# Mott polynomials

In mathematics the Mott polynomials sn(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

${\displaystyle e^{x({\sqrt {1-t^{2}}}-1)/t}=\sum _{n}s_{n}(x)t^{n}/n!.}$

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

${\displaystyle s_{0}(x)=1;}$
${\displaystyle s_{1}(x)=-{\frac {1}{2}}x;}$
${\displaystyle s_{2}(x)={\frac {1}{4}}x^{2};}$
${\displaystyle s_{3}(x)=-{\frac {3}{4}}x-{\frac {1}{8}}x^{3};}$
${\displaystyle s_{4}(x)={\frac {3}{2}}x^{2}+{\frac {1}{16}}x^{4};}$
${\displaystyle s_{5}(x)=-{\frac {15}{2}}x-{\frac {15}{8}}x^{3}-{\frac {1}{32}}x^{5};}$
${\displaystyle s_{6}(x)={\frac {225}{8}}x^{2}+{\frac {15}{8}}x^{4}+{\frac {1}{64}}x^{6};}$

The polynomials sn(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, ISSN 0950-1207, JSTOR 95868, doi:10.1098/rspa.1932.0044
• 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