Abstract: We state and prove forty hypergeometric (2F1) “strange ” series evaluations that were found by a systematic search for ‘nice ’ identities inspired by the simplified and annotated...
Proof of a Conjecture of Amitai Regev about Three-Rowed Young Tableaux (and much more!) (2008)
Shalosh B. Ekhad, Doron Zeilberger
Consider lattice paths in the three-dimensional cubic lattice, with unit positive steps, that always stay in the region x ≥ y ≥ z. Let Ab1,b2,b3(n) be the number of such walks of length n−(b1...
Dedicated to Two Combinatorial Giants: Gian-Carlo Rota (April 27, 1932- April 18, (2008)
Shalosh B. Ekhad, Doron Zeilberger, Rd Birthdays
respectively.
A High-School Algebra, Wallet-Sized Proof, of the Bieberbach Conjecture [After L. Weinstein] (2007)
Shalosh B. Ekhad, Doron Zeilberger
This paper makes explicit the power of formal math, that is (too) subtly hidden in Weinstein's original presentation.
A Wz Proof Of Ramanujan's Formula For (2007)
Shalosh Ekhad And, Shalosh B. Ekhad, Doron Zeilberger
0.37> \Gamma(3=2 + n) \Gamma(3=2)\Gamma(n + 1) = 1 X k=0 (\Gamma1) k (4k + 1) (1=2) 2 k (\Gamman) k k! 2 (3=2 + n) k : (3) To prove it for all positive integers n, we call the summand divided by...
There Are MORE THAN 2**(n/17) n-LETTER TERNARY SQUARE-FREE WORDS (2007)
Shalosh Ekhad And, Shalosh B. Ekhad, Doron Zeilberger, Improving On Brinkhuis
: We prove that the `connective constant' for ternary square-free words is at least 2 1=17 = 1:0416 : : :, improving on Brinkhuis and Brandenburg's lower bounds of 2 1=24 = 1:0293 : : : and...
Doron Zeilberger, David Bressoud, Gaurav Bhatnagar, Anders Bjorner, Jonathan Borwein, Francesco Brenti, ...
Two stones build two houses. Three build six houses. Four build four and twenty houses. Five build hundred and twenty houses. Six build Seven hundreds and twenty houses. Seven build five thousands...
A Proof of the Loehr-Warrington Amazing TEN to the Power n Conjecture (2005)
Ekhad, Shalosh B., Vatter, Vince, Zeilberger, Doron
We prove, via 30 seconds of Maple computation, that there are 10^n words in the alphabet {3,-2} of length 5n, sum 0, and such that every factor that sums to 0 and that starts with a 3 may not be...
A WZ proof of a curious identity (2003)
Shalosh B. Ekhad, Mohamud Mohammed
We give a short WZ-proof of a binomial coefficient identity due to Zhi-Wei Sun. In [3], [2], and [1] the identity m� (x + m + 1)(−1) i x + y + i y +2i m − i i i=0 m� x + i
The Number of Permutations With A Prescribed Number of 132 and 123 Patterns (1999)
Ekhad, Shalosh B., Robertson, Aaron, Zeilberger, Doron
Here we present the reasoning behind, and program to find, the generating functions for the number of permutations in the title. The article duals as the "accompanying" Maple package.
Proof of Conway's Lost Cosmological Theorem (1998)
Ekhad, Shalosh B., Zeilberger, Doron
John Horton Conway's Cosmological Theorem, about Audioactive sequences, for which no extant proof existed, is given a computer-generated proof, hopefully for good.
Curing the Andrews syndrom (1998)
Ekhad, Shalosh B., Zeilberger, Doron
George Andrews's recent challenge to automated identity-proving and the WZ method is dealt with. It is argued that the rivalry between the classical and automated approaches to hypergeometric sums is...
A binomial coefficient identity associated to a conjecture of Beukers (1998)
Ahlgren, Scott, Ekhad, Shalosh B., Ono, Ken, Zeilberger, Doron
A combinatorial identity that was needed in Ahlgren and Ono's proof of a certain congruence conjecture of Frits Beukers is stated, and a pointer to its WZ proof is given.
