A reciprocity theorem for certain q-series found in Ramanujan’s lost notebook

Three proofs are given for a reciprocity theorem for a certain q-series found in Ramanujan’s lost notebook. The first proof uses Ramanujan’s ₁ψ₁ summation theorem, the second employs an identity of N. J. Fine, and the third is combinatorial. Next, we show that the reciprocity theorem leads to a two variable generalization of the quintuple product identity. The paper concludes with an application to sums of three squares. Dedicated to Richard Askey on the occasion of his 70th birthday. 2000 Mathematics Subject Classification Primary—33D15 B. C. Berndt: Research partially supported by grant MDA904-00-1-0015 from the National Security Agency. A. J. Yee: Research partially supported by a grant from The Number Theory Foundation.     

The translation theorem

Summary We state and prove the Translation Theorem. Then we apply the Translation Theorem to Soare's Extension Theorem, weakening slightly the hypothesis to yield a theorem we call the Modified Extension Theorem. We use this theorem to reprove several of the known results about orbits in the lattice of recursively enumerable sets. It is hoped that these proofs are easier to understand than the old proofs.Mathematics subject classification (1991): 03D25The author was partially supported by a NSF Postdoctoral Fellowship and by the U.S. Army Research Office through the Mathematical Sciences Institute of Cornell University and wishes to thank Michael Stob and Rodney Downey for their help     

The Banach-Tarski theorem

This article was originally published in French in the February 1987 issue of Pour la Science. The translation was done by the author.     

The three-lines theorem

Letf(z) be an entire function represented by a Dirichlet series which is absolutely convergent in the finite plane and whose exponents _k 0; let M(x) be the exact supremum of ¦f (z)¦ on {z: Re z=x}. If we assume that F(x)=ln M(x) has a continuous second derivative, the three-lines theorem asserts that F(x) >- 0. In the paper, this theorem is supplemented by the assertion that for x + the upper limit of F(x) is larger than a positive constant which depends only on {_k}. In the case of positive coefficients of the series, the obtained bound cannot be improved.Translated from Matematicheskie Zametki, Vol. 15, No. 1, pp. 45–53, January, 1974.In conclusion the authors express their gratitude to I. V. Ostrovskii for valuable advice and assistance.     

The nonstable neighborhood theorem

Publications mathématiques de l'IHÉS -     

The Chung-Feller theorem revisited

In this note, we provide a direct and elegant bijective proof of Chung-Feller theorem.     

The SSA congruence theorem

Many teachers and textbooks mention ‘three and a half basic congruence theorems: SAS, ASA, SSS and HL for right triangles. Why do they neglect the SSA theorem? A recent article [1] investigated the SSA theorem, but it seems necessary to extend and, in one point, to correct the result given there.