Melham, Ray(State University of West Georgia, 2010)
Jacobi's two-square theorem states that the number of representations of a positive integer k as a sum of two squares, counting order and sign, is 4 times the surplus of positive divisors of k congruent to 1 modulo 4 ...
Melham, Ray(University of Waterloo, Ontario, Canada, 2009)
In 1987 Knopfmacher and Knopfmacher published new infinite product expansions for real numbers 0 < A < 1 and A > 1. They called these expansions Engel product expansions. At that time they had difficulty finding rational ...
The Fibonacci identity belongs to a family of identities where each identity contains only one product on the right side. In this paper we give this family together with two other such families. We also state two ...
There are many Fibonacci identities to be found in short informal articles in the early editions of The Fibonacci Quarterly. See, for example, (1) and (2). The aim of the authoprs was to gather Fibonacci identities from ...
We present identities that we feel can be regarded as higher order analogues of the well-known identity F; + F;+l = Hn+l. We give three theorems corresponding to the powers 4, 6, and 8. We also state two conjectures that ...