Plane Geometry: An Elementary School Textbook (ca. 2050) (1998)
Ekhad, Shalosh B., Zeilberger, Doron
In this reverse archeological find, we see how mathematics will be written in fifty years.
Amdeberhan, Tewodros, Ekhad, Shalosh B.
We give a generalization and a short mechanized proof of determinant conjectured by G. Kuperberg and J. Propp. Further generalizations and applications of the method to some q-analogues may be found...
CURING The ANDREWS SYNDROME (1998)
Shalosh Ekhad And, Shalosh B. Ekhad, Doron Zeilberger
: George Andrews's recent challenge to automated identity-proving and the WZ method is dealt with. It is argued that the rivalry between the classical and automated approaches to hypergeometric...
Proof of Conway's Lost Cosmological Theorem (1997)
Shalosh B. Ekhad, Doron Zeilberger
th(B i+1 )=length(B i ) ! . This is an immediate consequence of The Cosmological Theorem: There exists an integer N such that every string decays in at most N days to a compound of common and...
Tewodros Amdeberhan, Shalosh B. Ekhad, Greg Kuperberg
[P] have conjectured the following determinant identity:
(Communicated by Ronald Graham) (1997)
Shalosh B. Ekhad, Doron Zeilberger
existed, is given a new proof, this time hopefully for good. One of the most intriguing sequences [CG], [F], [SP], [V] is Conway’s [C] 1, 11, 21, 1211, 111221, 312211, 13112221, 1113213211,.... It...
Ekhad, Shalosh B., Zeilberger, Doron
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...
A Short Wz-Style Proof Of Abel's Identity (1995)
Shalosh B. Ekhad, John E. Majewicz
F10.08> r)Fn-1,k (r +1,s- 1) + (n - 1)(r + s)Fn-2,k (r +1,s- 1) = G n,k -G n,k+1 , (check!),wehavebysummingfromk =0tok = n, thanks to the telescoping on the right: an (r, s) - san-1 (r, s) - (n +...
Proof Of The Alternating Sign Matrix Conjecture (1995)
Doron Zeilberger, Gert Almkvist, Noga Alon, George Andrews, Dror Bar-natan, Francois Bergeron, ...
: The number of n 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...
Shalosh B. EKHAD, Doron Zeilberger
: We prove an explicit formula for the number of n \Theta n upper triangular matrices, over GF (q), whose square is the zero matrix. This formula was recently conjectured by Sasha Kirillov and Anna...
Ekhad, Shalosh B., Zeilberger, Doron
L. Weinstein's brilliant short proof of de Branges's Theorem is made even shorter by using computer algebra.
A high-school algebra wallet-sized proof, of the Bieberbach conjecture After L. Weinstein] (1993)
Ekhad, Shalosh B., Zeilberger, Doron
Weinstein's[2] brilliant short proof of de Branges'[1] theorem can be made yet much shorter(modulo routine calculations), completely elementary (modulo L\"owner theory), self contained(no need for...
A WZ proof of Ramanujan's Formula for Pi (1993)
Ekhad, Shalosh B., Zeilberger, Doron
Ramanujan's series for Pi, that appeared in his famous letter to Hardy, is given a one-line WZ proof.
George E. Andrews, Shalosh B. Ekhad, Doron Zeilberger
n\Gammak = 1; (a) n X k=0 2(\Gammaq n+1 ) k 1 + q k H k H n = n X k=\Gamman (\Gammaq) k 2 : (b) Proof: Let L 1 (n) and L 2 (n) be the left sides of (a) and (b) respectively, and let F 1 (n; k), and F...
Proof of Dougall’s Sum Identity (1988)
Shalosh B. Ekhad, Doron Zeilberger, C. Berndt
My guess is that, within fifty or hundred years (or it might be one hundred and fifty) computers will successfully compete with the human brain in doing mathematics, and that their mathematical style...