Question

Derive Rodrigue's formula: \begin{equation} P_n(x) = \frac{1}{2^n n!} \frac{d^n}{dx^n} (x^2 - 1)^n \end{equation}

          Derive Rodrigue's formula: \begin{equation*} P_n(x) = \frac{1}{2^n n!} \frac{d^n}{dx^n} (x^2 - 1)^n \end{equation*}

Derive Rodrigue's formula:
Pn(x) = (1)/(2^n n!)(d^n)/(dx^n) (x^2 - 1)^n

Added by Alex F.

Calculus: Early Transcendentals

James Stewart 8th Edition

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Solved by Expert Sandip Ranjan

Step 1

First, we need to define the Rodrigues formula, which expresses a polynomial $P(x)$ of degree $n$ in terms of its derivatives: $$P(x) = \frac{1}{2^n n!} \frac{d^n}{dx^n} \left[(x^2 - 1)^n\right]$$ Show more…

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Derive Rodrigue' $ formula: P(x) = d('-1) 2" n! dr"

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Question

Derive Rodrigue's formula: \begin{equation*} P_n(x) = \frac{1}{2^n n!} \frac{d^n}{dx^n} (x^2 - 1)^n \end{equation*}

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Derive Rodrigue's formula: \begin{equation} P_n(x) = \frac{1}{2^n n!} \frac{d^n}{dx^n} (x^2 - 1)^n \end{equation}