A note on generalized Sylvester polynomials (Q796708)

The generalized Sylvester polynomials \(\phi_ n(x;c)\) defined by \[ (1)\quad \sum^{\infty}_{n=0}\phi_ n(x;c)t^ n=(1-t)^{- c}e^{cxt} \] are essentially the Laguerre polynomials \[ L_ n^{(-x-n)}(cx), \] the connecting relation being \((2)\quad \phi_ n(x;c)=(-1)^ nL_ n^{(-x-n)}(cx).\) Making use of (2), the author derives various classes of generating functions for \(\phi_ n(x;c)\) by specializing much more general results given earlier by the reviewer and \textit{J.-L. Lavoie} [Indagationes Math. 37, 304-320 (1975; Zbl 0297.30004)], the reviewer [ibid. 42, 221-233, 234-246 (1980; Zbl 0415.33008)], and the reviewer and \textit{H. L. Manocha} [A treatise on generating functions (1984; Zbl 0535.33001), p. 450, Problem 21]. This last reference contains a systematic discussion of the subject-matter of this paper.