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14 Legendre and Related FunctionsNotation

§14.1 Special Notation

(For other notation see Notation for the Special Functions.)

x, y, τ real variables.
z=x+iy complex variable.
m, n unless stated otherwise, nonnegative integers, used for order and degree, respectively.
μ, ν general order and degree, respectively.
12+iτ complex degree, τ.
γ Euler’s constant (§5.2(ii)).
δ arbitrary small positive constant.
ψ(x) logarithmic derivative of gamma function (§5.2(i)).
ψ(x) dψ(x)/dx .
𝐅(a,b;c;z) Olver’s scaled hypergeometric function: F(a,b;c;z)/Γ(c).

Multivalued functions take their principal values (§4.2(i)) unless indicated otherwise.

The main functions treated in this chapter are the Legendre functions 𝖯ν(x), 𝖰ν(x), Pν(z), Qν(z); Ferrers functions 𝖯νμ(x), 𝖰νμ(x) (also known as the Legendre functions on the cut); associated Legendre functions Pνμ(z), Qνμ(z), 𝑸νμ(z); conical functions 𝖯12+iτμ(x), 𝖰12+iτμ(x), 𝖰^12+iτμ(x), P12+iτμ(x), Q12+iτμ(x) (also known as Mehler functions).

Among other notations commonly used in the literature Erdélyi et al. (1953a) and Olver (1997b) denote 𝖯νμ(x) and 𝖰νμ(x) by Pνμ(x) and Qνμ(x), respectively. Magnus et al. (1966) denotes 𝖯νμ(x), 𝖰νμ(x), Pνμ(z), and Qνμ(z) by Pνμ(x), Qνμ(x), 𝔓νμ(z), and 𝔔νμ(z), respectively. Hobson (1931) denotes both 𝖯νμ(x) and Pνμ(x) by Pνμ(x); similarly for 𝖰νμ(x) and Qνμ(x).