In mathematics the Mott polynomials sn(x) are polynomials introduced by N. F. Mott (1932, p. 442) who applied them to a problem in the theory of electrons. They are given by the exponential generating function
\( e^{x(\sqrt{1-t^2}-1)/t}=\sum_n s_n(x) t^n/n!. \)
Because the factor in the exponential has the power series
\( {\displaystyle {\frac {{\sqrt {1-t^{2}}}-1}{t}}=-\sum _{k\geq 0}C_{k}({\frac {t}{2}})^{2k+1}} \)
in terms of Catalan numbers \( C_{k} \) , the coefficient in front of \( x^{k} \) of the polynomial can be written as
\( {\displaystyle [x^{k}]s_{n}(x)=(-1)^{k}{\frac {n!}{k!2^{n}}}\sum _{n=l_{1}+l_{2}+\cdots l_{k}}C_{(l_{1}-1)/2}C_{(l_{2}-1)/2}\cdots C_{(l_{k}-1)/2}}, \)
according to the general formula for generalized Appell polynomials, where the sum is over all compositions \( {\displaystyle n=l_{1}+l_{2}+\cdots l_{k}} \) of n into k positive odd integers. The empty product appearing for \( {\displaystyle k=n=0} \) equals 1. Special values, where all contributing Catalan numbers equal 1, are
\( {\displaystyle [x^{n}]s_{n}(x)={\frac {(-1)^{n}}{2^{n}}}.} \)
\( {\displaystyle [x^{n-2}]s_{n}(x)={\frac {(-1)^{n}n(n-1)(n-2)}{2^{n}}}.} \)
By differentiation the recurrence for the first derivative becomes
\( {\displaystyle s'(x)=-\sum _{k=0}^{\lfloor (n-1)/2\rfloor }{\frac {n!}{(n-1-2k)!2^{2k+1}}}C_{k}s_{n-1-2k}(x).} \)
The first few of them are (sequence A137378 in the OEIS)
\( s_0(x)=1; \)
\( s_1(x)=-\frac{1}{2}x; \)
\( s_2(x)=\frac{1}{4}x^2; \)
\( s_3(x)=-\frac{3}{4}x-\frac{1}{8}x^3; \)
\( s_4(x)=\frac{3}{2}x^2+\frac{1}{16}x^4; \)
\( s_5(x)=-\frac{15}{2}x-\frac{15}{8}x^3-\frac{1}{32}x^5; \)
\( s_6(x)=\frac{225}{8}x^2+\frac{15}{8}x^4+\frac{1}{64}x^6; \)
The polynomials sn(x) form the associated Sheffer sequence for –2t/(1–t2) (Roman 1984, p.130). Arthur Erdélyi, Wilhelm Magnus, and Fritz Oberhettinger et al. (1955, p. 251) give an explicit expression for them in terms of the generalized hypergeometric function 3F0:
\( {\displaystyle s_{n}(x)=(-x/2)^{n}{}_{3}F_{0}(-n,{\frac {1-n}{2}},1-{\frac {n}{2}};;-{\frac {4}{x^{2}}})} \)
References
Erdélyi, Arthur; Magnus, Wilhelm; Oberhettinger, Fritz; Tricomi, Francesco G. (1955), Higher transcendental functions. Vol. III, McGraw-Hill Book Company, Inc., New York-Toronto-London, MR 0066496
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, doi:10.1098/rspa.1932.0044, ISSN 0950-1207, JSTOR 95868
Roman, Steven (1984), The umbral calculus, Pure and Applied Mathematics, 111, London: Academic Press Inc. [Harcourt Brace Jovanovich Publishers], ISBN 978-0-12-594380-2, MR 0741185, Reprinted by Dover, 2005
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