In mathematics the Mott polynomials sn(x) are polynomials given by the exponential generating function:
e x ( 1 − t 2 − 1 ) / t = ∑ n s n ( x ) t n / n ! . {\displaystyle 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
1 − t 2 − 1 t = − ∑ k ≥ 0 C k ( t 2 ) 2 k + 1 {\displaystyle {\frac {{\sqrt {1-t^{2}}}-1}{t}}=-\sum _{k\geq 0}C_{k}\left({\frac {t}{2}}\right)^{2k+1}}
in terms of Catalan numbers C k {\displaystyle C_{k}} , the coefficient in front of x k {\displaystyle x^{k}} of the polynomial can be written as
[ x k ] s n ( x ) = ( − 1 ) k n ! k ! 2 n ∑ n = l 1 + l 2 + ⋯ + l k C ( l 1 − 1 ) / 2 C ( l 2 − 1 ) / 2 ⋯ C ( l k − 1 ) / 2 {\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 n = l 1 + l 2 + ⋯ + l k {\displaystyle n=l_{1}+l_{2}+\cdots +l_{k}} of n {\displaystyle n} into k {\displaystyle k} positive odd integers. The empty product appearing for k = n = 0 {\displaystyle k=n=0} equals 1. Special values, where all contributing Catalan numbers equal 1, are
[ x n ] s n ( x ) = ( − 1 ) n 2 n . {\displaystyle [x^{n}]s_{n}(x)={\frac {(-1)^{n}}{2^{n}}}.}
[ x n − 2 ] s n ( x ) = ( − 1 ) n n ( n − 1 ) ( n − 2 ) 2 n . {\displaystyle [x^{n-2}]s_{n}(x)={\frac {(-1)^{n}n(n-1)(n-2)}{2^{n}}}.}
… excerpt ends here. Continue reading the full article.
