Friday, November 16, 2018

A bibasic Heine transformation formula

While studying chapter 1 of Andrews and Berndt's Lost Notebook, Part II, I stumbled upon a bibasic Heine's transformation. A special case is Heine's 1847 transformation. Other special cases include an identity of Ramanujan (c. 1919), and  a 1966 transformation formula of Andrews. Eventually, I realized that it follows from a Fundamental Lemma given by Andrews in 1966. Still, I'm happy to have rediscovered it. Using this formula one can find many identities proximal to Ramanujan's own $_2\phi_1$ transformations.

And of course, the multiple series extensions (some in this paper, and others appearing in another paper) are all new.

Here is a preprint.

Update (November 10, 2018). The multi-variable version has been accepted for publication in the Ramanujan Journal. This has been made open access. It is now available online, even though the volume and page number has not been decided yet. The title is: Heine's method and $A_n$ to $A_m$
transformation formulas.

Here is a reprint.


UPDATE (Feb 11, 2016). This has been published. Reference (perhaps to be modified later): A bibasic Heine transformation formula and Ramanujan's $_2\phi_1$ transformations, in Analytic Number Theory, Modular Forms and q-Hypergeometric Series, In honor of Krishna Alladi's 60th Birthday, University of Florida, Gainesville, Mar 2016,  G. E. Andrews and F. G. Garvan (eds.), 99-122 (2017)

The book is available here. The front matter from the Springer site.

UPDATE (June 16, 2016).  The paper has been accepted to appear in: Proceedings of the Alladi 60 conference held in Gainesville, FL. (Mar 2016), K. Alladi, G. E. Andrews and F. G. Garvan (eds.)

Here is a video of a talk I presented at the Alladi 60 Conference. March 17-21, 2016.


Monday, August 27, 2018

Mathematics and Life: A Speech



On August 27, 2018, I was invited by the Millennium School, Noida to their investiture ceremony. I have previously taught mathematics to Class 11 students in another campus of the school. At that time, I instituted a "Mathguru Prize" for one or two students who did well in mathematics in Grade 10. (The first winner was my student, Ayush Tripathi, who was in the first graduating batch of the school). Every year I buy some books to be given to the winner, with a copy for the school library. 

Here is a speech I gave at the occasion (with some editing). 

***

I am a mathematician, so I speak here only in terms of mathematics. Today we will be awarding the Mathguru prize to two very bright students. The first thing I wish to tell you is something they know very well.

To get 100% in math, you have to do two things. 
  1. Solve all the problems in the book
  2. Write the solutions in a way that others can understand it. Even you should be able to understand what you have written if you read the solution after 6 months. 
The second thing I wish to tell you is something which Professor Littlewood said. Littlewood was a famous mathematician, who played a big part in Ramanujan's life. He said that if you are trying to solve a really hard problem, then you may not make much progress in a year or two. But you will certainly make a lot of progress in 10 years or so.

Keep this in mind when choosing problems to solve.  Know that even if the problem is very tough, if you keep at it for years, you will make a lot of progress.

Finally, the third thing I wish to share are some words of George Polya, another famous mathematician. Polya said:
Beauty in mathematics is seeing the truth without effort.
So one must aspire to understand things so well, that we can see the beauty of it without any effort. The same goes when we are presenting something that we have understood. 

I speak in terms of mathematics, but I speak not only of mathematics. Much of what I said is applicable in other domains of life.

Best wishes and good luck to all of you, as you pursue your aspirations.

***

PS. I may have been influenced a bit in the way I phrased certain things by a book I just finished reading for the nth time. The book is called Shibumi, written by Trevanian. 

Sunday, July 08, 2018

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.