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Chebyshev equation (Definition)

Chebyshev's equation is the second order linear differential equation

$\displaystyle (1-x^2)\frac{d^2y}{dx^2} - x\frac{dy}{dx} + p^2y = 0$
where $ p$ is a real constant.

There are two independent solutions which are given as series by:

$\displaystyle y_1(x) = 1 - \tfrac{p^2}{2!}x^2 + \tfrac{(p-2)p^2(p+2)}{4!}x^4 - \tfrac{(p-4)(p-2)p^2(p+2)(p+4)}{6!}x^6 + \dotsb $
and
$\displaystyle y_2(x) = x - \tfrac{(p-1)(p+1)}{3!}x^3 + \tfrac{(p-3)(p-1)(p+1)(p+3)}{5!}x^5 - \dotsb $

In each case, the coefficients are given by the recursion

$\displaystyle a_{n+2} = \frac{(n-p)(n+p)}{(n+1)(n+2)} a_n $
with $ y_1$ arising from the choice $ a_0 = 1$, $ a_1 = 0$, and $ y_2$ arising from the choice $ a_0 = 0$, $ a_1 = 1$.

The series converge for $ \vert x\vert < 1$; this is easy to see from the ratio test and the recursion formula above.

When $ p$ is a non-negative integer, one of these series will terminate, giving a polynomial solution. If $ p \ge 0$ is even, then the series for $ y_1$ terminates at $ x^p$. If $ p$ is odd, then the series for $ y_2$ terminates at $ x^p$.

These polynomials are, up to multiplication by a constant, the Chebyshev polynomials. These are the only polynomial solutions of the Chebyshev equation.

(In fact, polynomial solutions are also obtained when $ p$ is a negative integer, but these are not new solutions, since the Chebyshev equation is invariant under the substitution of $ p$ by $ -p$.)



"Chebyshev equation" is owned by mclase.
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Other names:  Chebyshev differential equation
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Cross-references: invariant, negative, Chebyshev polynomials, multiplication, odd, even, polynomial, integer, ratio test, easy to see, converge, coefficients, series, solutions, independent, real, linear differential equation, second order
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This is version 3 of Chebyshev equation, born on 2002-11-21, modified 2002-11-21.
Object id is 3616, canonical name is ChebyshevEquation.
Accessed 7698 times total.

Classification:
AMS MSC34A30 (Ordinary differential equations :: General theory :: Linear equations and systems, general)

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