Ferrers function

In mathematics, Ferrers functions are certain special functions defined in terms of hypergeometric functions.[1] They are named after Norman Macleod Ferrers.

Definitions

P_{v}^{\mu }(x)=\left({\frac {1+x}{1-x}}\right)^{\mu /2}\cdot {\frac {F(v+1,-v;1-\mu ;1/2-x/2)}{\Gamma (1-\mu )}}

Q_{v}^{\mu }(x)=\cos(\mu \pi )\left({\frac {1+x}{1-x}}\right)^{\mu /2}{\frac {F(v+1,-v;1-\mu ;1/2-2/x)}{\Gamma (1-\mu )}}

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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

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