Continued fraction expansions for q-tangent and q-cotangent functions

1 Philippe Flajolet and continued fractions In a paper that was written on the occasion of Philippe Flajolet’s 50th birthday [26] and discussed his various research areas, we wrote about his contributions to continued fractions: Continued fractions The papers [8, 9, 10] deal with the interplay of continued fractions and combinatorics. Let us consider lattice paths, consisting of steps NORTHEAST, EAST, SOUTHEAST, starting at the origin, returning to the x-axis after n steps, and never being negative. The possible steps are denoted by the letters {a, b, c}, and an index i is additionally used when a step starts at altitude i. Thus, such a lattice path is a word in the variables {a0, a1, . . . , b0, b1, . . . , c1, . . . }. The set of all paths (a formal language) is given by the infinite continued fraction