This article has multiple issues. Please help improve it or discuss these issues on the talk page. (Learn how and when to remove these messages) This article needs additional or more specific categories. Please help out by adding categories to it so that it can be listed with similar articles. (November 2025) This article needs more citations. Please help improve this article by adding citations to reliable sources. Unsourced material may be challenged and removed. Find sources: "Mott polynomials" – news · newspapers · books · scholar · JSTOR (November 2025) (Learn how and when to remove this message) This article may be too technical for most readers to understand. Please help improve it to make it understandable to non-experts, without removing the technical details. (November 2025) (Learn how and when to remove this message) (Learn how and when to remove this message)

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

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

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

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)^[2]

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

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