({-}m, m),\, (0, {-}m)\},
\\
W_{(m_1,m_2)}(A_2)=\{
(m_1, m_2)^+,\ ({-}m_1, m_1{+}m_2)^-,\ (m_1{+}m_2, {-}m_2)^-,
\\
\phantom{W_{(m_1,m_2)}(A_2)=\{}
({-}m_2, {-}m_1)^-,\ ( {-}m_1{-}m_2, m_1)^+,\ (m_2, {-}m_1{-}m_2)^+\}.\end{gathered}$$ Suppose $x=(x_1,x_2)$ is given in the $\alpha$-basis, then the orbit functions assume the form $$\begin{gathered}
\begin{gathered}\label{C_S_for_A2}
C_{(0,0)}(x)=1,\quad
C_{(0,m)}(x)=\overline{C_{(m,0)}(x)}=e^{{-}2\pi i mx_1}{+}e^{2\pi i mx_1}e^{{-}2\pi i mx_2}{+}e^{2\pi i mx_2},
\\
C_{(m_1,m_2)}(x)
=e^{2\pi i m_1x_1}e^{2\pi i m_2x_2}{+}e^{{-}2\pi i m_1x_1}e^{2\pi i (m_1{+}m_2)x_2}
{+}e^{2\pi i (m_1{+}m_2)x_1}e^{{-}2\pi i m_2x_2}{+}
\\ \phantom{C_{(m_1,m_2)}(x)=}
e^{{-}2\pi i m_2x_1}e^{{-}2\pi i m_1x_2}{+}e^{{-}2\pi i (m_1{+}m_2)x_1}e^{2\pi i m_1x_2}
{+}e^{2\pi i m_2x_1}e^{{-}2\pi i (m_1{+}m_2)x_2},\\
S_{(m_1,m_2)}(x)
=e^{2\pi i m_1x_1}e^{2\pi i m_2x_2}{-}e^{{-}2\pi i m_1x_1}e^{2\pi i (m_1{+}m_2)x_2}
{-}e^{2\pi i (m_1{+}m_2)x_1}e^{{-}2\pi i m_2x_2}{-}
\\ \phantom{S_{(m_1,m_2)}(x)=}
e^{{-}2\pi i m_2x_1}e^{{-}2\pi i m_1x_2}{+}e^{{-}2\pi i (m_1{+}m_2)x_1}e^{2\pi i m_1x_2}
{+}e^{2\pi i m_2x_1}e^{{-}2\pi i (m_1{+}m_2)x_2}.\end{gathered}\end{gathered}$$ Using (\[subst\_exp\]) we have the following corresponding polynomials $$\begin{gathered}
P_{(0,0)}^C=1,\qquad
P_{0,m}^C=\overline{P_{0,m}^C}=y_1^{{-}m}{+}Y_1^{m}y_2^{{-}m}{+}y_2^{m},
\\
\begin{gathered}\label{Polyn_c-A2}
P_{(m_1,m_2)}^{C}
=y_1^{m_1}y_2^{m_2}{+}y_1^{-m_1}y_2^{(m_1{+}m_2)}{+}y_1^{(m_1{+}m_2)}y_2^{-m_2}{+}
\\ \phantom{P^{C(A_2)}=}
y_1^{-m_1}y_2^{-m_2}{+}y_1^{-(m_1{+}m_2)}y_2^{m_1}{+}y_1^{m_2}y_2^{-(m_1{+}m_2)},
\end{gathered}
\\
\begin{gathered}\label{Polyn_s-A2}
P_{(m_1,m_2)}^{S}
=y_1^{m_1}y_2^{m_2}{-}y_1^{-m_1}y_2^{(m_1{+}m_2)}{-}y_1^{(m_1{+}m_2)}y_2^{-m_2}{-}
\\ \phantom{P^{S(A_2)})=}
y_1^{-m_1}y_2^{-m_2}{+}y_1^{-(m_1{+}m_2)}y_2^{m_1}{+}y_1^{m_2}y_2^{-(m_1{+}m_2)}.
\end{gathered}\end{gathered}$$

The polynomials $e^+$ and $e^-$ given in (2.6) of [@Koornwinder1-4](III) coincide with those in (\[C\_S\_for\_A2\]) whenever the correspondence $\sigma=2\pi x_1$, $\tau=2\pi x_2$ is set up. So, both the orbit functions polynomials of $A_2$ and $e^{\pm}$ are orthogonal on the interior of Steiner’s hypocycloid.

It is noteworthy that the regular tessellation of the plane by equilateral triangles considered in [@Koornwinder1-4] is the standard tiling 