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Spin Legendre functions

We summarize here the recurrences used by the S2HAT Library to calculate the arbitrary spin, $s$, Legendre function.

The expressions presented hereafter come mostly from:

  • Vrashalovich, D.A., Moskalev, A.N., & Khersonskij, V.K., Quantum Theory of Angular Momentum (World Scientific, Singapore, 1988)
in a rendition of:
  • Lewis, A., Challinor, A., & Turok, N., Phys. Rev. D. 65, 023505.

For $m \ge \left\vert s\right\vert)$, the recurrence is wrt the polar, $\ell$, number for a fixed spin, $s$, and azimuthal, $m$, number,

\begin{displaymath} _s\lambda_{\ell+2\; m} = \left( x + {s\,m\over \left(\ell+2\... ...\ell+2\; m} \over C_{s\; \ell+1\; m}} \, _s\lambda_{\ell\; m}. \end{displaymath} (1)

Here,
\begin{displaymath} C_{s\;\ell\;m} \equiv \sqrt{{ \ell^2\;\left(4\,\ell^2-1\right)\over\left(\ell^2-m^2\right)\,\left(\ell^2-s^2\right)}}, \end{displaymath} (2)

and the starting values are,

$\displaystyle _s\lambda_{m-1\,m}$ $\textstyle =$ $\displaystyle 0,$ (3)
$\displaystyle _s\lambda_{m\,m}\ \ \ $ $\textstyle =$ $\displaystyle (-2)^m\,\sqrt{{\left(2m+1\right)!\over4\pi\left(m+s\right)!\,\lef... ..., \left(1-x\right)^{\left(m+s\right)/2}\,\left(1+x\right)^{\left(m-s\right)/2}.$ (4)

For the $m$ values such as $s > m \ge 0$, we first note that,

$\displaystyle _s\lambda_{\ell\;m}$ $\textstyle =$ $\displaystyle (-1)^{m+s}\,_m\lambda_{\ell\,s},$ (5)
$\displaystyle _{-s}\lambda_{\ell\;m}$ $\textstyle =$ $\displaystyle _{-m}\lambda_{\ell\,s},$ (6)

and then perform a recurrence as in Eq. [1], setting $m$ to $s$ and $s$ to either $m$ or $-m$ and iterating up to the required value of $\ell\ (\ge s \ge 0)$. For the positive spin we then fix the sign as indicated in Eq. [5].


Next: Spin transforms    Previous: Polarized transforms    Up: Algebraic Background


radek stompor 2009-10-15