Tesseral Harmonics

A similar table has been given in [70] on p. 238 (mind, there are errors in $Z^c_{41}$ , $Z^c_{43}$ , $Z^c_{52}$ ... in this reference)

$\displaystyle Z_{00}$	$\textstyle =$	$\displaystyle \frac{1}{\sqrt{4\pi}}$
$\displaystyle \hline Z^s_{11}$	$\textstyle =$	$\displaystyle \sqrt{\frac{3}{4\pi}}[y/r]$
$\displaystyle Z_{10}$	$\textstyle =$	$\displaystyle \sqrt{\frac{3}{4\pi}}[z/r]$
$\displaystyle Z^c_{11}$	$\textstyle =$	$\displaystyle \sqrt{\frac{3}{4\pi}}[x/r]$
$\displaystyle \hline Z^s_{22}$	$\textstyle =$	$\displaystyle \frac{1}{4}\sqrt{\frac{15}{\pi}}[2xy/r^2]$
$\displaystyle Z^s_{21}$	$\textstyle =$	$\displaystyle \frac{1}{2}\sqrt{\frac{15}{\pi}}[yz/r^2]$
$\displaystyle Z_{20}$	$\textstyle =$	$\displaystyle \frac{1}{4}\sqrt{\frac{5}{\pi}}[(3z^2-r^2)/r^2]$
$\displaystyle Z^c_{21}$	$\textstyle =$	$\displaystyle \frac{1}{2}\sqrt{\frac{15}{\pi}}[xz/r^2]$
$\displaystyle Z^c_{22}$	$\textstyle =$	$\displaystyle \frac{1}{4}\sqrt{\frac{15}{\pi}}[(x^2-y^2)/r^2]$
$\displaystyle \hline Z^s_{33}$	$\textstyle =$	$\displaystyle \sqrt{\frac{35}{32\pi}}[(3x^2y-y^3)/r^3]$
$\displaystyle Z^s_{32}$	$\textstyle =$	$\displaystyle \sqrt{\frac{105}{16\pi}}[2xyz/r^3]$
$\displaystyle Z^s_{31}$	$\textstyle =$	$\displaystyle \sqrt{\frac{21}{32\pi}}[y(5z^2-r^2)/r^3]$
$\displaystyle Z_{30}$	$\textstyle =$	$\displaystyle \sqrt{\frac{7}{16\pi}}[z(5z^2-3r^2)/r^3]$
$\displaystyle Z^c_{31}$	$\textstyle =$	$\displaystyle \sqrt{\frac{21}{32\pi}}[x(5z^2-r^2)/r^3]$
$\displaystyle Z^c_{32}$	$\textstyle =$	$\displaystyle \sqrt{\frac{105}{16\pi}}[(x^2-y^2)z/r^3]$
$\displaystyle Z^c_{33}$	$\textstyle =$	$\displaystyle \sqrt{\frac{35}{32\pi}}[(x^3-3xy^2)/r^3]$
$\displaystyle \hline Z^s_{44}$	$\textstyle =$	$\displaystyle \frac{3}{16}\sqrt{\frac{35}{\pi}}[4(x^3y-xy^3)/r^4]$
$\displaystyle Z^s_{43}$	$\textstyle =$	$\displaystyle \frac{3}{8}\sqrt{\frac{70}{\pi}}[(3x^2y-y^3)z/r^4]$
$\displaystyle Z^s_{42}$	$\textstyle =$	$\displaystyle \frac{3}{8}\sqrt{\frac{5}{\pi}}[2xy(7z^2-r^2)/r^4]$
$\displaystyle Z^s_{41}$	$\textstyle =$	$\displaystyle \frac{3}{4}\sqrt{\frac{5}{2\pi}}[yz(7z^2-3r^2)/r^4]$
$\displaystyle Z_{40}$	$\textstyle =$	$\displaystyle \frac{3}{16}\frac{1}{\sqrt{\pi}}[35z^4-30z^2r^2+3r^4)/r^4]$
$\displaystyle Z^c_{41}$	$\textstyle =$	$\displaystyle \frac{3}{4}\sqrt{\frac{5}{2\pi}}[xz(7z^2-3r^2)/r^4]$
$\displaystyle Z^c_{42}$	$\textstyle =$	$\displaystyle \frac{3}{8}\sqrt{\frac{5}{\pi}}[(x^2-y^2)(7z^2-r^2)/r^4]$
$\displaystyle Z^c_{43}$	$\textstyle =$	$\displaystyle \frac{3}{8}\sqrt{\frac{70}{\pi}}[(x^3-3xy^2)z/r^4]$
$\displaystyle Z^c_{44}$	$\textstyle =$	$\displaystyle \frac{3}{16}\sqrt{\frac{35}{\pi}}[(x^4-6x^2y^2+y^4)/r^4]$
$\displaystyle \hline Z^s_{55}$	$\textstyle =$	$\displaystyle \sqrt{\frac{693}{512\pi}}[(5x^4y-10x^2y^3+y^5)/r^5]$
$\displaystyle Z^s_{54}$	$\textstyle =$	$\displaystyle \sqrt{\frac{3465}{256\pi}}[4(x^3y-xy^3)z/r^5]$
