On ( p , q ) $(p,q)$ -classical orthogonal polynomials and their characterization theorems

Abstract

In this paper, we introduce a general (p, q)-Sturm-Liouville difference equation whose solutions are (p, q)-analogues of classical orthogonal polynomials leading to Jacobi, Laguerre, and Hermite polynomials as (p, q) → (1, 1). In this direction, some basic characterization theorems for the introduced (p, q)-Sturm-Liouville difference equation, such as Rodrigues representation for the solution of this equation, a general three-term recurrence relation, and a structure relation for the (p, q)-classical polynomial solutions are given.MSC: Primary 34B24; secondary 39A70

On the (p, q)–Chebyshev Polynomials and Related Polynomials

“…and obtain a generating function as are generated by (20), and then we will derive the result, which provides bilateral generating functions for (p, q)-Fibonacci polynomials and (p, q)-Chebyshev polynomials of the first kind given by (5).…”

“…Note that, for details of (p, q)-analysis, one can see [15][16][17][18] and p → 1 in these properties, we have the property of q-calculus in [19]. On the other hand, for more details related to (p, q)-orthogonal polynomials, readers look at the papers in [20,21].…”

Representation of ( p , q ) $(p,q)$ -Bernstein polynomials in terms of ( p ,

“…Very recently Milovanović et al [11] considered a -analog of the beta operators and using it proposed an integral modification of the generalized Bernstein polynomials. -analogs of classical orthogonal polynomials have been characterized in [12]. …”