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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). [ABSTRACT FROM AUTHOR]

In this paper we consider transformation formulas for We derive reciprocity theorems for the sums arising in these transformation formulas and investigate certain properties of them. With the help of the character analogues of the Euler-Maclaurin summation formula we establish integral representations for Hardy-Berndt character sums s3,p(d, c : χ) and s4,p(d, c : χ). [ABSTRACT FROM AUTHOR]

Mahmudov (2012, 2013) introduced and investigated some q-extensions of the q-Bernoulli polynomials 픅(ɑ) n,q(x, y) of order ɑ, the q- Euler polynomials E n,q(ɑ)(x, y) of order ɑ, andthe q-Genocchi polynomials G n,q(ɑ)(x, y) of order ɑ. In this paper, we give some identities for픅(ɑ) n,q(x, y), Gn,q(ɑ)(x, y), and En,q(ɑ)(x, y) and the recurrence relations between these polynomials. This is an analogous result to the q-extension of the Srivastava-Pint'er addition theorem in Mahmudov (2013). [ABSTRACT FROM AUTHOR]

In the present paper, the authors consider the Pν, 2-transform as a generalization of the Widder potential transform and the Glasser transform. The Pν, 2-transform is obtained as an iteration of the L2-transform with itself. Many identities involving these transforms are given. By making use of these identities, a number of new Parseval-Goldstein-type identities are obtained for these and many other well-known integral transforms. The identities proved in this paper are shown to give rise to useful corollaries for evaluating infinite integrals of special functions. Some examples are also given as illustration of the results presented here. [ABSTRACT FROM AUTHOR]

The article discusses congruences for generalized q-Bernoulli polynomials. It presents properties of p-adic q-L-function of two variables, wherein applications of such properties provide general classes of congruences for generalized q-Bernoulli polynomials. Bernoulli numbers are said to have originated from the study of finite sums of a given power of consecutive integers. The mathematical notation of Bernoulli numbers and polynomials and the definition of the Dirichlet L-function associated with X are illustrated.

The main purpose of this paper is to study generating functions of the q-Genocchi numbers and polynomials. We prove a new relation for the generalized q-Genocchi numbers, which is related to the q-Genocchi numbers and q-Bernoulli numbers. By applying Mellin transformation and derivative operator to the generating functions, we define q-Genocchi zeta and l-functions, which are interpolated q-Genocchi numbers and polynomials at negative integers. We also give some applications of the generalized q-Genocchi numbers. [ABSTRACT FROM AUTHOR]