{{Multiple issues|{{more categories|date=November 2025}} {{more sources|date=November 2025}}{{Technical|date=November 2025}} }} In mathematics the '''Mott polynomials''' ''s''<sub>''n''</sub>(''x'') are polynomials given by the exponential [[generating function]]: :<math> e^{x(\sqrt{1-t^2}-1)/t}=\sum_n s_n(x) t^n/n!.</math> ==Introduction== They were introduced by [[Nevill Francis Mott]] who applied them to a problem in the theory of electrons.<ref>{{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 }}</ref> ==Logic== Because the factor in the exponential has the power series :<math> \frac{\sqrt{1-t^2}-1}{t} = -\sum_{k\ge 0} C_k \left(\frac{t}{2}\right)^{2k+1}</math> in terms of [[Catalan number]]s <math>C_k</math>, the coefficient in front of <math>x^k</math> of the polynomial can be written as :<math>[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}</math>, according to the general formula for [[generalized Appell polynomials]], where the sum is over all [[Composition (combinatorics)|compositions]] <math>n=l_1+l_2+\cdots+l_k</math> of <math>n</math> into <math>k</math> positive odd integers. The [[empty product]] appearing for <math>k=n=0</math> equals 1. Special values, where all contributing Catalan numbers equal 1, are :<math> [x^n]s_n(x) = \frac{(-1)^n}{2^n}.</math> :<math> [x^{n-2}]s_n(x) = \frac{(-1)^n n(n-1)(n-2)}{2^n}.</math> <!-- For odd <math>n</math> and <math>k=1</math> :<math> [x]s_n(x) = -\frac{n!}{2^n}C_{(n-1)/2}</math> --> By differentiation the recurrence for the first derivative becomes <!-- :<math> s'(x) =- \sum_{m=0}^{n-1} \frac{n!}{m!2^{n-m}} C_{(n-1-m)/2}s_m(x)</math>, where the sum is over the <math>m</math> such that <math>(n-1-m)/2</math> is integer. --> :<math> 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).</math>
The first few of them are {{OEIS|A137378}} :<math>s_0(x)=1;</math> :<math>s_1(x)=-\frac{1}{2}x;</math> :<math>s_2(x)=\frac{1}{4}x^2;</math> :<math>s_3(x)=-\frac{3}{4}x-\frac{1}{8}x^3;</math> :<math>s_4(x)=\frac{3}{2}x^2+\frac{1}{16}x^4;</math> :<math>s_5(x)=-\frac{15}{2}x-\frac{15}{8}x^3-\frac{1}{32}x^5;</math> :<math>s_6(x)=\frac{225}{8}x^2+\frac{15}{8}x^4+\frac{1}{64}x^6;</math> ==Sheffer sequence== The polynomials ''s''<sub>''n''</sub>(''x'') form the associated [[Sheffer sequence]] for –2''t''/(1–t<sup>2</sup>)<ref>{{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.</ref> ==Generalized hypergeometric function== An explicit expression for them in terms of the [[generalized hypergeometric function]] <sub>3</sub>F<sub>0</sub>:<ref>{{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/}}</ref> :<math>s_n(x)=(-x/2)^n{}_3F_0(-n,\frac{1-n}{2},1-\frac{n}{2};;-\frac{4}{x^2})</math>
==References== {{reflist}}
[[Category:Polynomials]]