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Another aspect of the symmetric nature of toroidal functions under index
interchange is further demonstrated by using Whipple's formulae for
associated Legendre functions (§3.3.1 in Erdélyi et al. 1953).
We start with eqs. (8.2.7) and (8.2.8) in Abramowitz and Stegun (1965), namely,
 |
(28) |
and,
 |
(29) |
which are valid only for
(since the
Legendre function of the second kind becomes discontinuous on the cut)
and for all complex 
and 
except when
since the Gamma function has poles along the negative real axis.
By combining eqs. (28) and (29) with eqs. (20)--(23) along with the
negative argument condition for Gamma functions [eq. (6.1.17) of Abramowitz
& Stegun (1965)], namely,
 |
(30) |
we can write two general expressions which are valid for all
complex
and 
.
Here, for associated Legendre functions
of the first kind,
![\begin{displaymath}P_{\nu-\frac{1}{2}}^\mu(z)=
\frac{\sqrt{2}\ \Gamma(\mu-\nu+\f...
...-\frac{1}{2}}^\nu
\biggl(\frac{z}{\sqrt{z^2-1}}\biggr)\Biggr],
\end{displaymath}](img67.gif) |
(31) |
which is equivalent to,
![\begin{displaymath}Q_{\nu-\frac{1}{2}}^\mu(z)=
\frac{\mathrm{e}^{i\mu\pi}\Gamma(...
...\frac{1}{2}}^\nu
\biggl(\frac{z}{\sqrt{z^2-1}}\biggr)
\Biggr].
\end{displaymath}](img68.gif) |
(32) |
Now, we substitute 
and 
(with m and n being
positive integers) in (28) and (29) in order to derive the Whipple formulae
for associated toroidal functions
 |
(33) |
which is equivalent to,
 |
(34) |
These last two expressions allow us to express toroidal functions
of a certain kind (first or second, respectively) with argument hyperbolic
cosine, as a direct proportionality in terms of the toroidal function of the
other kind (second or first, respectively) with argument hyperbolic
cotangent.
