Mott polynomials

In mathematics the Mott polynomials sn(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
 * $$ \frac{\sqrt{1-t^2}-1}{t} = -\sum_{k\ge 0} C_k \left(\frac{t}{2}\right)^{2k+1}$$

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
 * $$s_0(x)=1;$$
 * $$s_1(x)=-\frac{1}{2}x;$$
 * $$s_2(x)=\frac{1}{4}x^2;$$
 * $$s_3(x)=-\frac{3}{4}x-\frac{1}{8}x^3;$$
 * $$s_4(x)=\frac{3}{2}x^2+\frac{1}{16}x^4;$$
 * $$s_5(x)=-\frac{15}{2}x-\frac{15}{8}x^3-\frac{1}{32}x^5;$$
 * $$s_6(x)=\frac{225}{8}x^2+\frac{15}{8}x^4+\frac{1}{64}x^6;$$

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

An explicit expression for them in terms of the generalized hypergeometric function 3F0:
 * $$s_n(x)=(-x/2)^n{}_3F_0(-n,\frac{1-n}{2},1-\frac{n}{2};;-\frac{4}{x^2})$$