The determinant of an elliptic, Sylvesteresque matrix

My second determinant project with Christian Krattenthaler.

The determinant of the Sylvester matrix corresponding to the polynomials
\[
x^2+2s_1x+s_1^2 = (x+s_1)^2
\]
and
\[
x^3+3 s_2 x^2 +3s_2^2 x + s_2^3 = (x+s_2)^3
\]
is given by
\[
\det
\begin{pmatrix}
1 & 2s_1 & s_1^2 & 0 & 0\\
0 & 1& 2s_1 & s_1^2 & 0\\
0 & 0 &1 & 2s_1 & s_1^2 \\
1 & 3s_2 &  3s_2^2 & s_2^3 & 0\\
0& 1 & 3s_2 &  3s_2^2 & s_2^3 \\
\end{pmatrix}
= (s_1-s_2)^6.
\]

The determinant is $0$ when $s_1$ and $s_2$ are both $1$. In general, if the determinant of a Sylvester matrix is $0$, then this indicates that the two polynomials have a common root.

Here is an abstract of our paper.

We evaluate the determinant of a matrix whose entries are  elliptic hypergeometric terms and whose form is reminiscent of Sylvester matrices. In particular, it generalizes the determinant evaluation above. A hypergeometric determinant evaluation of a matrix of this type has appeared in the context of approximation theory, in the work of  Feng, Krattenthaler and Xu.  Our determinant evaluation is an elliptic extension of their evaluation, which has two additional parameters (in addition to the base $q$ and nome $p$ found in elliptic hypergeometric terms).  We also extend the evaluation to a formula transforming an elliptic determinant into a multiple of another elliptic determinant. This transformation has two further parameters. The proofs of the determinant evaluation and the transformation formula require an elliptic determinant lemma due to Warnaar, and the application of two $C_n$ elliptic formulas that extend Frenkel and Turaev's $_{10}V_9$ summation formula and $_{12}V_{11}$ transformation formula, results due to Warnaar, Rosengren, Rains, and Coskun and Gustafson.   

This paper has been published in Sigma. Here is a link:
The determinant of an elliptic Sylvesteresque matrix (with Christian Krattenthaler), SIGMA, 14 (2018), 052, 15pp.

I presented this paper in Combinatory Analysis 2018, a conference in honor of George Andrews' 80th birthday conference. Here is a picture from Andrews' talk. (The picture inside the picture is of Freeman J. Dyson.)

Next I expect to present the same paper in a Summer Research Institute on $q$-series in the University of Tianjin, China.

A long version (with lots of background information) was presented in our "Arbeitsgemeinschaft "Diskrete Mathematik" (working group in Discrete Mathematics) Seminar, TU-Wien and Uni Wien, on Tuesday, June 5, 2018. 

How to discover the exponential function

Another article on the "How to discover/guess/prove/..." series written for a high school audience. The basic idea is to find a function whose derivative is itself, and to find the power series which satisfies this. Then messing with it to guess it must be the exponential function. No proofs, in fact, it is outrageously un-rigourous.  I hope the editor allows it.

I try to include only the most beautiful items, and state facts which I feel every high school student should know, even if they doesn't appear formally in the syllabus. 

Update (Nov 2017). The article was published in the November issue of At Right Angles. A nice surprise was Shailesh Shirali's companion article which gives some graphical intuition to complement the algebraic computations in my article. Here is the link to a reprint. 

Abstract

If a function is such that its derivative is the function itself, then what would it be? Some interesting mathematical objects  appear while trying to answer this question, including a power series, the irrational number $e$ and the exponential function $e^x$. The article ends with a beautiful formula that  connects $e$, $\pi$, the complex number $i=\sqrt{-1}$, $1$ and $0$.

Update: 15/June/2017. I was wondering what happened to this article, and the editor said he had sent some comments from the referee which were yet to be incorporated. I resent the article after incorporating the referee's comments, and now this article is slated to appear in the November issue of At Right Angles. Time to think about the next article in the series.

Here is a link to the updated preprint. Please do give comments.