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. 



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, April 30, 2020

Lest we forget them (Edited) - By P. K. Ghosh

An email Asoke and I sent with a link to P K Ghosh's article "Lest we forget them". I have previously written a book review of PKG's novel: On returning to desh.

IITs of India are known worldwide as excellent academic institutions. For several decades, IIT Kanpur was the leader in this group. Those who were fortunate to have studied there, would fondly remember the ambience, the Halls of Residence, the Central Library, the canteens, the lecture hall complex...and certainly the teachers. They were great men, and sought to ignite the spark in kindred spirits among younger souls. Among them, Prof. P. K. Ghosh stands out as one who always maintained a high standard in teaching and research, and demanded the same from his students. The attached booklet gives a glimpse of his time in the Chemistry Department at IIT Kanpur, where he and his students created one of the highest resolution optical spectrometers in the country, an unusual programmable microprocessor for training, data acquisition and display, wrote several monographs which have been praised internationally, among other things. Prof. Ghosh has recounted those days in a delightful booklet "Lest We Forget Them."

Who are the people whom he doesn't want us to forget? What else does he not wish to forget? The mood. The vision. The bravado. The students who will go on to achieve great success and renown.

For details, you must read the booklet.

Here is a link: 

Best wishes,

Asoke and Gaurav 

PS. We would appreciate it if viewers and readers can identify themselves and comment below. Any messages to PKG will be forwarded to him along with your contact details. I will have to `approve' the comment. This is to avoid spam.

PPS. This is an edited version. Some pictures and letters have been edited from here.If you know PKG personally, he may share with you the complete article. Please send a request for the same to Asoke. 

Asoke Chattopadhyay
Department of Chemistry
University of Kalyani
Kalyani 741235
(MSc integrated, IIT Kanpur) 
asoke dot chattopadhyay at gmail

Gaurav Bhatnagar
bhatnagarg at gmail com

Saturday, October 26, 2019

Mummy's 90th birthday

My mother, Dr. Hem Bhatnagar, completed 90 years on October 28, 2019.  We asked her family and friends if they wished to say something in tribute or regarding their association with her. Madhulika  and I have collated them as a book, which was presented to mummy on her birthday. It was meant to be a book of wishes, but it ended up presenting a very multidimensional view of her as a person. It is also inspirational, in parts, and has some interesting historical photographs too.

Click on this sentence to download the book. It will open up as a Google Document, you can save it.

The Sanskar calendar was also released on this occasion. It is made up of her paintings and writings. This will also be available for sale as the annual sanskara calendar. You can download it here

As a PDF this book is available for free. We have got a small number of copies printed for a very limited distribution, only for mom's sisters and brothers. If you want a printed copy for yourself, Sanskara Society will make it available at cost. If you want a copy for yourself, please write to with your request. The cost is 700 (USD 10) plus S/H. 

Tuesday, September 10, 2019

Thank you, Dick

From R to L (facing camera): Ae Ja Yee, Bruce Berndt, Dick Askey, Shaun Cooper, Michael Schlosser, and me
At Alladi 60 conference at a conference reception at the Alladi residence

Howard Cohl and Mourad Ismail created a Liber Amicorum (Friendship Book) to present to Richard Askey. Askey is not well, and he and his wife Liz have moved into a Hospice in Wisconsin. Askey is our leader, the leader of our field, and of the people in the field. 

UPDATE (October 9, 2019): Alex Berkovich and Howard Cohl informed that Dick is no more.

My entry for his book is here: Thank you, Dick

The title is appropriate. When Dick autographed my copy of the book, Special Functions by Andrews, Askey and Roy, he wrote "Thank you for your work, early and now". (Here is my book review of this book.) He gave extensive comments on receiving a draft copy of my unpublished book "Experience Mathematics" and tried to help me get it published. My paper "How to discover the Rogers-Ramanujan Identities" is essentially an expansion of something that took Dick a couple of paragraphs. 

