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'Chebyshev's equation' is the second order linear differential equation
:$(1-x^2)\; \{d^2\; y\; over\; d\; x^2\}\; -\; x\; \{d\; y\; over\; d\; x\}\; +\; p^2\; y\; =\; 0$
where p is a real constant. The equation is named after Russian mathematician Pafnuty Chebyshev.
The solutions are obtained by power series:
:$y\; =\; sum\_\{n=0\}^infty\; a\_nx^n$
where the coefficients obey the recurrence relation
:$a\_\{n+2\}\; =\; \{(n-p)\; (n+p)\; over\; (n+1)\; (n+2)\; \}\; a\_n.$
These series converge for x in $[-1,\; 1]$, 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
:
and
:a₀ = 0 ; a₁ = 1, leading to the solution
:
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 p^th degree polynomial (converging
everywhere, of course), and that polynomial is proportional to the p^th
Chebyshev polynomial.
:$T\_p(x)\; =\; (-1)^\{p/2\}\; F(x),$ if p is even
:$T\_p(x)\; =\; (-1)^\{(p-1)/2\}\; p\; G(x),$ if p is odd