Mott polynomials

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

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

Because the factor in the exponential has the power series

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

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

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

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)

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

${\displaystyle s_n(x)=(-x/2{)}^n{}_3{F}_0(-n,\frac{1-n}{2},1-\frac{n}{2};;-\frac{4}{x^2})}$