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Ferrers function

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Ferrers Function First Kind P(x)

Ferrers Function Second Kind Q(x)

Ferrers functions are defined in terms of hypergeometric functions^[1]

Ferrers function of the first kind

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

Ferrers function of the second kind

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

References

^ Frank J. Oliver, NIST Handbook of Mathematical Functions, p352-356,Cambridge University Press 2010

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