Construct a continuous generalisation of the big Apéry continued fraction

Construct a continued fraction depending on a variable z that specialises, at z = 0, to Apéry’s “big” continued fraction for ζ(3), analogous to the continuous generalisation obtained for the small Apéry case in Theorem 6.3.

Background

After successfully identifying a continuous generalisation for the small Apéry continued fraction (Section 6.2), the authors attempted to find an analogous generalisation for the big Apéry continued fraction associated with ζ(3).

They report that no such generalisation is present in their database and explicitly state they do not know how to find one, leaving the construction of such a continued fraction as an open task.