When I returned to mathematics, I hung out with him in many conferences, and he was very supportive, coming for my talks, making remarks. In general, he was very welcoming. I have missed him the last couple of years. 

Sunday, August 11, 2019

On Entry II.16.12 of Ramanujan

Bruce's Return gift: A daily reminder of Ramanujan

In June, 2019, I attended a very inspiring conference, held at the University of Illinois at Urbana-Champaign. This was to celebrate Bruce Berndt's 80th birthday and his retirement. Bruce is the greatest living expert on Ramanujan and one of the nicest people in the world. He is scholarly and an inspiration. His retirement means nothing, he is already traveling around the world and he said he is working on another book with Ae Ja Yee on partitions. My wishes to him were more for myself than for him: Wishing us many, many books and papers from you for many, many years!

Mourad Ismail and I have written a paper for the occasion. I presented it in Illinois in front of a small audience, which included Bruce. This is the second paper with Mourad on the subject of Ramanujan's continued fractions, using techniques he has taught me. I gave a talk on this continued fraction in 2016, in SLC 77. The talk is available here: Some of this material is included in the paper with Mourad. 

Here is a picture from the conference. (More pictures below)

Picture in front of Altgeld Hall, the iconic department of mathematics, UIUC

Here is the announcement of the paper from ArXiv. Click on the link in the title to get to the preprint on ArXiv.
Title: On Entry II.16.12: A continued fraction of Ramanujan
Authors: Gaurav Bhatnagar and Mourad E. H. Ismail
Categories: math.CA
Comments: 15 Pages
MSC-class: 33D45 (Primary), 30B70 (Secondary)
We study a continued fraction due to Ramanujan, that he recorded as Entry 12
in Chapter 16 of his second notebook. It is presented in Part III of Berndt's
volumes on Ramanujan's notebooks. We give two alternate approaches to proving Ramanujan's Entry 12, one using a method of Euler, and another using the theory of orthogonal polynomials. We consider a natural generalization of Entry 12 suggested by the theory of orthogonal polynom
Here is a picture taken at the Banquet.

With Bruce Berndt and Michael Schlosser in the Banquet Hall

Here is a picture from Bruce's office, which I saw courtesy Atul Dixit who had the keys.

A picture of some of the pictures in Bruce's office

I presented this work in the OPSFA2019 conference in Hagenberg, Austria. Here are the slides from the talk in Hagenberg. OPSFA2019 was an amazing conference. I got a chance to hear Christian's concert on the church organ, saw and heard Chihara, met Alan Sokal (among others),  hung out with Michael and Hjalmar and worked with Mourad on this paper and on our next joint paper. Here is a picture from Hagenberg.

The group photo from OPSFA

Sunday, March 24, 2019

Prime number conjectures from the Shapiro class structure

The first of hopefully many joint projects with my childhood friend Hartosh Singh Bal. For many reasons this has been a most exciting collaboration. For one thing, Hartosh and I have been discussing mathematical ideas since Class 11 in Modern School. So it was good to work on something which will lead to something new. For another, Shapiro was Hartosh's number theory Professor at NYU. And for three more reasons, you will have to look at the last section of this paper.

Here is the abstract:
The height $H(n)$ of  $n$, introduced by  Pillai in 1929,  is the  smallest positive integer $i$ such that the  $i$th iterate of Euler's totient function at $n$ is $1$.  H.  N. Shapiro (1943) studied the structure of the set of all numbers at a height. We provide a formula for the height function thereby extending a result of Shapiro. We list steps to generate numbers of any height which turns out to be a useful way to think of this construct. In particular, we extend some results of Shapiro regarding the largest odd numbers at a height. We present some theoretical and computational evidence to show that $H$ and its relatives are closely related to the important functions of number theory, namely $\pi(n)$ and the $n$th prime $p_n$. We conjecture formulas for $\pi(n)$ and $p_n$ in terms of the height function.  
Here is a link to a reprint of the paper.

Prime number conjectures from the Shapiro class structure (with Hartosh Singh Bal), 
UPDATE (Feb 14, 2020): The paper has appeared to appear in INTEGERS: Electronic Journal of Combinatorial Number Theory (Volume 20), #A11, 23pp.

