Showing posts with label Ramanujan. Show all posts
Showing posts with label Ramanujan. Show all posts

Monday, August 17, 2020

Andrews' approach to conjecture the Rogers-Ramanujan identities

 If something is worth doing, then I suppose it is worth doing again. 

I had previously written an article on how to discover the Rogers-Ramanujan identities. That was based on ideas of Dick Askey. In this talk I presented an introduction to partitions  presenting many results of Euler and ending with George Andrews' approach to discover the Rogers-Ramanujan identities. This approach was given in his Number Theory book, and it seems that it is not as well known as it should be. A notation that my collaborator Hartosh Singh Bal and I use to gain intuition is also explained here. 

 



Abstract


The two Rogers-Ramanujan identities were sent by Ramanujan to Hardy in a letter in 1913. As an example, here is the first Rogers-Ramanujan identity:

$$1+\frac{q}{(1-q)}+\frac{q^4}{(1-q)(1-q^2)}+\frac{q^9}{(1-q)(1-q^2)(1-q^3)}+\cdots $$

$$=\frac{1}{(1-q)(1-q^6)(1-q^{11})\dots}\times \frac{1}{(1-q^4)(1-q^{9})(1-q^{14})\dots}$$


They look less forbidding when interpreted in terms of partitions, which is how MacMahon considered them. A partition of a number  is a way of writing it as an unordered sum of other numbers. Unordered means that $2+3$ and $3+2$ are considered the same. For example, 

$$5 = 4+1 + 3+2 = 3+1+1 =  2+1+1+1$$ 

are partitions  of $5$. (Two partitions of $5$ are missing in this list; can you find them?) The theory of partitions is an attractive area of mathematics, where many complicated formulas are rendered completely obvious by making the `right picture'. However, while each side of the Rogers--Ramanujan identities are represented naturally in terms of partitions, they are still far from obvious.


In this talk, we will introduce partitions, explain how to enumerate them systematically, represent them graphically, and write their generating functions. We present an experimental approach  to discover the Rogers-Ramanujan identities. This approach is due to Professor George Andrews of Penn State University.


Thursday, January 31, 2019

Orthogonal polynomials associated with continued fractions

My first joint paper with Professor Mourad Ismail. This has been a most interesting collaboration. Mourad taught me what to do on the sidelines of a series of meetings around the world. Most of these meetings were organized by the OPSF activity group of SIAM, one of the most interesting and diverse group of mathematicians and physicists. It began with a couple of meetings in Maryland in July 2016. The next one was in Hong Kong in July 2017, followed by a week long visit of Mourad to Austria (Oct 2017), where he gave me an exclusive, one-on-one, tutorial. Next we met in a summer school on $q$-series in Tianjin university in July-Aug 2018. Finally, we finished up things when I met him in Baltimore at the joint AMS meetings (Jan 2019), followed by a weekend trip to Orlando, right after visiting UF in Gainesville.  This work was presented in Baltimore in a special session on continued fractions.

I hope it is the first in a series on Orthogonal Polynomials. There is much to learn and much to do.

Here is a link to the preprint on ArXiv.

***


Orthogonal polynomials associated with a continued fraction of Hirschhorn

Gaurav Bhatnagar and Mourad E. H. Ismail

Abstract

We study orthogonal polynomials associated with a continued fraction due to Hirschhorn.
Hirschhorn's continued fraction contains as special cases the famous Rogers--Ramanujan continued fraction and two of Ramanujan's generalizations. The orthogonality measure of the set of
polynomials obtained has an absolutely continuous component. We find generating functions, asymptotic formulas, orthogonality relations, and the Stieltjes transform of the measure. Using standard generating function techniques, we show how to obtain formulas for the convergents of Ramanujan's continued fractions, including a formula that Ramanujan recorded himself as Entry 16 in Chapter 16 of his second notebook.

***
Here is a picture of Mourad with me in Tianjin (July-Aug 2018).



The picture below is the conference group photo from Hong Kong (July 2017).


Mourad is seated in the front row second from the left. Many of the leading lights of the OPSF world are in this picture.


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.

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

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.

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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.)




Tuesday, May 26, 2015

How to Discover the Rogers-Ramanujan Identities

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.


Monday, December 22, 2014

How to prove Ramanujan's q-Continued Fractions

The photograph of a page in Ramanujan's Lost Notebook where he expanded a ratio of two series in terms of three continued fractions. These three are among the continued fractions explained in this paper.

