Mott polynomials

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

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

Introduction

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

Logic

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

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

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

$s_0(x)=1;$

$s_1(x)=-\frac{1}{2}x;$

$s_2(x)=\frac{1}{4}{x}^2;$

$s_3(x)=-\frac{3}{4}x-\frac{1}{8}{x}^3;$

$s_4(x)=\frac{3}{2}{x}^2+\frac{1}{16}{x}^4;$

$s_5(x)=-\frac{15}{2}x-\frac{15}{8}{x}^3-\frac{1}{32}{x}^5;$

$s_6(x)=\frac{225}{8}{x}^2+\frac{15}{8}{x}^4+\frac{1}{64}{x}^6;$

Sheffer sequence

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

Generalized hypergeometric function

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

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