A Proof that Euler Missed...

By A. Poorten. 2000

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Congruences for central binomial sums and finite polylogarithms (2010, background): “The former played a role in Apery’s celebrated proof of irrationality of ζ(3), see van der Poorten’s account of Apery’s proof [18] for a discussion of both formulas and relevant references.”
Parameterized Telescoping Proves Algebraic Independence of Sums (2008, unknown): “For the Apéry sum S(m) = ∑k=0 (m k )2(m+k k ) , see [24].”
Congruences for Catalan and Motzkin numbers and related sequences (2004, background): “For an exposition of Apéry’s work, see the article of van der Poorten [27].”
Criteria for irrationality of Euler’s constant (2002, unknown): “Apéry’s startling 1979 proof ([1], [7]) that ζ(3) is irrational, there have been attempts to extend it “up” to ζ(n) = ∑∞ k=1 k −n for odd n > 3, with recent progress by K.”
Euler's constant: Euler's work and modern developments (2013, background): “For odd zeta values, in 1978 Apéry ([10], [11]) established that ζ(3) is irrational; see van der Poorten [249] and Section 3.”
THE FERMAT CUBIC, ELLIPTIC FUNCTIONS, CONTINUED FRACTIONS, AND A COMBINATORIAL EXCURSION (2005, unknown): “This represents a great piece of Eulerian mathematics of which van der Poorten has written a particularly vivid account in [59].”