ORDINARY GENERATING FUNCTIONS OF BINARY PRODUCTS OF (p,q)-MODIFIED PELL NUMBERS AND k-NUMBERS AT POSITIVE AND NEGATIVE INDICES

Abstract

In this paper, we introduce a operator in order to derive some new symmetric properties of (p,q)-modified Pell numbers and we give some new generating functions of the products of (p,q)-modified Pell numbers with k-Fibonacci and k-Lucas numbers, k-Pell and k-Pell Lucas numbers, k-Jacobsthal and k-Jacobsthal Lucas numbers at positive and negative indices. By making use of the operator defined in this paper, we give some new generating functions of the products of (p,q)-modified Pell numbers with k-balancing and k-Lucas-balancing numbers.

Generating Functions of Even and Odd Gaussian Numbers and Polynomials

“…5. [17,18] Let n be positive integer and [19,20] Given a function f on , n the divided difference operator is defined as follows:…”

Some new properties of generalized ( p , q ) -numbers and generating functions for certain products

“…(1.1) and (1.2) we thus have Table 1 for the bivariate Mersenne polynomials {M n (x, y)} n∈N and bivariate Mersenne Lucas polynomials {m n (x, y)} n∈N . In modern science, there are a huge interest in (p, q)-numbers and thier properties in [8,10,14,18,19]. There are many generalizations of these numbers, the generalized (p, q)-Fibonacci numbers {f p,q,n (α, β, γ)} n∈N , generalized (p, q)-Pell numbers {l p,q,n (α, β, γ)} n∈N , and generalized (p, q)-Jacobsthal numbers {C p,q,n (α, β, γ)} n∈N [16] are one of them, f p,q,0 = α, f p,q,1 = β + γp, and f p,q,n = pf p,q,n−1 + qf p,q,n−2 , (…”