Tesseral Harmonics

Tesseral Harmonics as defined in terms of spherical harmonics by

$$Z_{n0} = Y_n^0$$ (175)

$$Z_{n\alpha}\equiv Z_{n,\alpha}^c = \frac{1}{\sqrt{2}}[Y_n^{-\alpha}+(-1)^{\alpha}Y_n^{\alpha}] \dots \alpha>0$$ (176)

$$Z_{n\alpha}\equiv Z_{n,\vert\alpha\vert}^s = \frac{i}{\sqrt{2}}[Y_n^{\alpha}-(-1)^{\alpha}Y_n^{-\alpha}] \dots \alpha<0$$ (177)

A similar table has been given in [62] on p. 238 (mind, there are errors in $Z^c_{41}$,$Z^c_{43}$,$Z^c_{52}$ ... in this reference)

$$\begin{eqnarray*} Z_{00}&=&\frac{1}{\sqrt{4\pi}}\\ \hline Z^s_{11}&=&\sqrt{\fra... ...1}{64}\sqrt{\frac{26}{231\pi}}[(x^6-15x^4y^2+15x^2y^4-y^6)/r^6] \end{eqnarray*}$$