Derivative Form of Associated Legendre Function of the First Kind

Theorem

Let $\map { {P_n}^m} x$ be an associated Legendre function of the first kind.

Then:

$\map { {P_n}^m} x = \dfrac {\paren {1 - x^2}^{m / 2} } {2^n n!} \dfrac {\d^{m + n} } {\d x^{m + n} } \paren {x^2 - 1}^n$

Proof

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Sources

• 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 26$: Associated Legendre Functions: Associated Legendre Functions of the First Kind: $26.2$
• 2009: Murray R. Spiegel, Seymour Lipschutz and John Liu: Mathematical Handbook of Formulas and Tables (3rd ed.) ... (previous) ... (next): $\S 28$: Legendre and Associated Legendre Functions: Associated Legendre Functions of the First Kind: $28.50.$