Ferrers function
From HandWiki
In mathematics, Ferrers functions are certain special functions defined in terms of hypergeometric functions.[1] They are named after Norman Macleod Ferrers[citation needed].
Definitions
When the order μ and the degree ν are real and x ∈ (-1,1)
- Ferrers function of the first kind
- [math]\displaystyle{ 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)} }[/math]
- Ferrers function of the second kind
- [math]\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{F\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{F\left(v+1,-v;1+\mu;\frac{1-x}2\right)}{\Gamma(1+\mu)}\right) }[/math]
See also
References
- ↑ Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, Ronald F. et al., eds. (2010), "Ferrers Function", NIST Handbook of Mathematical Functions, Cambridge University Press, ISBN 978-0-521-19225-5, http://dlmf.nist.gov/14.3.i
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