// Workers AI · dad joke modeDo Narumi polynomials have roots? They're always polynomial-ite.

In mathematics, Narumi polynomials are polynomials introduced by Narumi (1929)^[1] given by the generating function

\sum _{n=0}^{\infty }{\frac {s_{n}(x)}{n!}}t^{n}=\left({\frac {t}{\log(1+t)}}\right)^{a}(1+t)^{x}.

g(t)={\Big (}{\frac {e^{t}-1}{t}}{\Big )}^{-a},\quad f(t)=e^{t}-1.

References

↑ Narumi, S. (1929), "On a power series having only a finite number of algebraico logarithmic singularities on its circle of convergence.", Tôhoku Mathematical Journal, 30: 185–201, JFM 55.0185.03
↑ Boas, Ralph P.; Buck, R. Creighton (1964) [1958]. Polynomial Expansions of Analytic Functions. Ergebnisse der Mathematik und ihrer Grenzgebiete. 1. Folge. Vol. 19 (2nd ed.). Springer Berlin, Heidelberg. ISBN 978-3-662-23179-1. MR 0094466. {{cite book}}: ISBN / Date incompatibility (help)
↑ Roman, Steven (1984). The Umbral Calculus. London: Academic Press. ISBN 978-0-08-087430-2.

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