View source for Chebyshev equation

The linear homogeneous second-order ordinary differential equation

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or, in self-adjoint form,

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where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021860/c0218603.png" /> is a constant. Chebyshev's equation is a special case of the [[Hypergeometric equation|hypergeometric equation]].

The points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021860/c0218604.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021860/c0218605.png" /> are regular singular points (cf. [[Regular singular point|Regular singular point]]) of the Chebyshev equation. Substituting the independent variable

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reduces this equation to a corresponding linear equation with constant coefficients:

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so that Chebyshev's equation can be integrated in closed form. A fundamental systems of solutions to Chebyshev's equation on the interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021860/c0218609.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021860/c02186010.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021860/c02186011.png" /> is a natural number, consists of the [[Chebyshev polynomials|Chebyshev polynomials]] (of the first kind) of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021860/c02186012.png" />,

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and the functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021860/c02186014.png" />, which are related to Chebyshev polynomials of the second kind. The polynomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021860/c02186015.png" /> is a real solution to Chebyshev's equation on the entire real line, with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021860/c02186016.png" />. Chebyshev's equation is also studied in complex domains.