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

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A representation of -Bernstein polynomials in terms of -Jacobi polynomials is obtained.

Using the Sturm–Liouville theory in q-difference spaces, we prove the finite orthogonality of q-Pseudo Jacobi polynomials. Their norm square values are then explicitly computed by means of the Favard theorem.

In this paper, we introduce (p, q)-Sturm-Liouville problems and prove that their solutions are orthogonal with respect to a (p, q)-integral space. We then present some illustrative examples for this kind of problems and obtain the (p, q)-hypergeometric representation of the polynomial solutions together with their 3-term recurrence relations. We also compute the norm square value of the polynomial solutions and obtain their limiting cases in the sequel.