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, 135 (827), The Royal Society: 429–458, doi:10.1098/rspa.1932.0044, ISSN 0950-1207, JSTOR 95868
• 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