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In mathematics the Mott polynomials s n x ) are polynomials given by the exponential 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!.}
They were introduced by Nevill Francis Mott who applied them to a problem in the theory of electrons.
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
C
k
{\displaystyle C_{k}}
x
k
{\displaystyle x^{k}}
[
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 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}}
generalized Appell polynomials , where the sum is over all compositions
n
=
l
1
+
l
2
+
⋯
+
l
k
{\displaystyle n=l_{1}+l_{2}+\cdots +l_{k}}
n
{\displaystyle n}
k
{\displaystyle k}
k
=
n
=
0
{\displaystyle k=n=0}
[
x
n
]
s
n
(
x
)
=
(
−
1
)
n
2
n
.
{\displaystyle s_{n}(x)={\frac {(-1)^{n}}{2^{n}}}.}
[
x
n
−
2
]
s
n
(
x
)
=
(
−
1
)
n
n
(
n
−
1
)
(
n
−
2
)
2
n
.
{\displaystyle 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 in the OEIS )
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};}
The polynomials s n x ) form the associated Sheffer sequence for –2t /(1–t)
An explicit expression for them in terms of the generalized hypergeometric function 3 F0 :
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
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 ISBN 978-0-12-594380-2 MR 0741185 . Reprinted by Dover, 2005.
Erdélyi, Arthur; Magnus, Wilhelm ; Oberhettinger, Fritz ; Tricomi, Francesco G. (1955). Higher transcendental functions. Vol. III MR 0066496 .
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↑
, the coefficient in front of 
of the polynomial can be written as
, according to the general formula for 
of 
into 
positive odd integers. The empty product appearing for 
equals 1. Special values, where all contributing Catalan numbers equal 1, are
