Systematic identification of limits of modified Apéry-type continued fractions

Develop a systematic method to identify the limit of continued fractions obtained by non-integer shifts of the index (for example, replacing n with n + 1/2) and related finite coefficient modifications of Apéry-type continued fractions, taking into account that finite modifications determine the limit only up to a Möbius (rational) transformation.

Background

The paper studies two variations of Apéry’s continued fractions: shifting the index n to n + a (with particular interest in a = 1/2) and constructing “continuous” generalisations that depend on a variable z and specialise to Apéry’s continued fractions at z = 0.

While the shift itself is elementary, the authors emphasize that the challenging step is recognizing the limit value of the new continued fraction produced by such modifications. This difficulty is exacerbated because changing finitely many coefficients of a continued fraction alters its limit by a Möbius transformation, making the limit only well-defined up to such a transformation.

The authors note that their successful identifications relied on consulting a large database of continued fractions rather than any general procedure, and explicitly state that no systematic method for this recognition is known to them.