Mott polynomials


In mathematics the Mott polynomials sn(x) are polynomials given by the exponential generating function:

They were introduced by Nevill Francis Mott who applied them to a problem in the theory of electrons.[1]

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)

The polynomials sn(x) form the associated Sheffer sequence for –2t/(1–t2)[2]

An explicit expression for them in terms of the generalized hypergeometric function 3F0:[3]


  1. ^ 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 [442]. doi:10.1098/rspa.1932.0044. ISSN 0950-1207. JSTOR 95868.
  2. ^ Roman, Steven (1984). The umbral calculus. Pure and Applied Mathematics. Vol. 111. London: Academic Press Inc. [Harcourt Brace Jovanovich Publishers]. p. 130. ISBN 978-0-12-594380-2. MR 0741185. Reprinted by Dover, 2005.
  3. ^ Erdélyi, Arthur; Magnus, Wilhelm; Oberhettinger, Fritz [in German]; Tricomi, Francesco G. (1955). Higher transcendental functions. Vol. III. New York-Toronto-London: McGraw-Hill Book Company, Inc. p. 251. MR 0066496.