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

A positive integer n is called a balancing number if there exists a positive integer r such that 1 + 2 + ⋯ + (n-1) = (n+1) + (n+2) + ⋯ + (n+r). The corresponding value r is known as the balancer of n. If n is a balancing number, then 8n²+1 is a perfect square, and its positive square root is called a Lucas-balancing number. For any integer k ≥ 2, let ₙ⁽ᵏ⁾ ≥ -(k-2) denote k-generalized Fibonacci sequence which starts with 0, ,1(k terms) where each next term is the sum of the k preceding terms. In this paper, we investigate all balancing and Lucas-balancing numbers that can be expressed as the product of two k-generalized Fibonacci numbers.