arXiv subjects
Publications and source records attributed to Doron Zeilberger.
A determinant evaluation is proven, a special case of which establishes a conjecture of Bombieri, Hunt, and van der Poorten (Experimental Math\. {\bf 4} (1995), 87--96) that arose in the study of Thue's method of approximating algebraic numbers.
It is remarked that Dave Robbins's Expression for the integral of an alternant over the unit simplex (math.CO/9805108) follows immediately from an excercise in Polya-Szego.
It is way too soon to teach our computers how to become full-fledged humans. It is even premature to teach them how to become mathematicians, it is even unwise, at present, to teach them how to become combinatorialists. But the time is ripe to teach them how to become experts in a suitably defined and narrowly focused subarea of combinatorics. In this article, I will describe my efforts to teach my beloved computer, Shalosh B. Ekhad, how to be an enumerator of Wilf classes.
By practicing the philosophy of our beloved late master, Marco Schutzenberger, to whose memory this article is dedicated, we give an insightful bijective proof of the three-term recurrence satisfied by the Hipparchus-Schroeder numbers 1,1,3,11,45,197,903, ...
Based on the WZ method, some series acceleration formulas are given. These formulas allow to write down an infinite family of parametrized identities from any given identity of WZ type. Further, this family, in the case of the Riemann Zeta function, gives rise to many accelerated expressions for $ζ(3)$. The present method produced a formula, given at the end of the paper, that was used recently Sebastian Wedeniwski to compute Apery's constant to more than 32 million digits, holding the present record!
Using WZ forms, Apery-style proofs of the irrationality of the q-analogues of the Harmonic seires and Ln(2) are given. For the q-analogue of Ln(2), this method of proof produces an improved irrationality measure.
H. N. V. Temperley's method for counting vertically convex polyominoes is modified, generalized, and most importantly, programmed (in Maple).
Mills, Robbins, and Rumsey conjectured, and Zeilberger proved, that the number of alternating sign matrices of order $n$ equals $A(n):={{1!4!7! ... (3n-2)!} \over {n!(n+1)! ... (2n-1)!}}$. Mills, Robbins, and Rumsey also made the stronger conjecture that the number of such matrices whose (unique) `1' of the first row is at the $r^{th}$ column, equals $A(n) {{n+r-2} \choose {n-1}}{{2n-1-r} \choose {n-1}}/ {{3n-2} \choose {n-1}}$. Standing on the shoulders of A.G. Izergin, V. E. Korepin, and G. Kuperberg, and using in addition orthogonal polynomials and $q$-calculus, this stronger conjecture is proved.
We prove an explicit formula for the number of $n \times n$ upper triangular matrices, over $GF(q)$, whose square is the zero matrix. This formula was recently conjectured by Sasha Kirillov and Anna Melnikov[KM].
Dodgson's determinant condensation rule is shown to immediately imply the evaluation of MacMahon's determinant expression that leads to the Box Theorem.
Fred Galvin's amazing proof of the Dinitiz conjecture is used to illustrate the method of undetermined generalization and specialization.
The Bordelaise philosophy, or rather a juvenile version of it, is used to enumerate self avoiding walks in a $[0,1] \times (- \infty, \infty)$.
The number of $n \times n$ matrices whose entries are either -1, 0, or 1, whose row- and column- sums are all 1, and such that in every row and every column the non-zero entries alternate in sign, is proved to be $[1!4! >... (3n-2)!]/[n!(n+1)! ... (2n-1)!]$, as conjectured by Mills, Robbins, and Rumsey.
L. Weinstein's brilliant short proof of de Branges's Theorem is made even shorter by using computer algebra.
A tutorial on what later became to be known as WZ theory, as well as a motivated account of the seminal Gosper algorithm.
A generalization of the classical statistics ``maj'' and ``inv'' (the major index and number of inversions) on words is introduced, parameterized by arbitrary graphs on the underlying alphabet. The question of characterizing those graphs that lead to equi-distributed "inv" and "maj" is posed and answered.
How Enumerative Combinatorics met Special Functions, thanks to Joe Gillis
Capelli's and Turnbull's classical identities are given elegant combinatorial proofs.