A NOTE ON THE (p; q)-HERMITE POLYNOMIALS

In this paper, we introduce a new generalization of the Hermite polynomials via (p; q)-exponential generating function and investigate several properties and relations for mentioned polynomials including derivative property, explicit formula, recurrence relation, integral representation. We also de…ne a (p; q)analogue of the Bernstein polynomials and acquire their some formulas. We then provide some (p; q)hyperbolic representations of the (p; q)-Bernstein polynomials. In addition, we obtain a correlation between (p; q)-Hermite polynomials and (p; q)-Bernstein polynomials. 1. INTRODUCTION During the last three decades, applications of quantum calculus based on q-numbers have been studied and investigated succesfully, densely and considerably (see [7; 8]). In conjunction with the motivation and inspiration of these applications and introduction of the (p; q)-numbers, many mathematicians and physicists have extensively developed the theory of post quantum calculus based on (p; q)-numbers along the traditional lines of classical and quantum calculus. Agyuz et al. [2] presented some novel results of multiplications of (p; q)-Bernstein polynomials and derived several new relations with related to (p; q)-Gamma and (p; q)-Beta functions. Duran et al. [3] introduced a new class of Bernoulli, Euler and Genocchi polynomials based on the (p; q)-calculus and investigated their many properties involving addition theorems, di¤erence equations, derivative properties, recurrence relationships, and so on. Furthermore, they derived (p; q)-extension of Cheon’s main result and (p; q)-analogue of the Srivastava and Pintér’s addition theorem. Sadjang [9] investigated some properties of the (p; q)-derivative and the (p; q)-integration and presented two appropriate polynomials basis for the (p; q)-derivative, and then he derived various properties of these bases. As an application, he provided two (p; q)-Taylor formulas for polynomials. Furthermore, he stated the fundamental theorem of (p; q)-calculus and proved the formula of (p; q)-integration by part. The (p; q)-number is de…ned as [n]p;q = p q p q (0 < q < p 5 1) . Note that [n]p;q = p n 1 [n]q=p ; where [n]q=p stands for q-number known as [n]q=p = (q=p) 1 (q=p) 1 . One can see that (p; q)-number is closely related to q-number with this relation [n]p;q = p n 1 [n] q p . By appropriately using this obvious relation between the q-notation and its variant, the (p; q)-notation, most (if not all) of the (p; q)-results can be derived from the corresponding known q-results by merely changing the parameters and variables involved (see [4], [5]). The (p; q)-derivative operator Dp;q;xf (x) of a function f with respect to x given as Dp;q;xf (x) := Dp;qf (x) = f (px) f (qx) (p q)x (Dp;qf (x) when x 6= 0; f 0 (0) when x = 0) (1.1) is a lineer operator and satis…es the following property Dp;q (f (x) g (x)) = f (px)Dp;qg (x) + g (qx)Dp;qf (x) : (1.2) Corresponding Author 2010 Mathematics Subject Classi…cation11B68, 11B83, 81S40.