Its the 125th year of Ramanujan's birth, but how many of his formulas do you know? Here is an opportunity to get familiar with 9 of Ramanujan's continued fraction formulas. These include the three continued fractions that appear in the Lost Notebook in the above photograph. 

Abstract:
By using Euler's approach of using Euclid's algorithm to expand a power series into a continued fraction, we show how to derive Ramanujan's $q$-continued fractions in a systematic manner.

A Preprint of this expository paper is now available from arXiv. The latest version fixes a typo. The final version appears in this book. You may wish to buy/access the entire volume from the AMS, its really an amazing piece of work.

Update (Sept 7, 2018): I presented this topic in IISER, Mohali, after adding a few ideas from the recent joint work with Mourad Ismail. Here is the presentation.

Update (December 20, 2014): Published in Contemporary Mathematics: Ramanujan 125, K. Alladi, F. Garvan, A. J. Yee (eds.)  627, 49-68 (2014)

Update (July 17, 2013): Accepted for publication in the proceedings of Ramanujan 125, where I presented this paper.




Thursday, February 06, 2014

Interview in Annulus - Hindu College math department magazine

This is an email interview with Annulus, a magazine taken out by Acuity, the mathematics society run by mathematics students of Hindu College, Delhi University. Most of it was published in the magazine. I thought it was meant as a tribute to Saroj Bala Malik, who taught me 4 classes when I attended Hindu, from 1984-87, but they edited out the question about her in the final article.  


Tell us about your time at Hindu College. What made you pursue Mathematics here?

I wanted to become a mathematician. I joined Hindu college because it was the best college where I managed to get admission to study Math. It was the best! And I had the most wonderful time.

Hindu was a most liberal place, where a lot of leeway and freedom was given to students to figure out their own approach to life. Teachers did not impose upon us. The students came from all kinds of economic and social backgrounds, which was great for me, because I had gone to a somewhat elitist school. The cafeteria those days was really a park in front of the hostel, with a few chairs, but plenty of sunshine. A lot of people (even from other colleges) just hung around. A lot of time was spent in ‘CafĂ© Hons’.

The key highlights was the annual ‘trip’ which was 4-5 days of concentrated fun, followed by discussions of what all happened there for the rest of the year. Plus, of course, Mecca. One year in Mecca, we took out a daily magazine called ‘The Quark that Quakes’, consisting of mathematical puzzles, limericks and bad jokes. It was a big hit.

What career option were you looking for when you decided to take up Mathematics?

I wanted to be a research mathematician. I knew that all that I had studied in school was essentially stuff known to Newton and Archimedes many centuries ago. I wanted to reach the frontiers, whatever that meant. I had no idea what it takes to discover and prove your own theorem. I just thought it would be cool to have one I can call my own!

Questions on whether I could become rich, or even survive financially, didn’t really enter my head. Perhaps the practice—prevalent in Hindu—of treating our friends’ money and possessions as our own, contributed to this attitude.

How hard was it to make it to IIT Delhi back then? Any tips that you would like to share with the students?

I don’t recall having studied at all for the IIT entrance. The exam was so tricky that it was fair game for anyone. About 250 students took the test, and 20 were selected. My rank was 2, so I suppose I did quite well! The test required understanding the basic ideas/definitions rather than extensive knowledge of the subject. In fact, I recall that one of the questions was to state and prove your favorite theorem, so they were looking to see if you liked math and what you liked in it.

An idea that works for me is to find one book that gives a historical overview of a particular subject. After going through it in a week or two, I am able to understand what’s happening for the entire semester. There are books like this for algebra, analysis, complex analysis, number theory—you just have to find one that you like.

If you do this, then you can begin to appreciate the beauty of the subject, and are able to understand why you are doing what you are doing. The subject becomes easy, and you will be able to answer the kind of questions that examiners are looking for.  You will also be able to slog through the difficult theorems and proofs, because you have a sense of where you are going.

Any Dr. Saroj Bala Malik memories?

SBM has been one of the most influential teachers in my life. In our first class, she asked questions and I was one of the two or three students who answered her. The same day I met her at the bus stop, and she recognized me. I told her I want to do research in mathematics. And from that day on, she took it upon herself to help me in whichever way she can.

Our entire batch was her favorite. It wasn’t that she was all mushy or soft on us. She practiced what is called ‘tough love’. She worked hard at her teaching and demanded we work hard at our learning. She asked a lot of questions. She praised us when we could answer, and, well, took our trip, when we couldn’t. She went out of her way to fund our activities, and covered for us in case we got into trouble with other faculty members!