From left to right: Sonit, Hartosh, me, Punya in 1983 or so

Tuesday, February 12, 2019

An infinite family of Borwein-type + - - conjectures

Another collaboration with Michael Schlosser written to celebrate the 80th birthday of Professor George Andrews. The paper's opening paragraph talks about a very interesting event that took place  in a conference celebrating George's birthday in June 2018.

The so-called Borwein conjectures, due to Peter Borwein (circa 1990), were  popularized by Andrews. The first of these concerns the expansion of finite products of the form
into a power series in $q$ and the sign pattern displayed by the coefficients.  In June 2018, in a conference at Penn State celebrating Andrews' 80th birthday,  Chen Wang, a young Ph.D. student studying at the University of Vienna, announced that he has vanquished the first of the Borwein conjectures. In this paper, we  propose another set of Borwein-type conjectures.  The conjectures here are consistent with the first two Borwein conjectures as well as what is known about their refinement proposed by Andrews. At the same time, they do not appear to be very far from these conjectures in form and content.
Our first conjecture considers products of the form
\prod_{i=0}^{n-1} (1-q^{3i+1}) (1-q^{3i+2})
\prod_{j=1}^m \prod_{i=-n}^{n-1} (1-p^jq^{3i+1})(1- p^jq^{3i+2})
These are motivated by theta products. 

Here is a link to a preprint of the paper.
A partial theta function Borwein conjecture, by Gaurav Bhatnagar and Michael Schlosser.

UPDATE (September 16, 2019). The paper has been accepted to appear in the Andrews 80 Special Issue in the Annals of Combinatorics.

Here is a picture from a trip to Hong Kong for an OPSF meeting in June 2017.  From left to right: Heng Huat Chan (Singapore), Michael Schlosser (Vienna), Hjalmar Rosengren (Sweden), Shaun Cooper (New Zealand), me. A special team of Special Functions people from around the world!

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


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.


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.)

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
x^3+3 s_2 x^2 +3s_2^2 x + s_2^3 = (x+s_2)^3
is given by
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 \\
= (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. 

Thursday, June 15, 2017

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


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.

Tuesday, May 09, 2017

WP Bailey Lemmas (Elliptic, multivariable)

After many many years, Michael Schlosser and I wrote another joint paper. We first collaborated in 1995-96 when both of us were Ph.D. students or shortly thereafter. Our joint work was part of his thesis, and published in Constructive Approximation. This time around, I was his post-doc in Vienna from Feb 1, 2016 to Feb 28, 2017.

The picture was taken in Strobl, a favorite place for small meetings and conferences for Krattenthaler's group in the University of Vienna.

In this paper, we give multivariable extensions (over root systems) of the elliptic well-poised (WP) Bailey Transform and Lemma. In the classical (i.e. dimension = 1) case, this work was done by Spiridonov, who in turn extended the work of Andrews and Bailey. It is Andrews' exposition which we found very useful while finding generalizations. We used  previous $q$-Dougall summations due to Rosengren, and Rosengren and Schlosser, and found a few of our own along with some new elliptic Bailey $_{10}\phi_9$ transformation formulas, extending some fundamental formulas given in the classical case by Frenkel and Turaev in 1997. Along the way, we discovered a nice trick to generalize the theorem of my advisor, Steve Milne, that  I had named "Fundamental Theorem of $U(n)$ series" in my thesis.

Hopefully, there will be many more collaborative ventures in the near future.

Update (Mar 22, 2018): The paper has been published. Here is the reference and Link:
G. Bhatnagar and  M.J. Schlosser, Elliptic well-poised Bailey transforms and lemmas on root systems, SIGMA, 14 (2018), 025, 44pp.

Friday, April 07, 2017

Spiral Determinants

We consider Spiral Determinants of the kind
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:

Sunday, February 14, 2016

Analogues of a Fibonacci-Lucas Identity

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!