By Edgar Dehn

Meticulous and entire, this presentation is aimed toward upper-level undergraduate and graduate scholars. It explores the easy principles of algebraic concept in addition to Lagrange and Galois conception, concluding with the appliance of Galoisian idea to the answer of distinct equations. Many numerical examples, with whole suggestions. 1930 version.

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To values of n other than the nonnegative integers. In fact n! is thereby defined for all complex numbers n other than the negative integers. Some of the relationships between these three notations are Γ(n + 1) = n! = (1)n , (a)n = a(a + 1) · · · (a + n − 1) = (a + n − 1)! Γ(n + a) = . (a − 1)! Γ(a) (I) Gauss’s 2 F1 identity. If b is a nonpositive integer or c − a − b has positive real part, then Γ(c − a − b)Γ(c) a b ;1 = . 2F1 c Γ(c − a)Γ(c − b) (II) Kummer’s 2 F1 identity. If a − b + c = 1, then Γ( 2b + 1)Γ(b − a + 1) a b ; −1 = .

Now take a = b = r = n. ✷ That was a proof by generating functions, another of the popular tools used by the species Homo sapiens for the proof of identities before the computer era. Next we’ll show what a computerized proof of the same identity looks like. We preface it with some remarks about standardized proofs and certificates. Suppose we’re going to develop machinery for proving some general category of theorems, a category that will have thousands of individual examples. Then it would clearly be nice to have a rather standardized proof outline, one that would work on all of the thousands of examples.

So when your computer finds a WZ proof, it doesn’t have to recite the whole thing; it needs to describe only the rational function R(n, k) that applies to the particular identity that you are trying to prove. The rest of the proof is standardized. The rational function R(n, k) certifies the proof. Here is the standardized WZ proof algorithm: 1. Suppose that you wish to prove an identity of the form k t(n, k) = rhs(n), and let’s assume, for now, that for each n it is true that the summand t(n, k) vanishes for all k outside of some finite interval.