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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 a_{0} and a_{1},
leading to the two-dimensional space of solutions that arises from second order
differential equations. The standard choices are:

- a
_{0}= 1 ; a_{1}= 0, leading to the solution

- $F(x)\; =\; 1\; -\; frac\{p^2\}\{2!\}x^2\; +\; frac\{(p-2)p^2(p+2)\}\{4!\}x^4\; -\; frac\{(p-4)(p-2)p^2(p+2)(p+4)\}\{6!\}x^6\; +\; cdots$

- a
_{0}= 0 ; a_{1}= 1, leading to the solution

- $G(x)\; =\; x\; -\; frac\{(p-1)(p+1)\}\{3!\}x^3\; +\; frac\{(p-3)(p-1)(p+1)(p+3)\}\{5!\}x^5\; -\; cdots$

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 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

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Last updated on Monday August 06, 2007 at 12:23:46 PDT (GMT -0700)

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This article is licensed under the GNU Free Documentation License.

Last updated on Monday August 06, 2007 at 12:23:46 PDT (GMT -0700)

View this article at Wikipedia.org - Edit this article at Wikipedia.org - Donate to the Wikimedia Foundation

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