orthogonal polynomials in (-1,1) with a modified weight function

Question

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.

Answer 1

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.

Answer 2

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.