On some continued fraction expansions of the Rogers–Ramanujan type

Abstract

By guessing the relative quantities and proving the recursive relation, we present some continued fraction expansions of the Rogers-Ramanujan type. Meanwhile, we also give some J -fraction expansions for the q-tangent and q-cotangent functions.

Finite Rogers-Ramanujan type continued fractions

“…Since there are many Rogers-Ramanujan type identities and polynomials approximating them are not even unique, there might be additional additional results; compare our previous eort [9] for innite versions. Most polynomials from Sill's list [10] are, however, not expressable in terms of one summation and thus not candidates for the present approach.…”

How to prove Ramanujan’s 𝑞-continued fractions

“…The first paper with continued fractions of this kind was written by Selberg [51]. Further examples appear in Gu and Prodinger [29].…”

The continued fraction expansion of Gauss' hypergeometric function and a new application to the tangent function

“…We demonstrate how such people can also derive the continued fraction expansion for tan(nx), by using a technique that has produced many other beautiful expansions [3,4].…”