Peters polynomials

In mathematics, the Peters polynomials s_n(x) are polynomials studied by Peters (1956, 1956b) given by the generating function

\displaystyle \sum _{n=0}^{+\infty }s_{n}(x){\frac {t^{n}}{n!}}={\frac {(1+t)^{x}}{(1+(1+t)^{\lambda })^{\mu }}}

(Roman 1984, 4.4.6), (Boas & Buck 1958, p.37). They are a generalization of the Boole polynomials.

References

Boas, Ralph P.; Buck, R. Creighton (1958), Polynomial expansions of analytic functions, Ergebnisse der Mathematik und ihrer Grenzgebiete. Neue Folge., vol. 19, Berlin, New York: Springer-Verlag, ISBN 978-0-387-03123-1, MR 0094466
Peters, George Owen (1956), Schafer, Richard D. (ed.), "Boole polynomials of higher and negative orders", Bulletin of the A.M.S., 62 (1): 7, doi:10.1090/S0002-9904-1956-09972-0
Peters, George Owen (1956b), Schafer, Richard D. (ed.), "Boole polynomials and numbers of the second kind", Bulletin of the A.M.S., 62: 387, doi:10.1090/S0002-9904-1956-10046-3
Roman, Steven (1984), The umbral calculus, Pure and Applied Mathematics, vol. 111, London: Academic Press Inc. [Harcourt Brace Jovanovich Publishers], ISBN 978-0-12-594380-2, MR 0741185 Reprinted by Dover, 2005