In mathematics, '''Ferrers functions''' are certain special functions defined in terms of 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.<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
==References== {{reflist}}
Category:Special functions