Euler proved the following general continued fraction formula,

which automatically gives a representation for the *Riemann zeta function* . However, the convergence is rather slow. Apery found a much faster version for and proved its irrationality. The status for other odd *s*, however, remains open. While it has already been proved irrational for all even *s*, Wadim Zudilin nonetheless found an interesting faster version for .

First, consider the following known binomial sums,

*This nice pattern stops at s = 4*. (There are sums for other *s*, but they do not have this succinct form.) It is suggestive then that belongs to the same family and may share certain similarities.

Zudilim found that,

where, starting with *n* = 0,

(Note that the cfrac does not use .) Like Apery’s accelerated version for , the partial denominators use *addition*, in contrast to Euler’s form which has subtraction. Also, like its siblings, it obeys a recurrence relation,

Starting with , one gets the sequence,

which is not yet in the OEIS. In Theorem 4, Zudilin mentions they are positive rationals, but some *Mathematica* experimentation will show that apparently all the are integral. (*Proof anyone?*) Furthermore, they have the limit,

For more details, refer to Zudilin’s *An Apery-like difference equation for Catalan’s constant*. Now if only someone will do something more about …

Posted by Mason Korb on September 8, 2018 at 10:43 pm

Zudilin* found that,