On (p,q)-Bernoulli, (p,q)-Euler and (p,q)-Genocchi Polynomials

Abstract

In the present paper, we introduce a new kind of Bernoulli, Euler and Genocchi polynomials based on the (p; q)-calculus and investigate their some properties involving addition theorems, di¤erence equations, derivative properties, recurrence relationships, and so on. We also derive (p; q)-analogues of some known formulae belong to usual Bernoulli, Euler and Genocchi polynomials. Moreover, we get (p; q)-extension of Cheon's main result in [6]. Furthermore, we discover (p; q)-analogue of the main results given earlier by Srivastava and Pintér in [26].

Apostol type (p, q): Frobenius-Euler polynomials and numbers

“…Duran et al [7] defined Apostol type (p, q)-Bernoulli polynomials B (α) n (x, y; λ : p, q) of order α, the Apostol type (p, q)-Euler polynomials E (α) n (x, y; λ : p, q) of order α and the Apostol type (p, q)-Genocchi polynomials G (α) n (x, y; λ : p, q) of order α by the following generating functions:…”

Unified (<em>p, q</em>) -analog of Apostol type polynomials of order <em>a</em>

“…In recent years, Corcino [5] studied on the (p; q)-extension of the binomial coe¢ cients and also derived some properties parallel to those of the ordinary and q-binomial coe¢ cients, comprised horizontal generating function, the triangular, vertical, and the horizontal recurrence relations, and the inverse and the orthogonality relationships. Duran et al [6] considered (p; q)-analogs of Bernoulli polynomials, Euler polynomials and Genocchi polynomials and acquired the (p; q)-analogues of known earlier formulae. Duran and Acikgoz [7] gave (p; q)-analogue of the Apostol-Bernoulli, Euler and Genocchi polynomials and derived their some properties.…”

“…Duran et al [6] considered (p; q)-analogs of Bernoulli polynomials, Euler polynomials and Genocchi polynomials and acquired the (p; q)-analogues of known earlier formulae. Duran and Acikgoz [7] gave (p; q)-analogue of the Apostol-Bernoulli, Euler and Genocchi polynomials and derived their some properties. Gupta [10] proposed the (p; q)-variant of the Baskakov-Kantorovich operators by means of (p; q)-integrals and also analyzed some approximation properties of them.…”

“…Also q-analogs of Apostol type polynomials and numbers were introduced and discussed by several authors, see [4; 16; 20]. Moreover, (p; q)-analog of Apostol type polynomials and numbers were de…ned and investigated by Duran and Acikgoz in [7]. Uni…cation of classical polynomials and Apostol type polynomials were considered in [13; 14; 23-25].…”

“…(jzj < when = 1; jzj < jlog ( )j when 6 = 1) n (x; y; : p; q) of order and the Apostol-Genocchi polynomials G ( ) n (x; y; : p; q) of order are, respectively, de…ned by Duran and Acikgoz [7], as follows:…”