orthogonal polynomials in (-1,1) with a modified weight function I would like to find the orthogonal polynomial system   $ \{ P_n(x), n \in \textbf{Z} \} $ corresponding to the weight function $ w(x) = \frac { 1} {\sqrt{1-x^2} (1+\sqrt{1-x^2} ) } $  defined on the interval $(-1,1)$. Noticing that this $w(x)$ is a modified Chebyshev weight function, I am just wondering whether the solution to this problem studied previously. I am not sure about it.  Your responses are very much appreciated. 
 A: I think these polynomials are known.
First substitute $x = \cos(\theta)$, such that
$$
\int_0^{2\pi} p_m(\cos(\theta))p_n(\cos(\theta)) \frac{d\theta}{(1 + \sin(\theta))}.
$$
Now substitute $\nu = \sin(\theta)$, then 
$$
d\nu/d\theta = \cos(\theta) = \sqrt{(1 - \sin(\theta)^2)} = \sqrt{(1 - \nu^2)} = \sqrt{(1 - \nu)}\sqrt{(1 + \nu)},
$$
such that the orthogonality relation is equivalent to
$$
\int_{-1}^{1} p_m(\nu) p_n(\nu) (1 + \nu)^{-\frac{3}{2}}(1 - \nu)^{-\frac{1}{2}} d\nu.
$$
The weight $w(\nu) = (1 + \nu)^{-\frac{3}{2}}(1 - \nu)^{-\frac{1}{2}}$ corresponds to the Jacobi polynomials $P_m^{(-1/2, -3/2)}(\nu)$.
Under this identification you basically study well known polynomials. 
A: The integral $ \int_{x=-1}^{+1} \frac {dx} {\sqrt{1-x^2} (1 + \sqrt{1-x^2}) } = \int_{x=-1}^{0} \frac {dx} {\sqrt{1-x^2} (1 + \sqrt{1-x^2} )} + \int_{x=0}^{1} \frac {dx} {\sqrt{1-x^2} (1 + \sqrt{1-x^2} )} $.
Using  substitutions $ s \rightarrow -\sqrt{1-x^2}$ and $t \rightarrow\sqrt{1-x^2} $ respectively in to the first and second integrals on the right-hand side gives
$ \int_{x=-1}^{+1} \frac {dx} {\sqrt{1-x^2} (1 + \sqrt{1-x^2}) } = 
 - \int_{s=-1}^{0} \frac{ds}{(1+s)^{ \frac{1}{2}} (1-s)^{\frac{3}{2}}} +
  \int_{t=0}^{1} \frac{dt}{(1+t)^{ \frac{3}{2}} (1-t)^{\frac{1}{2}}} $ .
Thus, the resulting polynomial space appears to be
$L_2[(-1,1),w(x)] =L_2[(-1,0),w_1] \cup L_2[(0,1),w_2] $.
Where 
$ w(x) = \frac{1} { \sqrt{1-x^2} (1+\sqrt{1-x^2} ) }$  ,
$w_1(x) = \frac{-1} { (1+x)^{1/2} (1-x)^{3/2 } } $ and
$w_2(x) = \frac{1} { (1+x)^{3/2} (1-x)^{1/2 } } $.
So two different beta distributions for the half-intervals of $(-1,1)$, leading to two different orthogonal polynomial sequences there.
