- Source: Lommel polynomial
A Lommel polynomial Rm,ν(z) is a polynomial in 1/z giving the recurrence relation
J
m
+
ν
(
z
)
=
J
ν
(
z
)
R
m
,
ν
(
z
)
−
J
ν
−
1
(
z
)
R
m
−
1
,
ν
+
1
(
z
)
{\displaystyle \displaystyle J_{m+\nu }(z)=J_{\nu }(z)R_{m,\nu }(z)-J_{\nu -1}(z)R_{m-1,\nu +1}(z)}
where Jν(z) is a Bessel function of the first kind.
They are given explicitly by
R
m
,
ν
(
z
)
=
∑
n
=
0
[
m
/
2
]
(
−
1
)
n
(
m
−
n
)
!
Γ
(
ν
+
m
−
n
)
n
!
(
m
−
2
n
)
!
Γ
(
ν
+
n
)
(
z
/
2
)
2
n
−
m
.
{\displaystyle R_{m,\nu }(z)=\sum _{n=0}^{[m/2]}{\frac {(-1)^{n}(m-n)!\Gamma (\nu +m-n)}{n!(m-2n)!\Gamma (\nu +n)}}(z/2)^{2n-m}.}
See also
Lommel function
Neumann polynomial
References
Erdélyi, Arthur; Magnus, Wilhelm; Oberhettinger, Fritz; Tricomi, Francesco G. (1953), Higher transcendental functions. Vol II (PDF), McGraw-Hill Book Company, Inc., New York-Toronto-London, MR 0058756
Ivanov, A. B. (2001) [1994], "Lommel polynomial", Encyclopedia of Mathematics, EMS Press
Kata Kunci Pencarian:
- Lommel polynomial
- Lommel (disambiguation)
- Lommel function
- Eugen von Lommel
- Bessel function
- List of eponyms of special functions
- Bessel polynomials
- Neumann polynomial
- Hahn–Exton q-Bessel function
- Generalized hypergeometric function
