Mott polynomials

In mathematics the Mott polynomials s_n(x) are polynomials introduced by N. F. Mott (1932, p. 442) who applied them to a problem in the theory of electrons. They are given by the exponential generating function

e^{x(\sqrt{1-t^2}-1)/t}=\sum_n s_n(x) t^n/n!.

The first few of them are (sequence A137378 in the OEIS)

s_0(x)=1;

s_1(x)=-\frac{1}{2}x;

s_2(x)=\frac{1}{4}x^2;

s_3(x)=-\frac{3}{4}x-\frac{1}{8}x^3;

s_4(x)=\frac{3}{2}x^2+\frac{1}{16}x^4;

s_5(x)=-\frac{15}{2}x-\frac{15}{8}x^3-\frac{1}{32}x^5;

s_6(x)=\frac{225}{8}x^2+\frac{15}{8}x^4+\frac{1}{64}x^6;

The polynomials s_n(x) form the associated Sheffer sequence for –2t/(1–t²) (Roman 1984, p.130). Arthur Erdélyi, Wilhelm Magnus, and Fritz Oberhettinger et al. (1955, p. 251) give an explicit expression for them in terms of the generalized hypergeometric function ₃F₀.

References

Erdélyi, Arthur; Magnus, Wilhelm; Oberhettinger, Fritz; Tricomi, Francesco G. (1955), Higher transcendental functions. Vol. III, McGraw-Hill Book Company, Inc., New York-Toronto-London, MR 0066496
Mott, N. F. (1932), "The Polarisation of Electrons by Double Scattering", Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character, The Royal Society, 135 (827): 429–458, doi:10.1098/rspa.1932.0044, ISSN 0950-1207, JSTOR 95868
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

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