Definitions

# Chebyshev equation

[chuh-buh-shawf]
Chebyshev's equation is the second order linear differential equation

$\left(1-x^2\right) \left\{d^2 y over d x^2\right\} - x \left\{d y over d x\right\} + p^2 y = 0$

where p is a real constant. The equation is named after Russian mathematician Pafnuty Chebyshev.

The solutions are obtained by power series:

$y = sum_\left\{n=0\right\}^infty a_nx^n$

where the coefficients obey the recurrence relation

$a_\left\{n+2\right\} = \left\{\left(n-p\right) \left(n+p\right) over \left(n+1\right) \left(n+2\right) \right\} a_n.$

These series converge for x in $\left[-1, 1\right]$, as may be seen by applying the ratio test to the recurrence.

The recurrence may be started with arbitrary values of a0 and a1, leading to the two-dimensional space of solutions that arises from second order differential equations. The standard choices are:

a0 = 1 ; a1 = 0, leading to the solution
$F\left(x\right) = 1 - frac\left\{p^2\right\}\left\{2!\right\}x^2 + frac\left\{\left(p-2\right)p^2\left(p+2\right)\right\}\left\{4!\right\}x^4 - frac\left\{\left(p-4\right)\left(p-2\right)p^2\left(p+2\right)\left(p+4\right)\right\}\left\{6!\right\}x^6 + cdots$
and
a0 = 0 ; a1 = 1, leading to the solution
$G\left(x\right) = x - frac\left\{\left(p-1\right)\left(p+1\right)\right\}\left\{3!\right\}x^3 + frac\left\{\left(p-3\right)\left(p-1\right)\left(p+1\right)\left(p+3\right)\right\}\left\{5!\right\}x^5 - cdots$

The general solution is any linear combination of these two.

When p is an integer, one or the other of the two functions has its series terminate after a finite number of terms: F terminates if p is even, and G terminates if p is odd. In this case, that function is a pth degree polynomial (converging everywhere, of course), and that polynomial is proportional to the pth Chebyshev polynomial.

$T_p\left(x\right) = \left(-1\right)^\left\{p/2\right\} F\left(x\right),$ if p is even
$T_p\left(x\right) = \left(-1\right)^\left\{\left(p-1\right)/2\right\} p G\left(x\right),$ if p is odd

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