User:Egm6322.s12.team2.Xia/RP14.2

= R 14.2 - The relation of coefficient $$ h_n$$ of the polynomial $$ p_n$$=

Given
Using the following fomula

$$\displaystyle C_{n}=\frac{A_{n}}{A_{n-1}}\frac{h_{n}}{h_{n-1}}$$ (14.2.1) Where $$\displaystyle {\color{red}C_{n}} $$ is the coefficent of the following orthogonal polynomials $$\displaystyle {P_{n}(x)}$$  relations:

$$\displaystyle p_{n+1}(x)=(A_{n}x+B_{n})p_{n}(x)-{\color{red}C_{n}}p_{n-1}(x)$$

Find
Prove that

$$\displaystyle h_{n}=h_{0}\frac{A_{0}}{A_{n}}C_{1}C_{2}\ldots C_{n}$$ (14.2.2) where $$ h_n$$ represents the square of the magnitude of the polynomial $$ p_n$$.

Solution
From the 14.2.1, we have

$$\displaystyle \begin{array}{l} C_{1}=\frac{A_{1}}{A_{0}}\frac{h_{1}}{h_{0}}\\ \\ C_{2}=\frac{A_{2}}{A_{1}}\frac{h_{2}}{h_{1}}\\ \\ C_{3}=\frac{A_{3}}{A_{2}}\frac{h_{3}}{h_{2}}\\ \vdots\\ C_{n-1}=\frac{A_{n-1}}{A_{n-2}}\frac{h_{n-1}}{h_{n-2}}\\ \\ C_{n}=\frac{A_{n}}{A_{n-1}}\frac{h_{n}}{h_{n-1}} \end{array} $$ (14.2.3) then multiply left hand side by left hand side and right hand side by right hand side.

$$\displaystyle C_{1}\cdot C_{2}\cdot C_{3}\cdot\ldots C_{n-1}\cdot C_{n}=\frac{\bcancel{A_{1}}}{A_{0}}\frac{\bcancel{h_{1}}}{h_{0}}\cdot\frac{\cancel{A_{2}}}{\bcancel{A_{1}}}\frac{\cancel{h_{2}}}{\bcancel{h_{1}}}\cdot\frac{\bcancel{A_{3}}}{\cancel{A_{2}}}\frac{\bcancel{h_{3}}}{\cancel{h_{2}}}\cdot\cdots\cdot\frac{\bcancel{A_{n-1}}}{\cancel{A_{n-2}}}\frac{\bcancel{h_{n-1}}}{\cancel{h_{n-2}}}\cdot\frac{A_{n}}{\bcancel{A_{n-1}}}\frac{h_{n}}{\bcancel{h_{n-1}}}$$ (14.2.4) thus

$$\displaystyle C_{1}\cdot C_{2}\cdot C_{3}\cdot\ldots C_{n-1}\cdot C_{n}=\frac{1}{A_{0}h_{0}}\frac{A_{n}h_{n}}{1} $$ (14.2.5) and the equation 14.2.2 exists.