Doron Zeilberger

A holonomic systems approach to special functions identities

Abstract We observe that many special functions are solutions of so-called holonomic systems. Bernsteins deep theory of holonomic systems is then invoked to show that any identity involving sums and integrals of products of these special functions can be verified in a finite number of steps. This is partially substantiated by an algorithm that proves terminating hypergeometric series identities, and that is given both in English and in MAPLE.

The method of creative telescoping

An algorithm for definite hypergeometric summation is given. It is based, in a non-obvious way, on Gospers algorithm for definite hypergeometric summation, and its theoretical justification relies on Bernsteins theory of holonomic systems.

An algorithmic proof theory for hypergeometric (ordinary and q') multisum/integral identities

SummaryIt is shown that every ‘proper-hypergeometric’ multisum/integral identity, orq-identity, with a fixed number of summations and/or integration signs, possesses a short, computer-constructible proof. We give a fast algorithm for finding such proofs. Most of the identities that involve the classical special functions of mathematical physics are readily reducible to the kind of identities treated here. We give many examples of the method, including computer-generated proofs of identities of Mehta-Dyson, Selberg, Hille-Hardy,q-Saalschütz, and others. The prospect of using the method for proving multivariate identities that involve an arbitrary number of summations/integrations is discussed.

A fast algorithm for proving terminating hypergeometric identities

An algorithm for proving terminating hypergeometric identities, and thus binomial coefficients identities, is presented. It is based upon Gospers algorithm for indefinite hypergeometric summation. A MAPLE program implementing this algorithm succeeded in proving almost all known identities. Hitherto the proof of such identities was an exclusively human endeavor.

Resurrecting the asymptotics of linear recurrences

Abstract Once on the forefront of mathematical research in America, the asymptotics of the solutions of linear recurrence equations is now almost forgotten, especially by the people who need it most, namely combinatorists and computer scientists. Here we present this theory in a concise form and give a number of examples that should enable the practicing combinatorist and computer scientist to include this important technique in her (or his) asymptotics tool kit.

Random walk in a Weyl chamber

The classical Ballot problem that counts the number of ways of walking from the origin and staying within the wedge x 1 ≥x 2 ≥...≥x n (which is a Weyl chamber for the symmetric group), using positive unit steps, is generalized to general Weyl groups and general sets of steps

A Combinatorial Proof of Bass’s Evaluations of the Ihara-Selberg Zeta Function for Graphs

We derive combinatorial proofs of the main two evaluations of the Ihara-Selberg Zeta function associated with a graph. We give three proofs of the first evaluation all based on the algebra of Lyndon words. In the third proof it is shown that the first evaluation is an immediate consequence of Amitsurs identity on the characteristic polynomial of a sum of matrices. The second evaluation of the Ihara-Selberg Zeta function is first derived by means of a sign-changing involution technique. Our second approach makes use of a short matrix-algebra argument.

Theorems for a price: tomorrow’s semi-rigorous mathematical culture

ConclusionZeilberger has proved some breathtaking theorems [ZB], [Z3], and his W-Z method (joint with Wilf [WZ]) has been a godsend to me [A2] and an inspiration [A3]. However, there is not one scintilla of evidence in his accomplishments to support the coming “... metamorphosis to nonrigorous mathematics.”Until Zeilberger can provide identities which are (1) discovered by his computer, (2) important to some mathematical work external to pure identity tracking, and (3) too complicated to allow an actual proof using his algorithm, then he has produced exactly no evidence that his Brave New World is on its way.I regret feeling compelled to write this article. Unfortunately articles on why rigorous mathematics is dead create unintended side effects. We live in an age of rampant “educational reform.” Many proponents of mathematics education reform impugn the importance of proofs, and question whether there are right answers, etc. A wonderfully sane account of these problems has been given by H.-H. Wu [Wu1], [Wu2]. A much more disturbing account “Are proofs in high school geometry obsolete?” concludes Horgan’s article [H]. It is a disservice to mathematics inadvertently to provide unfounded ammunition for the epistemological relativists.If anyone reading this believes the last paragraph is rubbish because attempts (unknown to me) are currently underway to insert the Continuum Hypothesis or the Theory of Large Cardinals into the NCTM Standards for School Mathematics, please don”t write to tell me about them. I can take only so many shocks to my system.Finally, wisdom suggests that grand predictions of life in 2193 ought to be treated with scepticism. (“Next Wednesday’s meeting of the Precognition Society has been postponed due to unforeseen circumstances.”) A long-overdue analysis of some of our current prophets has been attempted by Max Dublin [Du]. Especially noteworthy is Dublin’s Chapter 5, “Futurehype in Education. “ I won’t give the plot away, but I recall the words of Claude Rains near the end ofCasablanca: “Round up the usual suspects!”

A proof of Julian West's conjecture that the number of two-stack-sortable permutations of length n is 2(3 n )!/(( n +1)!(2 n +1)!)

Abstract The Polya-Schutzenberger-Tutte methodology of weight enumeration, combined with about 10 hours of CPU time (of Maple running on Drexel Universitys Sun network) established Julian Wests conjecture that 2-stack-sortable permutations are enumerated by sequence # 651 in the Sloane listing.

Kathy O'Hara's constructive proof of the unimodality of the Gaussian polynomials

The best way to learn a topic is by teaching it. Similarly, the best way to understand a new proof is by writing an expository paper about it. This was the original motivation of the present paper. Once written, I thought that it would be nice if all the readers of this Monthly had the opportunity to savor this elegant proof. All the central ideas and constructions are O’Hara’s, but I have made a few minor improvements and shortcuts that I believe to make the argument clearer. I would like to thank Kathy O’Hara for stimulating conversations and correspondence.