# Ferrers function

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In [mathematics](/source/mathematics), '''Ferrers functions''' are certain [special functions](/source/special_functions) defined in terms of [hypergeometric functions](/source/hypergeometric_functions).<ref>{{dlmf|id=14.3.i |title=Ferrers Function}}</ref><ref>{{Cite web |title=DLMF: §14.3 Definitions and Hypergeometric Representations ‣ Real Arguments ‣ Chapter 14 Legendre and Related Functions |url=https://dlmf.nist.gov/14.3 |access-date=2025-03-17 |website=dlmf.nist.gov}}</ref>
They are named after [Norman Macleod Ferrers](/source/Norman_Macleod_Ferrers).<ref>Ferrers, Norman Macleod. ''An elementary treatise on spherical harmonics and subjects connected with them''. Macmillan and Company, 1877.</ref>

== Definitions ==
Define <math>\mu</math> the '''order''', and the <math>\nu</math> '''degree''' are real, and assume <math>x \in (-1, +1)</math>.
;Ferrers function of the first kind

: <math>P_v^\mu(x) = \left(\frac{1+x}{1-x}\right)^{\mu/2}\cdot\frac{{}_2F_1(v+1,-v;1-\mu;1/2-x/2)}{\Gamma(1-\mu)}                                        </math>

;Ferrers function of the second kind

: <math>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{{}_2F_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{{}_2F_1\left(v+1,-v;1+\mu;\frac{1-x}2\right)}{\Gamma(1+\mu)}\right)</math>

== See also ==
* [Legendre function](/source/Legendre_function)

==References==
{{reflist}}

Category:Special functions

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Adapted from the Wikipedia article [Ferrers function](https://en.wikipedia.org/wiki/Ferrers_function) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Ferrers_function?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.