Mott polynomials

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

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

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

Because the factor in the exponential has the power series

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

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

$[x^{k}]s_{n}(x)=(-1)^{k}{\frac {n!}{k!2^{n}}}\sum _{n=l_{1}+l_{2}+\cdots +l_{k}}C_{(l_{1}-1)/2}C_{(l_{2}-1)/2}\cdots C_{(l_{k}-1)/2}$, according to the general formula for generalized Appell polynomials, where the sum is over all compositions $n=l_{1}+l_{2}+\cdots +l_{k}$ of $n$ into $k$ positive odd integers. The empty product appearing for $k=n=0$ equals 1. Special values, where all contributing Catalan numbers equal 1, are

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

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

By differentiation the recurrence for the first derivative becomes

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

The first few of them are (sequence A137378 in the OEIS)

$s_{0}(x)=1;$

$s_{1}(x)=-{\frac {1}{2}}x;$

$s_{2}(x)={\frac {1}{4}}x^{2};$

$s_{3}(x)=-{\frac {3}{4}}x-{\frac {1}{8}}x^{3};$

$s_{4}(x)={\frac {3}{2}}x^{2}+{\frac {1}{16}}x^{4};$

$s_{5}(x)=-{\frac {15}{2}}x-{\frac {15}{8}}x^{3}-{\frac {1}{32}}x^{5};$

$s_{6}(x)={\frac {225}{8}}x^{2}+{\frac {15}{8}}x^{4}+{\frac {1}{64}}x^{6};$

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

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

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