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12. Start with the identities x2 1 P, (x) = xPn-1(x) + *P(x) n P, (x) = xP-1(x) + nP,-1(x ) (a) Show that (for n = 1, 2, 3, ...) + [P,(x)]² = 1- x? n2 1-x2 (P;(x)° + [P,(x)P= (P-1(x)]* + [P,-1(x)]* [P,-1(x)]² + [Pn-1(x)]? %3D nº

12. Start with the identities x2 1 P, (x) = xPn-1(x) + *P(x) n P, (x) = xP-1(x) + nP,-1(x ) (a) Show that (for n = 1, 2, 3, ...) + [P,(x)]² = 1- x? n2 1-x2 (P;(x)° + [P,(x)P= (P-1(x)]* + [P,-1(x)]* [P,-1(x)]² + [Pn-1(x)]? %3D nº

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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from book: SPECIAL FUNCTIONS OF MATHEMATICS FOR ENGINEERS.
12. Start with the identities x2 1 P, (x ) = xPp-1(x) + *P,-(x) P, (x) = xP-1(x ) + nP,-1(x ) (a) Show that (for n = 1, 2, 3,...) 1-x2 (P;(x)° + [P,(x)P= (P-1(x)]* + [P,-1(x)]* + [P,(x)]² = 1- x² n2 | [P,-1(x)]² + [Pn-1(x)]? %3D nº
Transcribed Image Text:12. Start with the identities x2 1 P, (x ) = xPp-1(x) + *P,-(x) P, (x) = xP-1(x ) + nP,-1(x ) (a) Show that (for n = 1, 2, 3,...) 1-x2 (P;(x)° + [P,(x)P= (P-1(x)]* + [P,-1(x)]* + [P,(x)]² = 1- x² n2 | [P,-1(x)]² + [Pn-1(x)]? %3D nº
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