Rad HAZU, Matematičke znanosti, Vol. 19 (2015), 1-12.

FORMULAS FOR QUADRATIC SUMS THAT INVOLVE GENERALIZED FIBONACCI AND LUCAS NUMBERS

Zvonko Čerin

Abstract.   We improve on Melham’s formulas in [10, Section 4] for certain classes of finite sums that involve generalized Fibonacci and Lucas numbers. Here we study the quadratic sums where products of two of these numbers appear. Our results show that most of his formulas are the initial terms of a series of formulas, that the analogous and somewhat simpler identities hold for associated dual numbers and that besides the alternation according to the numbers (-1)^n(n+1)/2 it is possible to get similar formulas for the alternation according to the numbers (-1)^n(n-1)/2. We also consider twelve quadratic sums with binomial coefficients that are products.

2010 Mathematics Subject Classification.   11B39, 11Y55, 05A19.

Key words and phrases.   (generalized) Fibonacci number, (generalized) Lucas number, factor, sum, alternating, binomial coefficient, product.

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