Expansions in series of Legendre functions

Abstract

The Legendre functions$P_\nu ^\mu (x)$occurring in this paper are sometimes called modlfied Legendre functions, or Legendre functions on the cut. They are defined in [2, p.143, eq. (6)] as

$$P_{v}^{x}(x) = \frac{1}{{\Gamma (1 - \mu )}}{{\left( {\frac{{1 + x}}{{1 - x}}} \right)}^{{\frac{1}{2}\mu }}}F\left( { - v;1 + v;1 - \mu ;\frac{{1 - x}}{2}} \right) $$

(5.1)

for -1 < x < 1, where F is Gauss’s hypergeometric function and μ and v are real or complex parameters. They satisfy the recurrence relation

$$ (\nu - \mu + 1)P_{{\nu + 1}}^{\mu }(x) + (\nu + \mu )P_{{\nu - 1}}^{\mu }(x) = (2\nu + 1)xP_{\nu }^{\mu }(x) $$

(5.2)

.