Ferrers function

In mathematics, Ferrers functions are certain special functions defined in terms of hypergeometric functions.^[1]^[2] They are named after Norman Macleod Ferrers.^[3]

Definitions

Define $μ$ the order, and the $ν$ degree are real, and assume $x \in (- 1, + 1)$ .

P_{v}^{μ} (x) = {(\frac{1 + x}{1 - x})}^{μ / 2} \cdot \frac{_{2} F_{1} (v + 1, - v; 1 - μ; 1 / 2 - x / 2)}{Γ (1 - μ)}

Q_{v}^{μ} (x) = \frac{π}{2 \sin (μ π)} (\cos (μ π) {(\frac{1 + x}{1 - x})}^{\frac{μ}{2}} \frac{_{2} F_{1} (v + 1, - v; 1 - μ; \frac{1 - x}{2})}{Γ (1 - μ)} - \frac{Γ (ν + μ + 1)}{Γ (ν - μ + 1)} {(\frac{1 - x}{1 + x})}^{\frac{μ}{2}} \frac{_{2} F_{1} (v + 1, - v; 1 + μ; \frac{1 - x}{2})}{Γ (1 + μ)})

^ Page Module:Citation/CS1/styles.css has no content.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.
^ Page Module:Citation/CS1/styles.css has no content."DLMF: §14.3 Definitions and Hypergeometric Representations ‣ Real Arguments ‣ Chapter 14 Legendre and Related Functions". dlmf.nist.gov. Retrieved 2025-03-17.
^ Ferrers, Norman Macleod. An elementary treatise on spherical harmonics and subjects connected with them. Macmillan and Company, 1877.