Let $\left( U _ { n } \right) _ { n \in \mathbb { N } }$ be the sequence of polynomials defined by $U _ { 0 } = 1 , U _ { 1 } = X - 1$ and $\forall n \in \mathbb { N } , U _ { n + 2 } = ( X - 2 ) U _ { n + 1 } - U _ { n }$, with the inner product $$( P \mid Q ) = \frac { 2 } { \pi } \int _ { 0 } ^ { 1 } P ( 4 x ) Q ( 4 x ) \frac { \sqrt { 1 - x } } { \sqrt { x } } \mathrm { ~d} x .$$ Deduce that $\left( U _ { n } \right) _ { n \in \mathbb { N } }$ is an orthogonal system and that, for all $n \in \mathbb { N } , \left\| U _ { n } \right\| = 1$. To calculate the value of $( U _ { m } \mid U _ { n } )$, one may perform the change of variable $x = \cos ^ { 2 } \theta$.
Let $\left( U _ { n } \right) _ { n \in \mathbb { N } }$ be the sequence of polynomials defined by $U _ { 0 } = 1 , U _ { 1 } = X - 1$ and $\forall n \in \mathbb { N } , U _ { n + 2 } = ( X - 2 ) U _ { n + 1 } - U _ { n }$, with the inner product
$$( P \mid Q ) = \frac { 2 } { \pi } \int _ { 0 } ^ { 1 } P ( 4 x ) Q ( 4 x ) \frac { \sqrt { 1 - x } } { \sqrt { x } } \mathrm { ~d} x .$$
Deduce that $\left( U _ { n } \right) _ { n \in \mathbb { N } }$ is an orthogonal system and that, for all $n \in \mathbb { N } , \left\| U _ { n } \right\| = 1$. To calculate the value of $( U _ { m } \mid U _ { n } )$, one may perform the change of variable $x = \cos ^ { 2 } \theta$.