Mott polynomials

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 : \frac{\sqrt{1-t^2}-1}{t} = -\sum_{k\ge 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}. For odd n and k=1 : [x]s_n(x) = -\frac{n!}{2^n}C_{(n-1)/2}

By differentiation the recurrence for the first derivative becomes : s'(x) =- \sum_{m=0}^{n-1} \frac{n!}{m!2^{n-m}} C_{(n-1-m)/2}s_m(x), where the sum is over the m such that (n-1-m)/2 is integer.

: 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 :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;

Sheffer sequence

The polynomials s**n(x) form the associated Sheffer sequence for –2t/(1–t2)

Generalized hypergeometric function

An explicit expression for them in terms of the generalized hypergeometric function 3F0: :s_n(x)=(-x/2)^n{}_3F_0(-n,\frac{1-n}{2},1-\frac{n}{2};;-\frac{4}{x^2})

References

References

1. (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.
2. (1984). "The umbral calculus". Academic Press Inc. [Harcourt Brace Jovanovich Publishers].
3. (1955). "Higher transcendental functions. Vol. III". McGraw-Hill Book Company, Inc..