The generalized Pell (p, i)-numbers and their Binet formulas, combinatorial representations, sums

Abstract

The theory of generalized Pell p-numbers was introduced by Stakhov and then have been studied by several authors. In this paper. we consider the usual Pell numbers and as similar to the Fibonacci p-numbers, we give fair generalization of the Pell numbers, which we call the generalized Pell (p, i)-numbers for 0 <= i <= p. First we give relationships between the generalized Pell (p, i)-numbers and give the generating matrices for these numbers. Also we derive the generalized Binet formulas, sums, combinatorial representations and generating function of the generalized Pell p-numbers. Also using matrix methods, we derive all explicit formula for file sums of the generalized Fibonacci p-numbers. Finally, we derive relationships between generalized Pell (p, i)-numbers and their sums and permanents of certain matrices. (c) 2007 Elsevier Ltd. All rights reserved.