GB

We consider Spiral Determinants of the kind
$$\text{det}\left(
\begin{matrix}
{16}&{15}&{14}&{13}\\
{5}&{4}&{3}&{12}\\
{6}&1&{2}&{11}\\
{7}&{8}&{9}&{10}
\end{matrix}
\right)
$$
and
$$\text{det}
\left(
\begin{matrix}
{17}&{16}&{15}&{14}&{13}\\
{18}&{5}&{4}&{3}&{12}\\
{19}&{6}&1&{2}&{11}\\
{20}&{7}&{8}&{9}&{10}\\
{21}&{22}&{23}&{24}&{25}
\end{matrix}
\right)
$$

where the entries spiral out from the center. Christian Krattenthaler, who is one of the greatest experts on determinants, tells the story of how he came across such determinants and how he went about discovering the formulas for such determinants. The preprint is available on arxiv.

I have wanted to work with Christian ever since my Ph.D. days, when I tried to generalize a matrix inversion due to him. Finally, we have a joint paper. This also means that my Erdos number has come down from 4 to 3.

The picture above is from Christian's course on "Bijections" which I had an opportunity to attend in the University of Vienna during the period October 2016 to January 2017.

Update: April 26, 2017 The paper has been accepted and will appear in Linear Algebra and its Applications. Here is a preprint on arxiv.
Update: May 10, 2017. The paper is published online. The reference is:
G. Bhatnagar and C. Krattenthaler, Spiral Determinants, Linear Algebra Appl., 529 (2017) 374-390.
Here is a link to the publisher's site: https://www.sciencedirect.com/science/article/pii/S0024379517302719

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!