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

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

Math Problem Book - Grade 6

Here is a book made from some problem sets I used to give Tejasi and her friends in my garage. The explanations have been added later on. Most of the kids who took these problem sets the sense that they began doing well in exams. Here is a collection of the problem sets for grade 6.

Click here to download/view Math Problem Book - Grade 6

Experience Mathematics

A book I wrote long ago. Recently, I re-edited it based on comments made by Professor Dick Askey. Looking for a publisher, but meanwhile here it is for my friends and their kids...

Click here to view the PDF File

The q-disease

Special Functions,
by Andrews, Askey and Roy.
Here's a belated review,
and a thank you.


Beauty in mathematics,
said Polya,
is seeing the truth
without effort.

in The Book
is as elegant,
as could be.

as simple,
as effortless,
as should be.

as beautiful,
as it is.


Design II

Design II
Ambigram by punyamishra

A great design:
Everything fits in nicely
into one complete whole.

Not a hair out of place,
and not one thing
than what is

The form
and the function,
made for each other.

GB #32


Ambigram by punyamishra

All Cretans are liars
said Epiminedes,
a Cretan,
a classic Paradox.

If Epiminedes tells the truth
then he must be lying.
And if he is lying,
he is telling the truth.


Ambigram by punyamishra

Watson and Crick
what fun it is
to be
a scientist.

What fun it is
to discover
something new.

What fun it is
to compete
with the best
and win.

Watson and Crick
the secret of life
GB #30


Ambigram by punyamishra

The Internet
inking pacts
across the world.

and connecting
all humanity.

all the dots
into one
continuous whole.

GB #29

Internet II

Internet II
Ambigram by punyamishra

Small knots
woven together
become a net.
Flexible, stable,
and very strong.

Small computers
become the Net.

the power
of communication,
of creativity,
of community.

Small individuals,
linked together
become the

GB #28

Douglas R Hofstadter

Ambigram by punyamishra

Douglas R Hofstadter
sides reversed is
Retdatsfoh R Salgoud
sides reversed is
Douglas R Hofstadter
is one Strange Loop.

Hofstadter wrote
Godel, Escher and Bach:
An eternal golden braid.

A personal review follows.


Originally uploaded by punyamishra

A dog,
wags his tail,
seeks attention.
And licks your face,
without asking for
your permission.

Comes up close
and becomes

And becomes
your best friend.

GB #26


Originally uploaded by punyamishra

looks the same
whichever way
you look at it.

Look down from the top
or up from bottom --
and find GOD.

Look in the mirror
and see GOD.

Just like the 0,
symbol for

But still,
to count from
one to infinity.

GB #25

I Love My India

Tejasi made this using My Paint Free, on my iPhone.