Pidduck polynomials

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

${\displaystyle \displaystyle \sum _{n}{\frac {s_{n}(x)}{n!}}t^{n}=\left({\frac {1+t}{1-t}}\right)^{x}(1-t)^{-1}}$

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

• Umbral calculus

• 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
• 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, 83 (563), The Royal Society: 347–356, Bibcode:1910RSPSA..83..347P, 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, 86 (588), The Royal Society: 396–405, Bibcode:1912RSPSA..86..396P, doi:10.1098/rspa.1912.0031, ISSN 0950-1207, JSTOR 93103
• 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 Publications, 2005

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