Convergence Aspects for q‐Appell Functions and q‐Lauricella Functions with Applications to the Ward q‐Addition

We first outline the foundations of the q‐umbral calculus, which consists of an infinite alphabet A of letters or umbrae, with two dual q‐additions, the Nalli‐Ward‐Alsalam (NWA) q‐addition and the Jackson‐Hahn‐Cigler (JHC) q‐addition. By using a certain q‐Stirling approximation, we show that the NWA decides the convergence region for 50% of the q‐Appell‐ and q‐Lauricella functions. This assumption is verified by numerical examples. In the process we investigate numerical aspects, including a local maximum, of the NWA q‐addition of two and three letters.