The complete Bell polynomials for certain arguments in terms of Stirling numbers of the first kind
The \(p\)-th exponential complete Bell polynomial \(Y_ p (x_ 1,\dots, x_ p)\) is defined by \[ \exp \biggl( \sum_{j=1}^ \infty x_ j {{t^ j} \over {j!}} \biggr)= 1+ \sum_{p=1}^ \infty Y_ p (x_ 1,\dots, x_ p) {{t^ p} \over {p!}}. \] Such polynomial for certain arguments, given essentially by finite sums of reciprocal powers with a real parameter, is expressed in terms of the coefficients \(S(p, q; \alpha)\) generated by the power series expansion of a Pochhammer symbol, i.e. \[ (1+\alpha+ x)_ q= \sum_{p=0}^ q S(p,q; \alpha) x^ p. \] This result generalizes an earlier representation, in terms of Stirling numbers, given by \textit{L. Comtet} [The art of finite and infinite expansions. (1974; Zbl 0283.05001)].
scientific article; zbMATH DE number 641403