In mathematics, Ferrers functions are certain special functions defined in terms of hypergeometric functions. They are named after Norman Macleod Ferrers.
Definitions
Define μ {\displaystyle \mu } the 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, Cambridge University Press, ISBN 978-0-521-19225-5, MR 2723248. 978-0-521-19225-5 ↩
"DLMF: §14.3 Definitions and Hypergeometric Representations ‣ Real Arguments ‣ Chapter 14 Legendre and Related Functions". dlmf.nist.gov. Retrieved 2025-03-17. https://dlmf.nist.gov/14.3 ↩
Ferrers, Norman Macleod. An elementary treatise on spherical harmonics and subjects connected with them. Macmillan and Company, 1877. ↩