A new family of symmetric and generating functions of binary products of (p,q)-numbers at consecutive and nonconsecutive terms

In this study, by making use of the symmetrizing operator ?k+1a1a2 we introduce a new theorem. By using this theorem we give a new class of generating functions of the products of (p, q)-Fibonacci numbers, (p, q)-Lucas numbers, (p, q)-Pell numbers, (p, q)-Pell Lucas numbers, (p, q)-Jacobsthal numbers and (p, q) -Jacobsthal Lucas numbers at consecutive and nonconsecutive terms and the products of these (p, q)-numbers with Mersenne numbers at consecutive and nonconsecutive terms.