Mott polynomials

Last updated May 23, 2024

In mathematics the Mott polynomialss_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!.

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

{\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 $C_{k}$ , 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'(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};

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,{\frac {1-n}{2}},1-{\frac {n}{2}};;-{\frac {4}{x^{2}}})

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References

↑ Mott, N. F. (1932). "The Polarisation of Electrons by Double Scattering". Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character. 135 (827): 429–458 [442]. doi: 10.1098/rspa.1932.0044 . ISSN 0950-1207. JSTOR 95868.
↑ Roman, Steven (1984). The umbral calculus. Pure and Applied Mathematics. Vol. 111. London: Academic Press Inc. [Harcourt Brace Jovanovich Publishers]. p. 130. ISBN 978-0-12-594380-2. MR 0741185. Reprinted by Dover, 2005.
↑ Erdélyi, Arthur; Magnus, Wilhelm; Oberhettinger, Fritz [in German]; Tricomi, Francesco G. (1955). Higher transcendental functions. Vol. III. New York-Toronto-London: McGraw-Hill Book Company, Inc. p. 251. MR 0066496.

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[1] Mott, N. F. (1932). "The Polarisation of Electrons by Double Scattering". Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character. 135 (827): 429–458 [442]. doi: 10.1098/rspa.1932.0044 . ISSN 0950-1207. JSTOR 95868.

[2] Roman, Steven (1984). The umbral calculus. Pure and Applied Mathematics. Vol. 111. London: Academic Press Inc. [Harcourt Brace Jovanovich Publishers]. p. 130. ISBN 978-0-12-594380-2. MR 0741185. Reprinted by Dover, 2005.

[3] Erdélyi, Arthur; Magnus, Wilhelm; Oberhettinger, Fritz [in German]; Tricomi, Francesco G. (1955). Higher transcendental functions. Vol. III. New York-Toronto-London: McGraw-Hill Book Company, Inc. p. 251. MR 0066496.

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