GB

Recently, in 2014, Sury published a Fibonacci-Lucas identity in the Monthly. It turned out that the identity had appeared earlier (as Identity 236 in Benjamin and Quinn's book: Proofs that count: The art of combinatorial proof). When I tried to prove it using my usual telescoping method, I found its connection with one of the oldest Fibonacci identities due to Lucas in 1876. I also found many generalizations and analogous identities for other Fibonacci type sequences and polynomials. This small paper has been accepted in the Fibonacci Quarterly.

Here is a link to a preprint: Analogues of a Fibonacci-Lucas Identity

Update: Its has appeared. The ref is: Analogues of a Fibonacci-Lucas identity, Fibonacci Quart., 54 (no. 2), 166-171, (2016)

I use the approach of my earlier paper on Telescoping: In praise of an elementary identity of Euler.

I am pleased, because I have thought of getting a paper in the Fibonacci Quarterly since I was in high school, and feel lucky I found something they found acceptable!

Dec 22, 2012: It is Ramanujan's 125th birthday, but how many of his famous identities do you know? Here we examine a method to conjecture two very famous identities that were conjectured by Ramanujan, and later found to be known to Rogers.

Here is a link: How to Discover the Rogers-Ramanujan Identities.

This was presented to a some high school math teachers in a conference. I tried to write it in a way that it could be understood by a motivated high school student.

Update (May 26, 2015): The article has been published. Here is a reference. Resonance, 20 (no. 5), 416-430, (May 2015).

Update (January 18, 2014): This article has been accepted for publication in Resonance, a popular science magazine aimed at the undergraduate level.