Beukers, F. A note on the irrationality of \(\zeta (2)\) and \(\zeta (3)\). (English) Zbl 0421.10023 Bull. Lond. Math. Soc. 11, 268-272 (1979). Page: −5 −4 −3 −2 −1 ±0 +1 +2 +3 +4 +5 Show Scanned Page Cited in 17 ReviewsCited in 88 Documents MSC: 11J81 Transcendence (general theory) 11M06 \(\zeta (s)\) and \(L(s, \chi)\) Keywords:double integral; Legendre polynomial; triple integral; irrationality; hypergeometric functions Citations:Zbl 0409.10028; Zbl 0401.10049 PDFBibTeX XMLCite \textit{F. Beukers}, Bull. Lond. Math. Soc. 11, 268--272 (1979; Zbl 0421.10023) Full Text: DOI Online Encyclopedia of Integer Sequences: Beukers integral int(int( -log(x*y) / (1-x*y) * P_n(2*x-1) * P_n(2*y-1) ,x=0..1,y=0..1)) = (A(n) + B(n)*zeta(3)) / A003418(n)^3. This sequence gives negated values of A(n). Beukers integral int(int( -log(x*y) / (1-x*y) * P_n(2*x-1) * P_n(2*y-1),x=0..1,y=0..1)) = (A(n) + B(n)*zeta(3)) / A003418(n)^3. This sequence gives values of B(n).