Exponential-type extended Riordan arrays and reciprocity law for generalized Stirling numbers

Abstract

We propose an exponential-type extended Riordan array (or exponential recursive matrix) with permitted negative indices using the Roman factorial. In this context, we establish a fundamental theory of the generalized Stirling numbers introduced by Hsu and Shiue [11]. Subsequently, we provide a lucid explanation for the reciprocity law among these numbers found by Choi et al. [5] and Maltenfort [19]. Moreover, our method reveals that interesting numbers emerge in the fourth quadrant of the proposed exponential recursive matrix.