

Generatingfunctionology (1989) [pdf] - octatoan
http://www.math.upenn.edu/~wilf/gfologyLinked2.pdf

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jordigh
Oh, man, this old classic.

Somewhat related, A = B is also a great book:

[http://www.math.upenn.edu/~wilf/AeqB.html](http://www.math.upenn.edu/~wilf/AeqB.html)

It's a very beautiful book that demonstrates how you can _prove_ certain
identities such as sin(a+b) = sin(a)cos(b) + cos(a)sin(b) by merely verifying
them at a few select values of a and b.

~~~
thomasahle
I thought A=B was just an algorithm for evaluating sums of binomial
coefficients?

~~~
DougMerritt
It's not "just" any small thing, it revolutionized the whole field. It is the
reason that mathematicians (and scientists and engineers) stopped the decades-
old common practice of circulating dusty facsimiles of hairy combinatorial
identities up through hypergeometric identities.

Some reviews:

> after 30 years of extraordinary efforts (largely the efforts of this book's
> authors), this problem [developing computer programs to simplify
> hypergeometric sums] is largely solved

And:

> If the complete automation of a major industry within discrete mathematics
> with relevance to computer science counts as the first miracle, this
> entertaining accessible exposition by the discoverers themselves counts as
> the second. ... Seldom do we find such a dramatic mathematical breakthrough
> placed within the reach of such a large audience so soon.

[http://www.math.upenn.edu/~wilf/AeqB.html](http://www.math.upenn.edu/~wilf/AeqB.html)

As to "an algorithm for evaluating sums of binomial coefficients", some of the
obvious kinds of sums have been trivially solvable since well before
computers, while the general case of partial sums are still beyond the state
of the art, so all in all, I think you're mixing all this up with something
else.

~~~
thomasahle
I guess "just" was the wrong word. What I meant was how it helps "prove
geometric identities by calculating a few values". I guess it's about series
representations.

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ambrop7
When in college I somehow managed to miss half of my combinatorics class.
Learned almost everything from this book then :)

