In mathematics the Mott polynomialssn(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
Because the factor in the exponential has the power series
in terms of Catalan numbers, the coefficient in front of of the polynomial can be written as
,
according to the general formula for generalized Appell polynomials,
where the sum is over all compositions of into positive odd integers. The empty product appearing for equals 1. Special values, where all contributing Catalan numbers equal 1, are
By differentiation the recurrence for the first derivative becomes
The first few of them are (sequence A137378 in the OEIS)
Erdélyi, Arthur; Magnus, Wilhelm; Oberhettinger, Fritz; Tricomi, Francesco G. (1955), Higher transcendental functions. Vol. III, McGraw-Hill Book Company, Inc., New York-Toronto-London, MR0066496
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): 429–458, doi:10.1098/rspa.1932.0044, ISSN0950-1207, JSTOR95868
Roman, Steven (1984), The umbral calculus, Pure and Applied Mathematics, vol. 111, London: Academic Press Inc. [Harcourt Brace Jovanovich Publishers], ISBN978-0-12-594380-2, MR0741185, Reprinted by Dover, 2005