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In mathematics , Ferrers functions are certain special functions defined in terms of hypergeometric functions .[ 1] [ 2] Norman Macleod Ferrers .[ 3]
Definitions
Define
μ
{\displaystyle \mu }
order , and the
ν
{\displaystyle \nu }
degree are real, and assume
x
∈
(
−
1
,
+
1
)
{\displaystyle x\in (-1,+1)}
Ferrers function of the first kind
P
v
μ
(
x
)
=
(
1
+
x
1
−
x
)
μ
/
2
⋅
2
F
1
(
v
+
1
,
−
v
;
1
−
μ
;
1
/
2
−
x
/
2
)
Γ
(
1
−
μ
)
{\displaystyle P_{v}^{\mu }(x)=\left({\frac {1+x}{1-x}}\right)^{\mu /2}\cdot {\frac {{}_{2}F_{1}(v+1,-v;1-\mu ;1/2-x/2)}{\Gamma (1-\mu )}}}
Ferrers function of the second kind
Q
v
μ
(
x
)
=
π
2
sin
(
μ
π
)
(
cos
(
μ
π
)
(
1
+
x
1
−
x
)
μ
2
2
F
1
(
v
+
1
,
−
v
;
1
−
μ
;
1
−
x
2
)
Γ
(
1
−
μ
)
−
Γ
(
ν
+
μ
+
1
)
Γ
(
ν
−
μ
+
1
)
(
1
−
x
1
+
x
)
μ
2
2
F
1
(
v
+
1
,
−
v
;
1
+
μ
;
1
−
x
2
)
Γ
(
1
+
μ
)
)
{\displaystyle Q_{v}^{\mu }(x)={\frac {\pi }{2\sin(\mu \pi )}}\left(\cos(\mu \pi )\left({\frac {1+x}{1-x}}\right)^{\frac {\mu }{2}}\,{\frac {{}_{2}F_{1}\left(v+1,-v;1-\mu ;{\frac {1-x}{2}}\right)}{\Gamma (1-\mu )}}-{\frac {\Gamma (\nu +\mu +1)}{\Gamma (\nu -\mu +1)}}\left({\frac {1-x}{1+x}}\right)^{\frac {\mu }{2}}\,{\frac {{}_{2}F_{1}\left(v+1,-v;1+\mu ;{\frac {1-x}{2}}\right)}{\Gamma (1+\mu )}}\right)}
See also
References
^ Olver, Frank W. J. ; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W., eds. (2010), "Ferrers Function" , NIST Handbook of Mathematical Functions ISBN 978-0-521-19225-5 MR 2723248 ^ "DLMF: §14.3 Definitions and Hypergeometric Representations ‣ Real Arguments ‣ Chapter 14 Legendre and Related Functions" . dlmf.nist.gov . Retrieved 2025-03-17 .^ Ferrers, Norman Macleod. An elementary treatise on spherical harmonics and subjects connected with them . Macmillan and Company, 1877.
