Enumerating path diagrams in connection with $q$-tangent and $q$-secant numbers

We enumerate height-restricted path diagrams associated with $q$-tangent and $q$-secant numbers by considering convergents of continued fractions, leading to expressions involving basic hypergeometric functions. Our work generalises some results by M. Josuat-Verg'es for unrestricted path diagrams [European Journal of Combinatorics 31 (2010) 1892].