New identities and relations derived from the generalized Bernoulli polynomials, Euler polynomials and Genocchi polynomials
© Kurt; licensee Springer. 2014
Received: 10 October 2013
Accepted: 3 December 2013
Published: 6 January 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
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 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 .
In this work, we give a different form of the analogue of the Srivastava-Pintér addition theorem.
2 Main theorems
Proof The proof of these propositions can be found from (1)-(6). □
If we take necessary operation, comparing the coefficients of , we have (13). □
From this last equality, we have (14). □
By using the Cauchy product, compression of the results, we have (15). □
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.
Comparing the coefficients of , we have (18). □
Comparing the coefficients of , we have (19). □
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.
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