Mathematics for Natural Scientists II by Lev Kantorovich

Author:Lev Kantorovich
Language: eng
Format: epub
Publisher: Springer International Publishing, Cham

Note that this recurrence relation is formally valid even for n = 0 as well if we postulate that P −1 = 0, in which case we simply get P 1(x) = xP 0(x) = x, the result we already knew.

This recurrent relation can be used to generate the functions P n (x). Indeed, using P 0 = 1 and P 1 = x, we have using n = 1 in the recurrence relation that 2P 2 + P 0 = 3xP 1, which gives

i.e. the same expression as obtained above using the direct method. All higher order functions (corresponding to larger values of n) are obtained in exactly the same way, i.e. using a very simple algebra.

Problem 4.24.

Show using the direct method (the Taylor’s expansion) that:

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