Following Helgaker et al. in "Molecular Electronic-Structure Theory" a real valued spherical harmonic GTO can be written as
$$\chi_{\alpha_{nl}lm}^{GTO}=N_{\alpha_{nl}lm}^{GTO}S_{lm}e^{-\alpha_{nl}r^2}$$
Where $N_{\alpha_{nl}lm}^{GTO}$ is a normalisation constant, and $S_{lm}$ is a real solid harmonic, which is the polynomial you want. The real solid harmonics obey the recurrence relations
$$S_{00}=1$$
$$S_{l+1,l+1}=\sqrt{2^{\delta_{l0}}{{2l+1}\over{2l+2}}}[xS_{l,l}-(1-\delta_{l0})yS_{l,-l}]$$
$$S_{l+1,-l-1}=\sqrt{2^{\delta_{l0}}{{2l+1}\over{2l+2}}}[yS_{l,l}+(1-\delta_{l0})xS_{l,-l}]$$
$$S_{l+1,m}={{(2l+1)zS_{lm}-\sqrt{(l+m)(l-m)}{r^2}S_{l-1,m}}\over{\sqrt{(l+m+1)(l-m+1)}}} $$
This is sufficient to evaluate the (form of the) polynomials.
Alternatively explicit formulae for the coefficients in the polynomials have been published in Schlegel, H.B. and Frisch, M.J. (1995), Transformation between Cartesian and pure spherical harmonic Gaussians. Int. J. Quantum Chem., 54: 83-87 . Again from Helgaker et al. these are given as follows: If $G_{lm}({\bf r},a,{\bf A})$ represents a spherical-harmonic GTO with exponent $a$ and centered at ${\bf R_A}$, and $G_{x,y,z}({\bf r},a,{\bf A})$ similarly for a Cartesian GTO then
$$G_{lm}({\bf r},a,{\bf A})=N^{S}_{lm}
\sum_{t=0}^{[(l-\left|m\right|)/2]}
\sum_{u=0}^{t}
\sum_{v=v_m}^{[\left|m\right|/2-v_m]+v_m}
C_{tuv}^{lm}G_{2t+\left|m\right|-2(u+v),2(u+v),l-2t-\left|m\right|}({\bf r},a,{\bf A})
$$
where
$$N^{S}_{lm}={1\over{2^{\left|m\right|}l!}}\sqrt{{2(l+\left|m\right|)!(l-\left|m\right|)!}\over{2^{}\delta_{0m}}}$$
$$C_{tuv}^{lm}=(-1)^{t+v-v_m}{1\over 4}^t
\binom{l}{t}
\binom{l-t}{\left|m\right|+t}
\binom{t}{u}
\binom{\left|m\right|}{2v}
$$
and
$$ v_m = 0 \space for \space m \ge 0, {1\over 2} \space for \space m<0$$
I strongly suggest you check the above formulae from the original sources for typos!
