November 2012
Intermediate to advanced
794 pages
22h 9m
English
book
Probability, Random Variables, and Random Processes: Theory and Signal Processing Applications
by John J. Shynk
Content preview from Probability, Random Variables, and Random Processes: Theory and Signal Processing Applications
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B.23 LAGUERRE POLYNOMIALS
Definition: Laguerre Polynomials Laguerre polynomials are defined by the following recursion:
(B.123) 
with initial conditions L0(x) = 1 and L1(x) = 1−x.
They correspond to solutions of the following differential equation as v is varied:
An explicit expression for the Laguerre polynomials is
(B.125) 
Figure B.19 shows plots of the first three polynomials:
Figure B.19 Laguerre polynomials Lv(x).
(B.126) 
(B.127) 
(B.128) 
Also included is L1/2(x) which is used for the mean and variance of the Rice random variable. It can be written in terms of the modified Bessel functions of the first kind as follows:
(B.129) 
The Laguerre polynomials, including fractional ...
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