### 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.

### 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 $0$.

### 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.

### Spiral Determinants

We consider Spiral Determinants of the kind
$$\text{det}\left( \begin{matrix} {16}&{15}&{14}&{13}\\ {5}&{4}&{3}&{12}\\ {6}&1&{2}&{11}\\ {7}&{8}&{9}&{10} \end{matrix} \right)$$
and
$$\text{det} \left( \begin{matrix} {17}&{16}&{15}&{14}&{13}\\ {18}&{5}&{4}&{3}&{12}\\ {19}&{6}&1&{2}&{11}\\ {20}&{7}&{8}&{9}&{10}\\ {21}&{22}&{23}&{24}&{25} \end{matrix} \right)$$
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: https://www.sciencedirect.com/science/article/pii/S0024379517302719

### 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!

### 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.

### Of Art and Math: A series of articles with Punya Mishra

 Right Angle: One of the many ambigrams made by Punya Mishra that appear in this series of articles appearing in "At Right Angles". All ambigrams are copyright Punya Mishra and cannot be used without permission.
Punya and I are writing a series of articles on the subject of ambigrams. All the ambigrams are made by Punya. For this series, he has been making many new ambigrams, which communicate mathematical ideas. Already, in the space of working on a few articles, it looks like he has made the largest number of mathematical ambigrams.

Here is a longer blog entry from Punya's blog, about this series of articles. His blog has further links to his amazing ambigrams.

Dec, 2015. I presented Punya's and my work in TIME 2015, in Baramati, Maharashtra in my talk: On Punya Mishra's Mathematical Ambigrams. This was the seventh edition of TIME, which stands for 'Technology and innovation in Math education'.

Mar, 2015. The fourth article is on Paradoxes. It is part 1 of 2 articles on this topic. Here we consider what TRUE and FALSE mean in the context of mathematics. Its an introduction to math philosophy. Again, it has many interesting ambigrams.

Feb 2015. The Michigan State Museum has launched an exhibit entitled "Deep Play: Creativity in Math and Art through Visual Wordplay." Check out: the exhibitions web-page

July 2014. The third article on Self-similarity. This one has some amazing ambigrams, and a graphic of the binary pascal's triangle I made many years ago.

Mar, 2014. The second article is on Introducing Symmetry. I think Punya outdid himself in some of the ambigrams here. The ambigram for sin (which is periodic, a sin wave, an odd function) and inverse (modeled on a hyperbola) and exp-log were my favorites. But this month's  puzzle ambigram is mind-blowing too.

Nov, 2013. The first article has come out. It is: Introducing Ambigrams. There is a hidden message in the article. See if you can find it.

### 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.

### How to discover 22/7 and other rational approximations to $\pi$

A short article written for a magazine that caters to high-school students. The basic idea is to use continued fractions found using Euclid's algorithm, and then to chop off the continued fraction to get rational approximations. Written at a high-school level. Some of the material was already present in my book Maths Concepts.

### 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.

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.

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!

### New website: Teaching Website

I have made a new website, that collates the math materials I keep creating and with information for my students.  It is available on
http://gb-teaching.blogspot.com. If you, or your child, is in high school, there are many materials available that may be useful. Eventually, I hope some of the new materials I am placing there also become a book or perhaps an e-book.

### Get Smart Maths Concepts now available as an e-book

Get Smart! Maths Concepts, published by Penguin in 2008 is now available as an e-book.

Check it out!
http://www.amazon.com/kindle/dp/B008ESLWRK/ref=rdr_kindle_ext_eos_detail

Or

http://www.flipkart.com/get-smart-maths-concepts/p/itmdumyhbgh6dyqh?pid=DGBDGGY4M2SVYDMH

### Arun Bhaiyya -- A limerick

Texas ka famous letter writer, our Arun Bhaiya,
He loves to cook large amounts of lobhia,
Eating those greens,
Keeps him full of beans--
But TT: he thinks its a taste worse than ghia!

Arun Kumar used to write an interesting "Letter from Austin" which was quite famous in the early days of the Internet. Even now, his letters to his mailing list "dakghar" are quite interesting and varied in the topics they cover. Now these letters are being recorded as his Facebook notes. This piece was inspired by one of his Facebook notes, where he describes his love for cooking (and eating) lobhia. TeeTee (TT for short) is his son. I hear he is quite a mathematician and a hockey player.

### How to Guess the Binomial Theorem for any index

Newton extended the Binomial Theorem to the case where the index is no longer a non-negative integer. Newton did not provide a proof of the general case, where the index is a real number. We too will not provide a proof, but will motivate Newton's Binomial Theorem by showing some of the clues that lead to the statement of the general case.

We wish to generalize the identity
$$(1+x)^n=\sum_{k=0}^n {n\choose k} x^k$$
by replacing $n$ by a real number $a$. On the LHS, there is no problem, since the product $(1+x)^a$ makes sense for $a$ a real number. But on the RHS, there are two problems:

1. The Binomial Coefficient ${n\choose k}$ is defined only when $n$ is a non-negative integer.
2. The index of summation goes from $0$ to $n$, and thus $n$ has to be a non-negative integer.

The problems are easily solved. Note that ${n\choose k}$ may be written as
\label{achoosek}
\frac{n(n-1)\cdots (n-k+1)}{k!},

and \eqref{achoosek} makes sense if we replace $n$ by $a$.
Further, note that when $k>n$, then \eqref{achoosek} reduces to $0$. So we may as well write the Binomial Theorem as
$$(1+x)^n=\sum_{k=0}^{\infty} \frac{n(n-1)\cdots (n-k+1)}{k!} x^k.$$
Since all the terms of this series where $k$ is bigger than $n$ reduce to $0$, the series reduces to the finite sum of the familiar Binomial Theorem for non-negative integral index.

However, if we replace $n$ by a real number $a$, we may have to deal with an infinite series, and we need conditions for it to converge. It turns out the series converges whenever $|x|<1$. So finally, we are ready to state the Binomial Theorem for real index.
\begin{align}
(1+x)^a&=&\sum_{k=0}^{\infty} \frac{a(a-1)\cdots (a-k+1)}{k!} x^k, \text{ for $|x|<1$}\label{binseries} \\
&=& 1+ax+\frac{a(a-1)}{2!}x^2+\frac{a(a-1)(a-2)}{3!}x^3+\cdots\notag
\end{align}

The conditions we need on $x$ are motivated by an example of the Binomial Theorem for real index that we have already seen. Recall the formula
$$\sum_{k=0}^\infty {x^k} = \frac{1}{1-x}, \text{ for |x|<1. }$$
for the sum of the geometric series with first term $1$ and common ratio $x$. This formula is a special case of \eqref{binseries}, where $a=-1$.