Mott polynomials

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In mathematics the Mott polynomials s_n(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!.}

They were introduced by Nevill Francis Mott who applied them to a problem in the theory of electrons.^[1]

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}}}.}

By differentiation the recurrence for the first derivative becomes

s ′ ( x ) = − ∑ k = 0 ⌊ ( n − 1 ) / 2 ⌋ n ! ( n − 1 − 2 k ) ! 2 2 k + 1 C k s n − 1 − 2 k ( x ) . {\displaystyle 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 (sequence A137378 in the OEIS)

s 0 ( x ) = 1 ; {\displaystyle s_{0}(x)=1;}

s 1 ( x ) = − 1 2 x ; {\displaystyle s_{1}(x)=-{\frac {1}{2}}x;}

s 2 ( x ) = 1 4 x 2 ; {\displaystyle s_{2}(x)={\frac {1}{4}}x^{2};}

s 3 ( x ) = − 3 4 x − 1 8 x 3 ; {\displaystyle s_{3}(x)=-{\frac {3}{4}}x-{\frac {1}{8}}x^{3};}

s 4 ( x ) = 3 2 x 2 + 1 16 x 4 ; {\displaystyle s_{4}(x)={\frac {3}{2}}x^{2}+{\frac {1}{16}}x^{4};}

s 5 ( x ) = − 15 2 x − 15 8 x 3 − 1 32 x 5 ; {\displaystyle s_{5}(x)=-{\frac {15}{2}}x-{\frac {15}{8}}x^{3}-{\frac {1}{32}}x^{5};}

s 6 ( x ) = 225 8 x 2 + 15 8 x 4 + 1 64 x 6 ; {\displaystyle s_{6}(x)={\frac {225}{8}}x^{2}+{\frac {15}{8}}x^{4}+{\frac {1}{64}}x^{6};}

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

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

s n ( x ) = ( − x / 2 ) n 3 F 0 ( − n , 1 − n 2 , 1 − n 2 ; ; − 4 x 2 ) {\displaystyle s_{n}(x)=(-x/2)^{n}{}_{3}F_{0}(-n,{\frac {1-n}{2}},1-{\frac {n}{2}};;-{\frac {4}{x^{2}}})}