However, there were a few rocky moments too. I used to organize a weekly puzzle contest. Every week, I would post a new puzzle, along with the answer of the previous puzzle, and the names of those who got it right. All went well for a few weeks. Until one day, when SBM got (in my view) the wrong answer! Her view was that the question was wrongly worded. She demanded that I correct my mistake. We fought long and hard. It wasn’t pretty—but it was interesting, and kind of fun!

The question above was not included in the printed interview.

Tell us about your time at IIT Delhi.

I spent only a year in IIT Delhi. IIT was mostly about very brilliant lecturers and a fun hostel life. But I did not learn much there, because I did not work very hard. Most of the time I was busy applying abroad. I got a scholarship, and left without finishing my MSc. But there was one important aspect of my year at IIT. I met the person whom I eventually married. So all-in-all it turned out to be a good year!

What was the experience at Ohio State University like?

Ohio State was truly the best educational experience possible. There were many famous mathematicians who taught me, among the best people in their area. My Ph.D. advisor was Steve Milne, who had given the first combinatorial proof of the Rogers-Ramanujan identities, thereby solving a long-standing problem. My story with him was similar to SBMs. He gave a talk about his area, and showed how he had extended a famous result of Ramanujan. Right after his talk I went and told him I wanted to work with him for my Ph.D.

The biggest truth I learnt at Ohio State was that mathematics is learnt by doing mathematics. Your professor can be the most brilliant lecturer in the world (or not), but you will learn only if you do all the problems of the textbook on your own.

In our department, there were people from all over the world; plus, I interacted with hundreds of American students as their Teaching Assistant. Living in the US, with enough money to have some fun, and hanging out with many people of many different countries—I think that was the most amazing and enjoyable part of doing a Ph.D. in the US.

From Modern School to Ohio State University, how has Mathematics shaped your life?

When I was in class 11, I took a Math Olympiad exam, where I happened to crack a problem I had never seen before. And I felt wonderful! I had got an exhilarating high, and it happened because I got a creative idea in mathematics. I figured that I want to have this feeling again and again, for the rest of my life. So I decided to become a mathematician.

From Modern to Hindu and IIT, and on to Ohio State, I stayed with this for nearly 15 years.

But I forgot about this after returning to India after my Ph.D. After a year in ISI, Delhi, I took up a job in the industry and thought I cannot pursue math any more. This went on for a few years, and I was totally miserable, and didn’t know why. Then one fine day I got a project to write a math book, and got reminded about this exhilarating feeling again!

That is when I realized that math is what keeps me happy. Now, despite a full time job, I look to do something mathematical, whether it is research, teaching, writing books, articles or papers, or even reading math books.  The thrill that comes from solving a math problem—especially a tough math problem—has never gone away. That is what keeps me happy.

The generation of today is somewhat reluctant to pursue Mathematics as a subject. What will be your advice to the students who are looking to or currently pursuing Mathematics?

My advice would be to do as much as you can handle, and then a little more. If you cannot do math just for the love of it, then consider the following 5 things that a math education does:

#1: It teaches you to question. 

Why prove theorems, when they have been proved a million times before? Because, as our teachers tell us, you need to see for yourself that the theorem is true. This is so unlike the real world, where often people tend to prove things to you by intimidation, or by asserting their authority. However, unless you question things, you will not get creative ideas. And in math, we question everything!

#2: It teaches you to reason. 

We learn to apply logic to prove theorems. In the real world, people frequently confuse a statement with its converse, and don’t believe that if ‘A or B’ is true, then both ‘A and B’ could also be true! Your capability to reason correctly and think clearly will quickly get you noticed.

#3: It teaches you to communicate clearly. 

The practice of understanding mathematical definitions and proving theorems teaches us that words have a precise meaning. Being able to communicate clearly is perhaps the most important requirement for success.

#4: It teaches you to think abstractly. 

As you grow in responsibility in an organization, you need to deal with a large number of facts. However, the time to deal with them is finite. At this time the ability to think abstractly becomes hugely important. Abstraction is a key requirement of any leadership position whether it is in academia, industry or the government!

#5. It gives you confidence. 

If you have done well in mathematics, or even reasonably well, you should take a huge amount of confidence from this. For someone who is a master of epsilon-delta proofs, point-set topology, or abstract algebra, most management or technical problems at the workplace are a piece of cake!

