Pidduck polynomials

In mathematics, the Pidduck polynomials s_n(x) are polynomials introduced by Pidduck (1910, 1912) given by the generating function

$\displaystyle \sum_n \frac{s_n(x)}{n!}t^n = \left(\frac{1+t}{1-t}\right)^x(1-t)^{-1}$

(Roman 1984, 4.4.3), (Boas & Buck 1958, p.38)

See also

• Umbral calculus

References

• Boas, Ralph P.; Buck, R. Creighton (1958), Polynomial expansions of analytic functions, Ergebnisse der Mathematik und ihrer Grenzgebiete. Neue Folge. 19, Berlin, New York: Springer-Verlag, MR 0094466
• Pidduck, F. B. (1910), "On the Propagation of a Disturbance in a Fluid under Gravity", Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character (The Royal Society) 83 (563): 347–356, doi:10.1098/rspa.1910.0023, ISSN 0950-1207, JSTOR 92977
• Pidduck, F. B. (1912), "The Wave-Problem of Cauchy and Poisson for Finite Depth and Slightly Compressible Fluid", Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character (The Royal Society) 86 (588): 396–405, doi:10.1098/rspa.1912.0031, ISSN 0950-1207, JSTOR 93103
• Roman, Steven (1984), The umbral calculus, Pure and Applied Mathematics 111, London: Academic Press Inc. [Harcourt Brace Jovanovich Publishers], ISBN 978-0-12-594380-2, MR 741185 Reprinted by Dover, 2005