A Lommel polynomial ''R''
''m'',ν(''z''), introduced by , is a polynomial in 1/''z'' giving the
recurrence relation
In mathematics, a recurrence relation is an equation according to which the nth term of a sequence of numbers is equal to some combination of the previous terms. Often, only k previous terms of the sequence appear in the equation, for a parameter ...
:
where ''J''
ν(''z'') is a
Bessel function
Bessel functions, first defined by the mathematician Daniel Bernoulli and then generalized by Friedrich Bessel, are canonical solutions of Bessel's differential equation
x^2 \frac + x \frac + \left(x^2 - \alpha^2 \right)y = 0
for an arbitrary ...
of the first kind.
They are given explicitly by
:
See also
*
Lommel function
The Lommel differential equation, named after Eugen von Lommel, is an inhomogeneous form of the Bessel differential equation:
: z^2 \frac + z \frac + (z^2 - \nu^2)y = z^.
Solutions are given by the Lommel functions ''s''μ,ν(''z'') and ''S' ...
*
Neumann polynomial In mathematics, the Neumann polynomials, introduced by Carl Neumann for the special case \alpha=0, are a sequence of polynomials in 1/t used to expand functions in term of Bessel functions.
The first few polynomials are
:O_0^(t)=\frac 1 t,
:O_1^(t ...
References
*
*
*{{citation, first=Eugen von , last=Lommel, title=Zur Theorie der Bessel'schen Functionen
, journal =Mathematische Annalen
, publisher =Springer , place=Berlin / Heidelberg
, volume =4, issue= 1 , year= 1871
, doi =10.1007/BF01443302
, pages =103–116
Polynomials
Special functions