Neumann polynomials 
are polynomials in 
for nonnegative integers 
and 
. For 
, they are defined by the sum
 |
(1)
|
where 
is the floor function. They obey the recurrence
relation
![O_n(x)=-n/(n-2)O_(n-2)(x)+(2n)/xO_(n-1)(x)+(2(n-1))/((n-2)x)sin^2[1/2(n-1)pi]](/images/equations/NeumannPolynomial/NumberedEquation2.svg) |
(2)
|
for 
.
They have the integral representation
 |
(3)
|
and the generating function
 |
(4)
|
for 
,
where 
is a Bessel function of the first kind
(Gradshteyn and Ryzhik 2000, p. 990). They obey the Neumann
differential equation.
The first few Neumann polynomials are given by
 |  |  |
(5)
|
 |  |  |
(6)
|
 |  |  |
(7)
|
 |  |  |
(8)
|
 |  |  |
(9)
|
The coefficients of 
in increasing power
of 
,
with rows read for 
,
1, ..., form OEIS A057869.
and 
."
§8.59 in