q-Stirling Numbers, an Umbral Approach

Three different approaches to q-difference operators are given, the first one applies to C(q)[x] and the last two to C(q)[q x]. For the first one (Hahn–Cigler), definitions and basic formulas for the two q-Stirling numbers are given. For the second (Carlitz–Gould), and third approach (Jackson), the respective q-Taylor formulas are used to find a q-binomial coefficient identity. Three different formulas for Carlitz ’ q-analogue of sums of powers are found. The first one uses a double sum for q-Stirling numbers. The last two are multiple sums with q-binomial coefficients.