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.


Keywords:
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


Monday, August 02, 2010

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 benefited...in 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

Sunday, August 01, 2010

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

Friday, June 04, 2010

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.

Everything
in The Book
is as elegant,
as could be.

Everything
as simple,
as effortless,
as should be.

Everything
as beautiful,
as it is.

~*~*~*~*~*~

Thursday, May 14, 2009

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
more
than what is
needed.

The form
and the function,
made for each other.

GB #32

Sunday, April 05, 2009

Paradox


Paradox
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.

Saturday, April 04, 2009

Watson-Crick


Watson-Crick
Ambigram by punyamishra


Watson and Crick
show
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
discovered
the secret of life
itself.
GB #30

Friday, February 27, 2009

Internet


Internet
Ambigram by punyamishra


The Internet
inking pacts
across the world.

Linking
and connecting
all humanity.

Shakily
connecting
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
inter-connected
become the Net.

Unleashing
the power
of communication,
of creativity,
and
of community.

Small individuals,
linked together
become the
Internet.

GB #28

Friday, February 20, 2009

Douglas R Hofstadter


Sides-reversed-is
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.

Saturday, February 14, 2009

Dog


Dog
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
personal.

And becomes
your best friend.

GB #26

God


God
Originally uploaded by punyamishra

GOD
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
nothingness.

But still,
required,
to count from
one to infinity.

GB #25