On balancing and Lucas-balancing numbers expressible as product of two k-Fibonacci numbers

Abstract

A positive integer n is called a balancing number if there exists a positive integer r such that \(1 + 2 + \cdots + (n-1) = (n+1) + (n+2) + \cdots + (n+r)\). The corresponding value r is known as the balancer of n. If n is a balancing number, then \(8n^{2}+1\) is a perfect square, and its positive square root is called a Lucas-balancing number. For any integer \(k \ge 2\), let \(\{F_{n}^{(k)} \}_{n \ge -(k-2)}\) denote the k-generalized Fibonacci sequence, which starts with \(0, \dots ,1\)(k terms) where each next term is the sum of the preceding k terms. In this paper, we investigate all balancing and Lucas-balancing numbers that can be expressed as a product of two k-generalized Fibonacci numbers.