Mott polynomials

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

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

They were introduced by Nevill Francis Mott who applied them to a problem in the theory of electrons.{{cite journal | last1=Mott | first1=N. F. | title=The Polarisation of Electrons by Double Scattering | jstor=95868 | year=1932 | journal=Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character | issn=0950-1207 | volume=135 | issue=827 | pages=429–458 [442] | doi=10.1098/rspa.1932.0044| doi-access=free | bibcode=1932RSPSA.135..429M }}

Because the factor in the exponential has the power series

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

in terms of Catalan numbers $C\_k$, the coefficient in front of $x^k$ of the polynomial can be written as

:$[x^k]\; s\_n(x)\; =(-1)^k\backslash frac\{n!\}\{k!2^n\}\backslash sum\_\{n=l\_1+l\_2+\backslash cdots\; +l\_k\}C\_\{(l\_1-1)/2\}C\_\{(l\_2-1)/2\}\backslash cdots\; C\_\{(l\_k-1)/2\}$, according to the general formula for generalized Appell polynomials, where the sum is over all compositions $n=l\_1+l\_2+\backslash cdots+l\_k$ 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

:$[x^n]s\_n(x)\; =\; \backslash frac\{(-1)^n\}\{2^n\}.$

:$[x^\{n-2\}]s\_n(x)\; =\; \backslash frac\{(-1)^n\; n(n-1)(n-2)\}\{2^n\}.$

By differentiation the recurrence for the first derivative becomes

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

The first few of them are {{OEIS|A137378}}

:$s\_0(x)=1;$

:$s\_1(x)=-\backslash frac\{1\}\{2\}x;$

:$s\_2(x)=\backslash frac\{1\}\{4\}x^2;$

:$s\_3(x)=-\backslash frac\{3\}\{4\}x-\backslash frac\{1\}\{8\}x^3;$

:$s\_4(x)=\backslash frac\{3\}\{2\}x^2+\backslash frac\{1\}\{16\}x^4;$

:$s\_5(x)=-\backslash frac\{15\}\{2\}x-\backslash frac\{15\}\{8\}x^3-\backslash frac\{1\}\{32\}x^5;$

:$s\_6(x)=\backslash frac\{225\}\{8\}x^2+\backslash frac\{15\}\{8\}x^4+\backslash frac\{1\}\{64\}x^6;$

The polynomials s_n(x) form the associated Sheffer sequence for –2t/(1–t²){{cite book | last1=Roman | first1=Steven | title=The umbral calculus | url=https://books.google.com/books?id=JpHjkhFLfpgC | publisher=Academic Press Inc. [Harcourt Brace Jovanovich Publishers] | location=London | series=Pure and Applied Mathematics | isbn=978-0-12-594380-2 | mr=741185 | year=1984 | volume=111 |page=130}} Reprinted by Dover, 2005.

An explicit expression for them in terms of the generalized hypergeometric function ₃F₀:{{cite book | last1=Erdélyi | first1=Arthur | last2=Magnus | first2=Wilhelm | author2-link=Wilhelm Magnus | last3=Oberhettinger | first3=Fritz |author3-link=:de:Fritz Oberhettinger |last4=Tricomi | first4=Francesco G. | title=Higher transcendental functions. Vol. III | publisher=McGraw-Hill Book Company, Inc. |place= New York-Toronto-London | mr=0066496 | year=1955 |page=251 |url=https://authors.library.caltech.edu/43491/}}

:$s\_n(x)=(-x/2)^n\{\}\_3F\_0(-n,\backslash frac\{1-n\}\{2\},1-\backslash frac\{n\}\{2\};;-\backslash frac\{4\}\{x^2\})$