In mathematics the Mott polynomials s_n(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

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

Because the factor in the exponential has the power series

${\displaystyle {\frac ((\sqrt {1-t^{2))}-1}{t))=-\sum _{k\geq 0}C_{k}\left({\frac {t}{2))\right)^{2k+1))$

in terms of Catalan numbers ${\displaystyle C_{k))$, the coefficient in front of ${\displaystyle x^{k))$ of the polynomial can be written as

${\displaystyle [x^{k}]s_{n}(x)=(-1)^{k}{\frac {n!}{k!2^{n))}\sum _{n=l_{1}+l_{2}+\cdots +l_{k))C_{(l_{1}-1)/2}C_{(l_{2}-1)/2}\cdots C_{(l_{k}-1)/2))$,

according to the general formula for generalized Appell polynomials, where the sum is over all compositions ${\displaystyle n=l_{1}+l_{2}+\cdots +l_{k))$ of ${\displaystyle n}$ into ${\displaystyle k}$ positive odd integers. The empty product appearing for ${\displaystyle k=n=0}$ equals 1. Special values, where all contributing Catalan numbers equal 1, are

${\displaystyle [x^{n}]s_{n}(x)={\frac {(-1)^{n)){2^{n))}.}$

${\displaystyle [x^{n-2}]s_{n}(x)={\frac {(-1)^{n}n(n-1)(n-2)}{2^{n))}.}$

By differentiation the recurrence for the first derivative becomes

${\displaystyle s'(x)=-\sum _{k=0}^{\lfloor (n-1)/2\rfloor }{\frac {n!}{(n-1-2k)!2^{2k+1))}C_{k}s_{n-1-2k}(x).}$

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 s_n(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:

${\displaystyle s_{n}(x)=(-x/2)^{n}{}_{3}F_{0}(-n,{\frac {1-n}{2)),1-{\frac {n}{2));;-{\frac {4}{x^{2))})}$