In short, a good mathematical education gives you an unfair advantage in the real world. So if you can handle it, go for it!


Friday, September 16, 2011

The q-analog of the Gamma Function


I have begun reading Bruce Berndt's "Ramanujan's Notebooks", Part III. Here is a small morsel from Ramanujan's table: Entry 1(ii) of Chapter 16 of his Notebooks. Its a discovery proof of the limit of the $q$-Gamma function, as $q$ goes to 1. In my humble opinion, this is easier than the usual proof (due to Gosper) which appears in Gasper and Rahman.

Download PDF or read below.

The $q$-analog of the Gamma Function

The objective of this note is to show how to arrive at the definition of the $q$-analog of the Gamma function. To do so, we "discover" the limit:
\begin{equation}\label{entry1ii} \newcommand{\qrfac}[2]{{\left({#1}; q\right)_{#2}}} \lim_{q\to 1} \frac{\qrfac{q}{\infty}}{(1-q)^x \qrfac{q^{x+1}}{\infty}}= \Gamma (x+1).
\end{equation}
Recall the limit definition of the Gamma function (from, for example Rainville [5, p. 11]):
$$\Gamma(x+1):=\lim_{n\to \infty} \frac{n! n^x}{(x+1)(x+2)\cdots (x+n)}.$$
To derive \eqref{entry1ii}, we find a $q$-analog of this limit. To that end, we use:
  1.  $\displaystyle \lim_{q\to 1} \frac{\qrfac{q}{n}}{(1-q)^n} = n!$ 
  2. $\displaystyle \lim_{q\to 1} \left(\frac{1-q^n}{1-q}\right)^x =n^x$ 
  3. $\displaystyle \lim_{q\to 1} \frac{\qrfac{q^{x+1}}{n}}{(1-q)^n}=(x+1)(x+2)\cdots (x+n)$  
 Thus, we have
 \begin{align*}\require{cancel} \Gamma(x+1)&= \lim_{n\to \infty} \frac{n! n^x}{(x+1)(x+2)\cdots (x+n)}\cr
& = \lim_{n\to \infty} \lim_{q\to 1} \frac{(1-q)^n}{\qrfac{q^{x+1}}{n}}\cdot \frac{\qrfac{q}{n}}{(1-q)^n} \cdot \left(\frac{1-q^n}{1-q}\right)^x\cr &= \lim_{q\to 1}\lim_{n\to \infty} \frac{\cancel{(1-q)^n}}{\qrfac{q^{x+1}}{n}}\cdot \frac{\qrfac{q}{n}}{\cancel{(1-q)^n}} \cdot \left(\frac{1-q^n}{1-q}\right)^x\cr &= \lim_{q\to 1} \frac{\qrfac{q}{\infty}}{\qrfac{q^{x+1}}{\infty}} \frac{1}{(1-q)^x}. \end{align*}
Here, we assume that the limits can be interchanged, and $|q|<1$. This completes the derivation of \eqref{entry1ii}.

 Given the relation \eqref{entry1ii}, we can define the $q$-Gamma function, for $|q|<1$, as \begin{equation}\label{qgammadef} \Gamma_q (x)= \frac{\qrfac{q}{\infty}}{(1-q)^{x-1} \qrfac{q^{x}}{\infty}}.
\end{equation}

Remarks. The proof by Gosper, reported by Andrews [1] and reproduced in Gasper and Rahman [4] uses Euler's Product definition of the Gamma Function. Equation \eqref{entry1ii} is Entry 1(ii) in Berndt [2, ch.16]. The limit definition is entry 2293 in Carr's book [3], so Ramanujan had access to it.

References
  1. G. E. Andrews, $q$-Series: Their development and application in analysis, number theory, combinatorics, physics and computer algebra, NSF CBMS Regional Conference Series, 66 1986.
  2. B. C. Berndt, Ramanujan's Notebooks, Part III, Springer Verlag, New York, 1991.
  3. G. S. Carr, Formulas and Theorems of Pure Mathematics, 2nd ed., Chelsea, NY, 1970.
  4. G. Gasper and M. Rahman, Basic Hypergeometric Series, Encyclopedia of Mathematics And Its Applications 35, Cambridge University Press, Cambridge, 1990; Second Ed. (2004). 
  5. E. D. Rainville, Special Functions, Chelsea, NY (1960).