- Open Access
New identities and relations derived from the generalized Bernoulli polynomials, Euler polynomials and Genocchi polynomials
Advances in Difference Equations volume 2014, Article number: 5 (2014)
In this article, we give some identities for the q-Bernoulli polynomials, q-Euler polynomials and q-Genocchi polynomials and recurrence relations between these polynomials in (Mahmudov in Discrete Dyn. Nat. Soc. 2012:169348, 2012; Mahmudov in Adv. Differ. Equ. 2013:1, 2013).
MSC:05A10, 11B65, 28B99, 11B68.
1 Introduction, definitions and notations
In the usual notations, let and denote, respectively, the classical Bernoulli and Euler polynomials of degree n in x, defined by the generating functions
where and are, respectively, the Bernoulli and Euler numbers of order n.
Carlitz first extended the classical Bernoulli polynomials and numbers, Euler polynomials and numbers . There are numerous recent investigations on this subject by many authors. Cheon , Kurt , Luo , Luo and Srivastava , Srivastava et al. [6, 7], Tremblay et al. , and Mahmudov [9, 10].
Throughout this paper, we always make use of the following notation: ℕ denotes the set of natural numbers and ℂ denotes the set of complex numbers.
The q-numbers and q-factorial are defined by
respectively, where , , . The q-polynomials coefficient is defined by
The q-analogue of the function is defined by
The q-binomial formula is known as
The q-exponential functions are given by
From these forms, we easily see that . Moreover, , , where is defined by
The above q-standard notation can be found in .
Mahmudov defined and studied properties of the following generalized q-Bernoulli polynomials of order α and q-Euler polynomials of order α as follows .
Let , and . The q-Bernoulli numbers and polynomials in x, y of order α are defined by means of the generating functions
The q-Euler numbers and polynomials in x, y of order α are defined by means of the generating functions
The q-Genocchi numbers and polynomials in x, y of order α are defined by means of the generating functions
It is obvious that
From (2), (4) and (6), it is easy to check that
In this work, we give a different form of the analogue of the Srivastava-Pintér addition theorem.
More precisely, we prove
2 Main theorems
Proposition 2.1 The generalized q-Bernoulli and q-Euler polynomials satisfy the following relations:
Proposition 2.2 For , the following relations hold true:
Proof The proof of these propositions can be found from (1)-(6). □
Theorem 2.3 The generalized q-Genocchi polynomials satisfy the following recurrence relation:
Proof In (6) for , we take the q-derivative of the generalized q-Genocchi polynomials according to t. We note that
If we take necessary operation, comparing the coefficients of , we have (13). □
Theorem 2.4 There is the following relation for the q-Genocchi polynomials:
Proof From (6) and , we have
Thus, we obtain
From this last equality, we have (14). □
Theorem 2.5 There is the following identity for the q-Genocchi polynomials:
Proof From , we write as
By using the Cauchy product, compression of the results, we have (15). □
Theorem 2.6 There are the following relationships for the q-Genocchi polynomials:
Proof Proof of (16), we write
Comparing the coefficients of , we have (16). The proof of (17) is similar to that of (16). □
3 Explicit relation between the q-Bernoulli polynomials and q-Genocchi polynomials
In this section, we prove two interesting relations between the q-Bernoulli polynomials of order α and the q-Genocchi polynomials of order α.
Theorem 3.1 There is the following relation between q-Genocchi polynomials and q-Bernoulli polynomials
Proof From (6), we deduce that
Comparing the coefficients of , we have (18). □
Theorem 3.2 There is the following relation between q-Bernoulli polynomials and q-Genocchi polynomials:
Proof From (2), we obtain
Comparing the coefficients of , we have (19). □
Carlitz L: Expansions of q -Bernoulli numbers. Duke Math. J. 1958, 25: 355-364. 10.1215/S0012-7094-58-02532-8
Cheon GS: A note on the Bernoulli and Euler polynomials. Appl. Math. Lett. 2003, 16: 365-368. 10.1016/S0893-9659(03)80058-7
Kurt B:A further generalization of the Bernoulli polynomials and on the 2D-Bernoulli polynomials . Appl. Math. Sci. 2010, 233: 3005-3017.
Luo QM: Some results for the q -Bernoulli and q -Euler polynomials. J. Math. Anal. Appl. 2010, 363: 7-18. 10.1016/j.jmaa.2009.07.042
Luo QM, Srivastava HM: q -Extensions of some relationships between the Bernoulli and Euler polynomials. Taiwan. J. Math. 2011, 15: 241-247.
Srivastava HM, Choi J: Series Associated with the Zeta and Related Functions. Kluwer Academic, London; 2011.
Srivastava HM, Pintér A: Remarks on some relationships between the Bernoulli and Euler polynomials. Appl. Math. Lett. 2004, 17: 375-380. 10.1016/S0893-9659(04)90077-8
Tremblay R, Gaboury S, Fugére BJ: A new class of generalized Apostol-Bernoulli polynomials and some analogues of the Srivastava-Pintér addition theorems. Appl. Math. Lett. 2011, 24: 1888-1893. 10.1016/j.aml.2011.05.012
Mahmudov NI: q -Analogues of the Bernoulli and Genocchi polynomials and the Srivastava-Pintér addition theorems. Discrete Dyn. Nat. Soc. 2012., 2012: Article ID 169348 10.1155/2012/169348
Mahmudov NI: On a class of q -Bernoulli and q -Euler polynomials. Adv. Differ. Equ. 2013., 2013: Article ID 1 10.1186/1687-1847-2013-1
This paper was supported by the Scientific Research Project Administration of Akdeniz University. The author is grateful to the referees for valuable comments. Proceedings of 2nd International Eurasian Conference on Mathematical Sciences and Applications.
The author declares that they have no competing interests.
About this article
Cite this article
Kurt, V. New identities and relations derived from the generalized Bernoulli polynomials, Euler polynomials and Genocchi polynomials. Adv Differ Equ 2014, 5 (2014). https://doi.org/10.1186/1687-1847-2014-5
- Bernoulli numbers and polynomials
- Genocchi polynomials
- generating function
- generalized Bernoulli polynomials
- generalized Genocchi polynomials
- q-Bernoulli polynomials
- q-Genocchi polynomials