Chebyshev Polynomial of the Second Kind

A modified set of Chebyshev polynomials defined by a slightly different generating function. They arise in the development of four-dimensional spherical harmonics in angular momentum theory. They are a special case of the Gegenbauer polynomial with . They are also intimately connected with trigonometric multiple-angle formulas. The Chebyshev polynomials of the second kind are denoted , and implemented in the Wolfram Language as ChebyshevU[n, x]. The polynomials are illustrated above for and , 2, ..., 5.

The first few Chebyshev polynomials of the second kind are

			(1)
			(2)
			(3)
			(4)
			(5)
			(6)
			(7)

When ordered from smallest to largest powers, the triangle of nonzero coefficients is 1; 2; , 4; , 8; 1, , 16; 6, , 32; ... (OEIS A053117).

The defining generating function of the Chebyshev polynomials of the second kind is

			(8)
			(9)

for and . To see the relationship to a Chebyshev polynomial of the first kind , take of equation (9) to obtain

			(10)
			(11)

Multiplying (◇) by then gives

(12)

and adding (12) and (◇) gives

			(13)
			(14)

This is the same generating function as for the Chebyshev polynomial of the first kind except for an additional factor of in the denominator.

The Rodrigues representation for is

(15)

The polynomials can also be defined in terms of the sums

			(16)
			(17)

where is the floor function and is the ceiling function, or in terms of the product

(18)

(Zwillinger 1995, p. 696).

also obey the interesting determinant identity

(19)

The Chebyshev polynomials of the second kind are a special case of the Jacobi polynomials with ,

			(20)
			(21)

where is a hypergeometric function (Koekoek and Swarttouw 1998).

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

(22)

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

(23)

which can also be written

(24)

where is a Chebyshev polynomial of the first kind. Note that is therefore not a polynomial.

The triangle of resultants is given by , , , , , ... (OEIS A054376).

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