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Chebyshev polynomials of the first and second kind. Chebyshev’s differential equation. Recurrence formulas. Generating function.

Chebyshev’s differential equation. The equation

1)        (1 - x²)y" - xy' + n²y = 0                 n = 0, 1, 2, .......

Chebyshev polynomials of the first kind. The polynomials given by

First few Chebyshev polynomials of the first kind

Generating function for T_n(x)

Miscellaneous formulas

Recurrence formula for T_n(x)

Orthogonality of T_n(x)

Orthogonal series

where

Chebyshev polynomials of the second kind. The polynomials given by

First few Chebyshev polynomials of the second kind

Generating function for U_n(x)

Miscellaneous formulas

Recurrence formula for U_n(x)

Orthogonality of U_n(x)

Orthogonal series

where

Relationships between T_n(x) and U_n(x)

General solution of Chebyshev’s differential equation. The general solution of Chebyshev’s differential equation is

                                          Source: Spiegel. Mathematical Handbook of Formulas and Tables. (Schaum)

References.

1. Spiegel. Mathematical Handbook of Formulas and Tables. (Schaum)