# Mott polynomials

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In mathematics the **Mott polynomials** *s**n*(*x*) are polynomials given by the exponential [generating function](/source/Generating_function):

- e x ( 1 − t 2 − 1 ) / t = ∑ n s n ( x ) t n / n ! . {\displaystyle e^{x({\sqrt {1-t^{2}}}-1)/t}=\sum _{n}s_{n}(x)t^{n}/n!.}

## Introduction

They were introduced by [Nevill Francis Mott](/source/Nevill_Francis_Mott) who applied them to a problem in the theory of electrons.[1]

## Logic

Because the factor in the exponential has the power series

- 1 − t 2 − 1 t = − ∑ k ≥ 0 C k ( t 2 ) 2 k + 1 {\displaystyle {\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](/source/Catalan_number) C k {\displaystyle C_{k}} , the coefficient in front of x k {\displaystyle x^{k}} of the polynomial can be written as

- [ x k ] s n ( x ) = ( − 1 ) k n ! k ! 2 n ∑ n = l 1 + l 2 + ⋯ + l k C ( l 1 − 1 ) / 2 C ( l 2 − 1 ) / 2 ⋯ C ( l k − 1 ) / 2 {\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](/source/Generalized_Appell_polynomials), where the sum is over all [compositions](/source/Composition_(combinatorics)) n = l 1 + l 2 + ⋯ + l k {\displaystyle n=l_{1}+l_{2}+\cdots +l_{k}} of n {\displaystyle n} into k {\displaystyle k} positive odd integers. The [empty product](/source/Empty_product) appearing for k = n = 0 {\displaystyle k=n=0} equals 1. Special values, where all contributing Catalan numbers equal 1, are

- [ x n ] s n ( x ) = ( − 1 ) n 2 n . {\displaystyle [x^{n}]s_{n}(x)={\frac {(-1)^{n}}{2^{n}}}.}

- [ x n − 2 ] s n ( x ) = ( − 1 ) n n ( n − 1 ) ( n − 2 ) 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

- s ′ ( x ) = − ∑ k = 0 ⌊ ( n − 1 ) / 2 ⌋ n ! ( n − 1 − 2 k ) ! 2 2 k + 1 C k s n − 1 − 2 k ( x ) . {\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](https://oeis.org/A137378) in the [OEIS](/source/On-Line_Encyclopedia_of_Integer_Sequences))

- s 0 ( x ) = 1 ; {\displaystyle s_{0}(x)=1;}

- s 1 ( x ) = − 1 2 x ; {\displaystyle s_{1}(x)=-{\frac {1}{2}}x;}

- s 2 ( x ) = 1 4 x 2 ; {\displaystyle s_{2}(x)={\frac {1}{4}}x^{2};}

- s 3 ( x ) = − 3 4 x − 1 8 x 3 ; {\displaystyle s_{3}(x)=-{\frac {3}{4}}x-{\frac {1}{8}}x^{3};}

- s 4 ( x ) = 3 2 x 2 + 1 16 x 4 ; {\displaystyle s_{4}(x)={\frac {3}{2}}x^{2}+{\frac {1}{16}}x^{4};}

- s 5 ( x ) = − 15 2 x − 15 8 x 3 − 1 32 x 5 ; {\displaystyle s_{5}(x)=-{\frac {15}{2}}x-{\frac {15}{8}}x^{3}-{\frac {1}{32}}x^{5};}

- s 6 ( x ) = 225 8 x 2 + 15 8 x 4 + 1 64 x 6 ; {\displaystyle 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](/source/Sheffer_sequence) for –2*t*/(1–t2)[2]

## Generalized hypergeometric function

An explicit expression for them in terms of the [generalized hypergeometric function](/source/Generalized_hypergeometric_function) 3F0:[3]

- s n ( x ) = ( − x / 2 ) n 3 F 0 ( − n , 1 − n 2 , 1 − n 2 ; ; − 4 x 2 ) {\displaystyle s_{n}(x)=(-x/2)^{n}{}_{3}F_{0}(-n,{\frac {1-n}{2}},1-{\frac {n}{2}};;-{\frac {4}{x^{2}}})}

## References

1. **[^](#cite_ref-1)** Mott, N. F. (1932). ["The Polarisation of Electrons by Double Scattering"](https://doi.org/10.1098%2Frspa.1932.0044). *Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character*. **135** (827): 429–458 [442]. [Bibcode](/source/Bibcode_(identifier)):[1932RSPSA.135..429M](https://ui.adsabs.harvard.edu/abs/1932RSPSA.135..429M). [doi](/source/Doi_(identifier)):[10.1098/rspa.1932.0044](https://doi.org/10.1098%2Frspa.1932.0044). [ISSN](/source/ISSN_(identifier)) [0950-1207](https://search.worldcat.org/issn/0950-1207). [JSTOR](/source/JSTOR_(identifier)) [95868](https://www.jstor.org/stable/95868).

1. **[^](#cite_ref-2)** Roman, Steven (1984). [*The umbral calculus*](https://books.google.com/books?id=JpHjkhFLfpgC). Pure and Applied Mathematics. Vol. 111. London: Academic Press Inc. [Harcourt Brace Jovanovich Publishers]. p. 130. [ISBN](/source/ISBN_(identifier)) [978-0-12-594380-2](https://en.wikipedia.org/wiki/Special:BookSources/978-0-12-594380-2). [MR](/source/MR_(identifier)) [0741185](https://mathscinet.ams.org/mathscinet-getitem?mr=0741185). Reprinted by Dover, 2005.

1. **[^](#cite_ref-3)** Erdélyi, Arthur; [Magnus, Wilhelm](/source/Wilhelm_Magnus); [Oberhettinger, Fritz](https://de.wikipedia.org/wiki/Fritz_Oberhettinger) [in German]; Tricomi, Francesco G. (1955). [*Higher transcendental functions. Vol. III*](https://authors.library.caltech.edu/43491/). New York-Toronto-London: McGraw-Hill Book Company, Inc. p. 251. [MR](/source/MR_(identifier)) [0066496](https://mathscinet.ams.org/mathscinet-getitem?mr=0066496).

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Adapted from the Wikipedia article [Mott polynomials](https://en.wikipedia.org/wiki/Mott_polynomials) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Mott_polynomials?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.