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

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.