Mott polynomials

In mathematics the Mott polynomials s_n(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

[math]\displaystyle{ e^{x(\sqrt{1-t^2}-1)/t}=\sum_n s_n(x) t^n/n!. }[/math]

Because the factor in the exponential has the power series

[math]\displaystyle{ \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 numbers [math]\displaystyle{ C_k }[/math], the coefficient in front of [math]\displaystyle{ x^k }[/math] of the polynomial can be written as

[math]\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} }[/math],

according to the general formula for generalized Appell polynomials, where the sum is over all compositions [math]\displaystyle{ n=l_1+l_2+\cdots+l_k }[/math] of [math]\displaystyle{ n }[/math] into [math]\displaystyle{ k }[/math] positive odd integers. The empty product appearing for [math]\displaystyle{ k=n=0 }[/math] equals 1. Special values, where all contributing Catalan numbers equal 1, are

[math]\displaystyle{ [x^n]s_n(x) = \frac{(-1)^n}{2^n}. }[/math]

[math]\displaystyle{ [x^{n-2}]s_n(x) = \frac{(-1)^n n(n-1)(n-2)}{2^n}. }[/math]

By differentiation the recurrence for the first derivative becomes

[math]\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). }[/math]

The first few of them are (sequence A137378 in the OEIS)

[math]\displaystyle{ s_0(x)=1; }[/math]

[math]\displaystyle{ s_1(x)=-\frac{1}{2}x; }[/math]

[math]\displaystyle{ s_2(x)=\frac{1}{4}x^2; }[/math]

[math]\displaystyle{ s_3(x)=-\frac{3}{4}x-\frac{1}{8}x^3; }[/math]

[math]\displaystyle{ s_4(x)=\frac{3}{2}x^2+\frac{1}{16}x^4; }[/math]

[math]\displaystyle{ s_5(x)=-\frac{15}{2}x-\frac{15}{8}x^3-\frac{1}{32}x^5; }[/math]

[math]\displaystyle{ s_6(x)=\frac{225}{8}x^2+\frac{15}{8}x^4+\frac{1}{64}x^6; }[/math]

The polynomials s_n(x) form the associated Sheffer sequence for –2t/(1–t²) (Roman 1984). 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:

[math]\displaystyle{ s_n(x)=(-x/2)^n{}_3F_0(-n,\frac{1-n}{2},1-\frac{n}{2};;-\frac{4}{x^2}) }[/math]

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 
• 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 
• Roman, Steven (1984), The umbral calculus, Pure and Applied Mathematics, 111, London: Academic Press Inc. [Harcourt Brace Jovanovich Publishers], Reprinted by Dover, 2005, ISBN 978-0-12-594380-2, https://books.google.com/books?id=JpHjkhFLfpgC