Monday, June 13, 2016

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 yet to be written down) are all new.

Here is a preprint.

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.


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!

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.


Tuesday, March 31, 2015

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.

Updates

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'.

July, 2015. The fifth article is Part 2 of 2 on the subject of paradoxes. It covers self-reference, Russell's Paradox and visual paradoxes. This article includes a 'new paradox', a version of Jourdain's card paradox by Punya. 

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.

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 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 (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.


Monday, March 31, 2014

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.

Update: March 2014. This article is published in "At Right Angles." Here is a link to the published article.

Saturday, March 15, 2014

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. 


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 $0$.

Here is a link to a preprint. Please do give comments.

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!


Monday, July 29, 2013

New website: The ABC of Math

I have made a new website, that collates the math materials I keep creating. It is available on
http://www.abc-of-math.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.

Monday, April 02, 2012

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.

Friday, September 16, 2011

How to Guess the Binomial Theorem for any index


Download PDF



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
\begin{equation}\label{achoosek}
\frac{n(n-1)\cdots (n-k+1)}{k!},
\end{equation}
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 $a=-1$.

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).

Wednesday, August 03, 2011

Precalculus by Askey and Wu

Many years back, Professor Richard Askey sent me hard copies of some notes he had made, with a supplement by  H. Wu. I think these notes are just wonderful, and am placing them here.

Review of Pre-Calculus by Richard Askey and H. Wu
Precalculus - Further Notes by Richard Askey

Wednesday, April 27, 2011

Punya - A nearly Palindromic poem

Punya
~

Love
your palindromes,
everything
the poetry,
ambigrammed symmetry.
An inspiration-
your blog.
Hope you keep up with
your slog.
the perspiration:
programmed asymmetry,
Ambi-poettary,
anything
but palindromes!
love

~
GB


~~
This was inspired by a blog post by Punya, about an 8th grader who loved his palindromic poetry. The associated facebook page attracted some comments admiring Punya, including a little palindromic poem by me. 


While I am not an 8th grader anymore, I do feel many times that I am still in 12th grade. 
So, I thought of a  
nearly-palindromic poem. 

So near a palindromic poem, yet far from it. The first of its kind. Enjoy. Or not.