$\displaystyle Z^s_{53}$	$\textstyle =$	$\displaystyle \sqrt{\frac{385}{512\pi}}[(3x^2y-y^3)(9z^2-r^2)/r^5]$
$\displaystyle Z^s_{52}$	$\textstyle =$	$\displaystyle \sqrt{\frac{1155}{64\pi}}[2xy(3z^3-zr^2)/r^5]$
$\displaystyle Z^s_{51}$	$\textstyle =$	$\displaystyle \sqrt{\frac{165}{256\pi}}[y(21z^4-14z^2r^2+r^4)/r^5]$
$\displaystyle Z_{50}$	$\textstyle =$	$\displaystyle \sqrt{\frac{11}{256\pi}}[(63z^5-70z^3r^2+15zr^4)/r^5]$
$\displaystyle Z^c_{51}$	$\textstyle =$	$\displaystyle \sqrt{\frac{165}{256\pi}}[x(21z^4-14z^2r^2+r^4)/r^5]$
$\displaystyle Z^c_{52}$	$\textstyle =$	$\displaystyle \sqrt{\frac{1155}{64\pi}}[(x^2-y^2)(3z^3-zr^2)/r^5]$
$\displaystyle Z^c_{53}$	$\textstyle =$	$\displaystyle \sqrt{\frac{385}{512\pi}}[(x^3-3xy^2)(9z^2-r^2)/r^5]$
$\displaystyle Z^c_{54}$	$\textstyle =$	$\displaystyle \sqrt{\frac{3465}{256\pi}}[(x^4-6x^2y^2+y^4)z/r^5]$
$\displaystyle Z^c_{55}$	$\textstyle =$	$\displaystyle \sqrt{\frac{693}{512\pi}}[(x^5-10x^3y^2+5xy^4)/r^5]$
$\displaystyle \hline Z^s_{66}$	$\textstyle =$	$\displaystyle \frac{231}{64}\sqrt{\frac{26}{231\pi}}[(6x^5y-20x^3y^3+6xy^5)/r^6]$
$\displaystyle Z^s_{65}$	$\textstyle =$	$\displaystyle \sqrt{\frac{9009}{512\pi}}[(5x^4y-10x^2y^3+y^5)z/r^6]$
$\displaystyle Z^s_{64}$	$\textstyle =$	$\displaystyle \frac{21}{32}\sqrt{\frac{13}{7\pi}}[4(x^3y-xy^3)(11z^2-r^2)/r^6]$
$\displaystyle Z^s_{63}$	$\textstyle =$	$\displaystyle \frac{1}{32}\sqrt{\frac{2730}{\pi}}[(3x^2y-y^3)(11z^3-3zr^2)/r^6]$
$\displaystyle Z^s_{62}$	$\textstyle =$	$\displaystyle \frac{1}{64}\sqrt{\frac{2730}{\pi}}[2xy(33z^4-18z^2r^2+r^4)/r^6]$
$\displaystyle Z^s_{61}$	$\textstyle =$	$\displaystyle \frac{1}{8}\sqrt{\frac{273}{4\pi}}[yz(33z^4-30z^2r^2+5r^4)/r^6]$
$\displaystyle Z_{60}$	$\textstyle =$	$\displaystyle \frac{1}{32}\sqrt{\frac{13}{\pi}}[(231z^6-315z^4r^2+105z^2r^4-5r^6)/r^6]$
$\displaystyle Z^c_{61}$	$\textstyle =$	$\displaystyle \frac{1}{8}\sqrt{\frac{273}{4\pi}}[xz(33z^4-30z^2r^2+5r^4)/r^6]$
$\displaystyle Z^c_{62}$	$\textstyle =$	$\displaystyle \frac{1}{64}\sqrt{\frac{2730}{\pi}}[(x^2-y^2)(33z^4-18z^2r^2+r^4)/r^6]$
$\displaystyle Z^c_{63}$	$\textstyle =$	$\displaystyle \frac{1}{32}\sqrt{\frac{2730}{\pi}}[(x^3-3xy^2)(11z^3-3zr^2)/r^6]$
$\displaystyle Z^c_{64}$	$\textstyle =$	$\displaystyle \frac{21}{32}\sqrt{\frac{13}{7\pi}}[(x^4-6x^2y^2+y^4)(11z^2-r^2)/r^6]$
$\displaystyle Z^c_{65}$	$\textstyle =$	$\displaystyle \sqrt{\frac{9009}{512\pi}}[(x^5-10x^3y^2+5xy^4)z/r^6]$
$\displaystyle Z^c_{66}$	$\textstyle =$	$\displaystyle \frac{231}{64}\sqrt{\frac{26}{231\pi}}[(x^6-15x^4y^2+15x^2y^4-y^6)/r^6]$

$\displaystyle Z_{n0}$	$\textstyle =$	$\displaystyle Y_n^0$	(212)
$\displaystyle Z_{n\alpha}\equiv Z_{n,\alpha}^c$	$\textstyle =$	$\displaystyle \frac{1}{\sqrt{2}}[Y_n^{-\alpha}+(-1)^{\alpha}Y_n^{\alpha}] \dots \alpha>0$	(213)
$\displaystyle Z_{n\alpha}\equiv Z_{n,\vert\alpha\vert}^s$	$\textstyle =$	$\displaystyle \frac{i}{\sqrt{2}}[Y_n^{\alpha}-(-1)^{\alpha}Y_n^{-\alpha}] \dots \alpha<0$	(214)