Apéry Limits and Special Values of L-functions

and then showing that bn/an converges to ζ(3) fast enough to ensure irrationality of ζ(3) (see [5]). Another remarkable discovery of Apéry is that an and bn satisfy the recursive relation (n+2)un+2− (34n+153n+231n+117)un+1+(n+1)un = 0 (un = an or bn). Thus, if we set A(t) = ∑∞ n=0 ant n and B(t) = ∑∞ n=0 bnt , then the functions A(t) and B(t) satisfy the differential equations (1) (1− 34t+ t)θA+ (3t − 51t)θA+ (3t − 27t)θA+ (t − 5t)A = 0 and (1− 34t+ t)θB + (3t − 51t)θB + (3t − 27t)θB + (t − 5t)B = 6t, where θ denotes the differential operator td/dt. Apéry also had an analogous result for ζ(2) = π/6. He showed that if {an} and {bn} are sequences of rational numbers satisfying the recursive relation (n+ 2)un+2 − (11n + 33n+ 25)un+1 − (n+ 1)un = 0 (un = an or bn) with the initial values a−1 = 0, a0 = 1, b0 = 0, b1 = 5,