Mott polynomials

In mathematics the Mott polynomials s_n(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 s_n(x) form the associated Sheffer sequence for –2t/(1–t²)

Generalized hypergeometric function

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

References