How do we argue that the polynomials obtained from Q_{n}(x) = 1/(2 ^ n * n!) * (d ^ n)/(d * x ^ n) * (x ^ 2 - 1) ^ n for different values of n, form an orthogonal set.

To argue that the polynomials \( Q_n(x) \) obtained from the expression

\[
Q_n(x) = \frac{1}{2^n \cdot n!} \frac{d^n}{dx^n} \left( (x^2 - 1)^n \right)
\]

for different values of \( n \) form an orthogonal set, we need to demonstrate that these polynomials satisfy the condition of orthogonality under some inner product defined on a suitable function space. Here's how you can approach the argument:

### 1. Identify the Inner Product
For orthogonal polynomials, the inner product of two functions \( f(x) \) and \( g(x) \) is typically defined as:

\[
\langle f, g \rangle = \int_{a}^{b} w(x) f(x) g(x) \, dx
\]

where \( w(x) \) is a weight function, and \( [a, b] \) is the interval over which the polynomials are defined. 

### 2. Recognize the Polynomials as a Special Set
The polynomials \( Q_n(x) \) are the Legendre polynomials \( P_n(x) \). They can be identified as such because the expression

\[
Q_n(x) = \frac{1}{2^n \cdot n!} \frac{d^n}{dx^n} \left( (x^2 - 1)^n \right)
\]

matches the Rodrigues' formula for Legendre polynomials:

\[
P_n(x) = \frac{1}{2^n n!} \frac{d^n}{dx^n} \left( (x^2 - 1)^n \right)
\]

Thus, \( Q_n(x) \) are precisely the Legendre polynomials.

### 3. Orthogonality Condition of Legendre Polynomials
The Legendre polynomials \( P_n(x) \) are known to be orthogonal with respect to the weight function \( w(x) = 1 \) on the interval \( [-1, 1] \). Specifically:

\[
\int_{-1}^{1} P_m(x) P_n(x) \, dx = \begin{cases} 
0 & \text{if } m \neq n \\
\frac{2}{2n+1} & \text{if } m = n 
\end{cases}
\]

### 4. Conclusion
Since \( Q_n(x) \) are Legendre polynomials \( P_n(x) \), they inherit the orthogonality property. Therefore, the polynomials \( Q_n(x) \) form an orthogonal set on the interval \( [-1, 1] \) with respect to the inner product defined by:

\[
\langle f, g \rangle = \int_{-1}^{1} f(x) g(x) \, dx
\]

This establishes that the polynomials obtained from \( Q_n(x) \) are indeed an orthogonal set.

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Would you like more details or have any further questions?

Here are 5 related questions to expand your understanding:

1. What are some applications of orthogonal polynomials in numerical analysis?
2. How do the Legendre polynomials relate to solving differential equations?
3. Can the weight function \( w(x) \) in the inner product be different for other orthogonal polynomials?
4. What is the significance of the Rodrigues' formula in generating orthogonal polynomials?
5. How would you generalize the concept of orthogonality to polynomials defined on different intervals or with different weight functions?

**Tip:** Orthogonal polynomials are crucial in approximating functions, especially in methods like least squares approximation and Gaussian quadrature.

Explore the orthogonality of polynomials Q_n(x) obtained from the expression (x^2 - 1)^n for different values of n. Understand how these polynomials form an orthogonal set under specific inner product definitions. Suitable for advanced undergraduate students and researchers interested in orthogonal polynomials and their properties.

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