In mathematics the Mott polynomials sn(x) are polynomials given by the exponential generating function:
Introduction They were introduced by Nevill Francis Mott who applied them to a problem in the theory of electrons. Logic 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
Sheffer sequence The polynomials sn(x) form the associated Sheffer sequence for –2t/(1–t2) Reprinted by Dover, 2005. Generalized hypergeometric function An explicit expression for them in terms of the generalized hypergeometric function 3F0:
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