Mott polynomials explained

In mathematics the Mott polynomials s_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

(\sqrt{ 1- t ^{2})/(-1 }{t}) = - \sum _{k\ge} C _{k} \left( (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 (n!)/(k! 2 ^{n})} \sum _{n = l _{1+l} _{2+ …} + l _{k}} C _{(l _{1-1)/2}} C _{(l _{2-1)/2}} … 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} 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)} = (( -1 ) ^{n})/(2 ^{n}) . [x ^{n-2} ]s _{n(x)} = (( -1 ) ^{n} n(n -1 )(n -2 ))/(2 ^{n}) . By differentiation the recurrence for the first derivative becomess'(x) =- \sum ^{\lfloor (n -1 )/ 2 \rfloor} _{k =0} (n!)/((n -1-2 k)! 2 ^{2k+1}) C _{k} s _{n-1-2k} (x).

The first few of them are

s _{0(x)=1;} s _{1 (x) =- (1)/(2)} x; s _{2 (x) = (1)/(4)} x ^{2;} s _{3 (x) =- (3)/(4)} x - (1)/(8) x ^{3;} s _{4 (x) = (3)/(2)} x ^{2+ (1)/(16)} x ^{4;} s _{5 (x) =- (15)/(2)} x - (15)/(8) x ^{3- (1)/(32)} x ^{5;} s _{6 (x) = (225)/(8)} x ^{2+ (15)/(8)} x ^{4+ (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{}} _{3 F} , 1- _{0 ( - n, (1- n)/(2)} (n)/(;; - 2) (4)/(x ^{2}) )

Notes and References

1. Mott . N. F. . The Polarisation of Electrons by Double Scattering . 95868 . 1932 . Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character . 0950-1207 . 135 . 827 . 429–458 [442] . 10.1098/rspa.1932.0044. free .
2. Book: Roman . Steven . The umbral calculus . Academic Press Inc. [Harcourt Brace Jovanovich Publishers] . London . Pure and Applied Mathematics . 978-0-12-594380-2 . 741185 . 1984 . 111 . 130. Reprinted by Dover, 2005.
3. Book: Erdélyi . Arthur . Magnus . Wilhelm . Wilhelm Magnus . Oberhettinger . Fritz .

   de:Fritz Oberhettinger

   . Tricomi . Francesco G. . Higher transcendental functions. Vol. III . McGraw-Hill Book Company, Inc. . New York-Toronto-London . 0066496 . 1955 . 251 .