## Chebyshev Polynomial of the Second Kind

A modified set of Chebyshev Polynomials defined by a slightly different Generating Function. Used to develop four-dimensional Spherical Harmonics in angular momentum theory. They are also a special case of the Ultraspherical Polynomial with . The Chebyshev polynomials of the second kind are illustrated above for and , 2, ..., 5.

The defining Generating Function of the Chebyshev polynomials of the second kind is

 (1)

for and . To see the relationship to a Chebyshev Polynomial of the First Kind (), take ,
 (2)

Multiply (2) by ,
 (3)

and take (3) minus (2),

 (4)

The Rodrigues representation is
 (5)

The polynomials can also be written
 (6)

where is the Floor Function and is the Ceiling Function, or in terms of a Determinant
 (7)

The first few Polynomials are

Letting allows the Chebyshev polynomials of the second kind to be written as

 (8)

The second linearly dependent solution to the transformed differential equation is then given by
 (9)

which can also be written
 (10)

where is a Chebyshev Polynomial of the First Kind. Note that is therefore not a Polynomial.

See also Chebyshev Approximation Formula, Chebyshev Polynomial of the First Kind, Ultraspherical Polynomial

References

Abramowitz, M. and Stegun, C. A. (Eds.). Orthogonal Polynomials.'' Ch. 22 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 771-802, 1972.

Arfken, G. Chebyshev (Tschebyscheff) Polynomials'' and Chebyshev Polynomials--Numerical Applications.'' §13.3 and 13.4 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 731-748, 1985.

Rivlin, T. J. Chebyshev Polynomials. New York: Wiley, 1990.

Spanier, J. and Oldham, K. B. The Chebyshev Polynomials and .'' Ch. 22 in An Atlas of Functions. Washington, DC: Hemisphere, pp. 193-207, 1987.