the desired result.
For , we apply the same process. So, we omit it. □
We set the following differential equation:
Theorem 2.3
Proof We make some arrangement (9) and obtain
Therefore,
From the above equation, we get
By comparing the coefficients of on both sides of the above equation, we have the desired result. □
Remark 2.4 In Theorem 2.3, we obtain a convolution formula for the polynomials . If we substitute into Theorem 2.3, then we get a convolution formula for the Euleriantype numbers (cf.[10,13]).
Higherorder partial differential equation for is given by the following theorem.
where
Proof Taking vth derivative of the operator , with respect to x, we obtain the following higherorder partial differential equation:
Using Theorem 2.2, we get
Thus, we get the desired result. □
3 Matrix representations of partial Hecketype operators
In this section, we give some numerical examples for the matrix representations of the operator . For the basis , our matrix representations contain Bernoulli polynomials and Euler polynomials for the operators and , respectively. Therefore, we need the following lemmas.
Proofs of Lemma 3.1 and Lemma 3.2 have been given by many authors (among others) (cf.[2,4,8,10]).
In a special case, substituting into (iii) in Lemma 1.1 and using Lemma 3.1, we get
According to the above equation, we are ready to give the main result of this section by the following theorem.
Theorem 3.3The matrixcorresponding to the operator (restricted to) in the basisis represented by Bernoulli polynomials as follows:
Setting (iii) in Lemma 1.1 and using Lemma 3.2, we obtain
If , then we obtain another main result by the following theorem.
Theorem 3.4Letabe an odd number. The matrixcorresponding to the operator (restricted to) in the basisis represented by Euler polynomials as follows:
4 Some applications of total Hecketype operators
In this section, we give some applications related to eigenvalues for the total Hecketype operators of and . We derive many new identities which are related not only to the total Hecketype operators, but also to the Riemann zeta function, the Hurwitz zeta function, Bernoulli and Euler numbers, Euler identities and the convolution of Bernoulli and Euler numbers and polynomials.
Throughout this section, we use the following notation:
The partial zeta function is defined by
where , and () (cf.[4,8,10,12,14]).
Theorem 4.1The polynomialsare eigenfunctions for the operatorswith eigenvalues, that is,
whereis a partial zeta function.
Proof
Therefore,
Substituting , and into the above equation, after using Theorem 1.3, we arrive at the desired result. □
Theorem 4.2Letwith. Then we have
Proof Putting in Theorem 1.3 and using
we have
We recall from the definition of and that we have
(cf. [[10], p.96]). Combining (11) and (12), we get
If we replace n by 2n in the above equation, we obtain
From the work of Srivastava and Choi [[4], p.98], we recall that
By substituting (15) and (16) into (14), after some elementary calculations, we arrive at the desired result. □
Proof Combining (14) and (16), we easily complete the proof of the theorem, that is,
□
By using (10) and (17), we obtain a convolution formula (Euler identity) for Bernoulli numbers.
Proof Since the lefthand sides of (10) and (17) are equal, the righthand sides of (10) and (17) must be equal. Thus, we obtain
After some elementary calculation in the above equation, we get the desired result. □
Observe that the proof of (18) is also given in [4].
(cf. [[4], p.131]). By using (14) and (19), we obtain
Thus, the proof is completed. □
Proof Consider that n is replaced by in (13), we have
(cf. [[4], p.99, Eq. (21)]). Hence, we have
Thus, the proof is completed. □
Proof Note that, for all , we have
(cf. [[4], p.99, Eq. (22)]). By using (21) and (24), we have
Thus, the proof is completed. □
Proof Substituting into Theorem 1.3 and by , we have
If n is replaced by 2n in the above equation, we get
By using (16), we have
Thus, the proof is completed. □
Theorem 4.9Letwith. Then we have
Proof By using (26), (15) and (16), we have
Thus, the proof is completed. □
Proof By using (26) and (19), we have
Thus, the proof is completed. □
Proof By replacing n by in (25), we have
By substituting (29) into (22), we get
Thus, the proof is completed. □
Proof By using (29) and (24), we have
Thus, the proof is completed. □
By comparing (20) and (23) or (28) and (30), we arrive at the following result.
Corollary 4.13
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors completed the paper together. All authors read and approved the final manuscript.
Acknowledgements
Dedicated to Professor Hari M Srivastava.
All authors are partially supported by Research Project Offices Akdeniz Universities. We would like to thank the referees for their valuable comments.
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