On generalized exponential Euler polynomials (Q2266092)

Certain polynomials generated by the relation \[ ((1-t)/(e^ z- t))^{r+1}e^{xz}=\sum^{\infty}_{n=0}A_{n,r}(x,t)z^ n/n!,\quad t\neq 1, \] where r is a non-positive integer, have been investigated by \textit{T. N. E. Greville}, \textit{I. J. Schoenberg}, and \textit{A. Sharma} [J. Approximation Theory 17, 200-221 (1976; Zbl 0332.41004)] in connection with the theory of cardinal splines. If \(r+1\) is replaced by \(\alpha\), an arbitrary real number, then the so-called generalized exponential Euler polynomials, introduced here, are generated. The object of this present paper is to study these generalized polynomials from the point of view of the theory of special functions using the umbral calculus of \textit{G.-C. Rota} [Finite operator calculus (1975; Zbl 0328.05007)]. Interesting results by way of recurrence relations, operational representations and umbral compositions are deduced.