Geodesic
Continued fraction expansions for q-tangent and q-cotangent functions
Discrete mathematics & theoretical computer science, Tome 12 (2010) no. 2.

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For 3 different versions of q-tangent resp. q-cotangent functions, we compute the continued fraction expansion explicitly, by guessing the relative quantities and proving the recursive relation afterwards. It is likely that these are the only instances with a ''nice'' expansion. Additional formulae of a similar type are also provided. 
@article{DMTCS_2010_12_2_a3,
     author = {Prodinger, Helmut},
     title = {Continued fraction expansions for q-tangent and q-cotangent functions},
     journal = {Discrete mathematics & theoretical computer science},
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     volume = {12},
     number = {2},
     year = {2010},
     doi = {10.46298/dmtcs.492},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.46298/dmtcs.492/}
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Prodinger, Helmut. Continued fraction expansions for q-tangent and q-cotangent functions. Discrete mathematics & theoretical computer science, Tome 12 (2010) no. 2. doi : 10.46298/dmtcs.492. http://geodesic.mathdoc.fr/articles/10.46298/dmtcs.492/

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