Advances in Difference Equations
Volume 2010 (2010), Article ID 143521, 11 pages
doi:10.1155/2010/143521
Research Article

Elementary Proof of Yu. V. Nesterenko Expansion of the Number Zeta(3) in Continued Fraction

Leonid Gutnik

Moscow State Institute of Electronics and Mathematics, Russia

Received 12 August 2009; Revised 10 December 2009; Accepted 10 January 2010

Academic Editor: Binggen Zhang

Copyright © 2010 Leonid Gutnik. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Yu. V. Nesterenko has proved that ζ(3)=b0+a1|/|b1+⋯+aν|/|bν+⋯, b0=b1=a2=2, a1=1,b2=4, b4k+1=2k+2, a4k+1=k(k+1), b4k+2=2k+4, and a4k+2=(k+1)(k+2) for k∈ℕ; b4k+3=2k+3, a4k+3=(k+1)2, and b4k+4=2k+2, a4k+4=(k+2)2 for k∈ℕ0. His proof is based on some properties of hypergeometric functions. We give here an elementary direct proof of this result.