A hypergeometric proof that

A 21st Century Proof of Dougall’s Hypergeometric Sum Identity

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 will be rather different from ours. Fairly long computational verifications (numerical or combinatorical) will not bother them at all, and this should lead not just to different sorts of proofs, bu...

Proof of the Wilf–Zeilberger Conjecture for Mixed Hypergeometric Terms

In 1992, Wilf and Zeilberger conjectured that a hypergeometric term in several discrete and continuous variables is holonomic if and only if it is proper. Strictly speaking the conjecture does not hold, but it is true when reformulated properly: Payne proved a piecewise interpretation in 1997, and independently, Abramov and Petkovšek in 2002 proved a conjugate interpretation. Both results addre...

The Proof That N

is similar to the proof of Theorem 2.4 ((rst direction), together with the second part of Lemma 4.3 that guarantees that D 1 (e) = D 1 (e f) D 1 (e g). Using the last two theorems, we get Let us now brieey discuss the case of computing general relations and not necessarily functions. The equality in Lemma 4.3 part (1), does not hold anymore. However, by FKN91] the two sides cannot be too far. A...

Formalizing a Proof that e is Transcendental

A transcendental number is one that is not the root of any non-zero polynomial having integer coefficients. It immediately follows that no rational number q is transcendental, since q can be written as a/b where a and b are integers, and thus q is a root of bx − a. Furthermore, the transcendentals are a proper subset of the irrationals, since for example the irrational √ 2 is a root of x − 2. T...

A Sane Proof that COLk ࣘ COL3