Chebyshev Differential Equation

A Chebyshev differential equation is given by

(1)

for . The Chebyshev differential equation has regular singular points at , 1, and . A power series solution about uses the expansions

Substituting these expansions into (1) gives

(5)

(6)

(7)

(8)

(9)

(10)

(11)

and equating the remaining coefficients gives

(12)

for , 3, ....

The equations for and are special cases, so the recurrence relation

(13)

holds for , 1, .... It determines all coefficients from and . For the even-indexed coefficients,

and for the odd-indexed coefficients,

Both product formulas hold for , 1, ..., with the products taken to be 1 when .

For even , the coefficients can also be expressed using the gamma function as

			(20)

and for odd as

			(21)

At parameter values where the gamma function expressions are indeterminate, their limits give the finite products above. The general solution is obtained by combining the even and odd parts of the power series,

(22)

The change of variables , with , transforms (1) into

(23)

(24)

For , the limiting form is

(25)

For , an equivalent form of the general solution is

			(26)

where and are Chebyshev polynomials of the first kind and Chebyshev polynomials of the second kind, respectively, extended to noninteger orders when necessary. Another equivalent form of the general solution for is

(27)

See also