Have I mentioned that Christian Krattenthaler is the world’s leading authority on determinants? One benefit of such a reputation is people from all walks of mathematics write to him to take his help in proving things involving determinants and matrices. When those cannot be obtained as special cases of known determinants, or if they are unable to do so, they write to him.

This paper arose from a similar situation. Two mathematicians Feng and Xu wrote to Christian. They wanted to prove a certain matrix is invertible. To prove that, they could show the determinant is non-zero. Small experiments led to guessing a formula for the determinant. Christian asked me to try proving it, but I got nowhere. Then he proved it. In my opinion, this spoilt the feng shui of the paper (since it became Feng-Krattenthaler-Xu!).

At any rate, this was a hypergeometric determinant, and their proof used a complicated multiple series identity. By following this proof, we could extend it to an elliptic determinant identity. In fact, we also got a transformation formula for determinants which was quite beautiful.

A very, very special case of our determinant is what is called a Sylvester matrix. Here is a screen grab from the paper, which immortalises my friendship with Christian.

Hint. The cleverness in the paper came from getting the above past the referee. The mathscinet reviewer totally missed the joke. (Requires access to mathscinet).

This paper has been published. Here is the link: The determinant of an elliptic Sylvesteresque matrix (Gaurav Bhatnagar and Christian Krattenthaler), SIGMA, 14 (2018), 052, 15pp.

Further notes

• 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.)

• I presented 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. 
• See here for my first collaboration with Christian.