Several new aspects on q-Horn and related triple functions in the spirit of Karlsson

We first introduce a notation for multiple (n ≥ 3) q-hypergeometric functions, where negative values of summation indices are allowed. Then we extend the notation for q-Horn functions to include tilde values corresponding to powers of 2. Karlssons reduction formulas are correspondingly q-deformed by using these notations. A formula for sums of inverse q-shifted factorials is used to find further formulas. The second part of the paper is devoted to convergence aspects for q-Horn functions and ’abnormal’ q-Horn functions. It turns out that some simple estimates for convergence can be made in the q-case, these are then supplemented with tables of numerical values. It is shown that the convergence regions are significantly increased in the q-case, and we compare with convergence regions in the ordinary case.