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The generating function w (x, t) = (1– 2xt + t2)-1 is given. (a) Show that it satisfies the identity Əw + 2(t – x)w = 0 at (1- 2xt + t2).

The generating function w (x, t) = (1– 2xt + t2)-1 is given. (a) Show that it satisfies the identity Əw + 2(t – x)w = 0 at (1- 2xt + t2).

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.
8. The generating function w (x, t) = (1– 2xt + t²)-1 is given. (a) Show that it satisfies the identity %3D (1– 2xt + t2) + 2(t – x)w = 0 at %3D (b) Substitute the series in problem 7 into the identity in (a) and derive the recurrence formula (for n = 1, 2, 3, ...) Un+1(x) – 2xU,(x) + Un-1(x) = 0
Transcribed Image Text:8. The generating function w (x, t) = (1– 2xt + t²)-1 is given. (a) Show that it satisfies the identity %3D (1– 2xt + t2) + 2(t – x)w = 0 at %3D (b) Substitute the series in problem 7 into the identity in (a) and derive the recurrence formula (for n = 1, 2, 3, ...) Un+1(x) – 2xU,(x) + Un-1(x) = 0
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