Friday, September 16, 2011

How to Guess the Binomial Theorem for any index

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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
\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.
(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

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.

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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).
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}}.

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.

  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


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


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.

Monday, April 25, 2011

Identities and Mathematical Intuition: Talk in DPS - Dwarka to DPS Math Teachers

On April 18th, I gave a talk on Identities to Delhi Public School (DPS) Math teachers  attending a training conference/workshop. The teachers were from DPSs all over the country and teach in senior school (XIth-XIIth).

The overall idea of the talk was to organize information about identities according to the three kinds of mathematical intuition I have spoken about earlier. The three kinds of mathematical intuition are: Symbolic, graphical or physical intuition, and structural intuition. These are motivated by the following quote:
…some mathematicians are more endowed with the talent of making pictures, others with that of juggling symbols and yet others with the ability of picking a flaw in an argument.
~Gian Carlo Rota 

Tuesday, March 08, 2011

Sunil Mittal

A schoolboy, named Sunil Mittal,
What goes through 
his adolescent mind?

Here's a clue:

The movies playing in his brain
and the color of his uniform,
are both the same.

They are Blue!

Sunil is a friend from modern school. This one came up on FB as a comment on a discussion. 

Wednesday, February 09, 2011

In Praise of an Elementary Identity of Euler

After many years, a new math paper. Its mostly a survey of my favorite identities, but has some new identities too. The new results have been checked (as typeset in the paper) using Maxima.  I have tried to write the first few sections so that  anyone can read and appreciate it.
I would appreciate any comments, typos, etc.

Update (March 16, 2011):
Presentation from: Georgia Southern q-Series conference, March 15. Here is a link.

Update (June 11, 2011): The paper is published by the Electronic J. of Combinatorics, Vol 18 (2), P13 44 pp. Download.

Telescoping, Fibonacci Numbers,  Pell Numbers, Derangements, Hypergeometric Series, Fibonacci Polynomials,  q-Fibonacci Numbers,  q-Pell numbers, Basic Hypergeometric Series, q-series, Binomial Theorem, q-Binomial Theorem, Chu--Vandermonde sum, q-Chu--Vandermonde sum, Pfaff--Saalschutz sum, q-Pfaff--Saalschutz sum, q-Dougall summation, very-well-poised 6 phi 5 sum, Generalized Hypergeometric Series, WZ Method

Sunday, February 06, 2011

My Mathematical Forefathers

From time to time, I look at the Mathematics Genealogy Project, and search for my own mathematical tree. I was happy to note that I am a direct descendant of Gauss and of Leibnitz. What I noticed today, was that I am a mathematical cousin of Saroj Malik, my teacher in Hindu College, who taught me abstract algebra and elementary number theory. We branch out at Gauss.

Here is the complete list of my mathematical forefathers.

  • Friedrich Leibnitz
  • Jakob Thomasius
  • Otto Mencke
  • Johann Christoph Wichmannshausen
  • Christian August Hausen
  • Abraham Gotthelf Kästner
  • Johann Friedrich Pfaff
  • Carl Friedrich Gauß
  • Christoph Gudermann
  • Karl Theodor Wilhelm Weierstraß
  • Leo Königsberger
  • Georg Alexander Pick
  • Charles Loewner
  • Adriano Mario Garsia
  • Stephen Carl Milne
  • Gaurav Bhatnagar