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)
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(2)
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(3)
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(4)
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(5)
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(6)
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(7)
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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)
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(9)
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for and . To see the relationship to a Chebyshev polynomial of the first kind , take of equation (9) to obtain
(10)
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(11)
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Multiplying (◇) by then gives
(12)
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and adding (12) and (◇) gives
(13)
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(14)
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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)
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The polynomials can also be defined in terms of the sums
(16)
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(17)
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where is the floor function and is the ceiling function, or in terms of the product
(18)
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(Zwillinger 1995, p. 696).
also obey the interesting determinant identity
(19)
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The Chebyshev polynomials of the second kind are a special case of the Jacobi polynomials with ,
(20)
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(21)
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where is a hypergeometric function (Koekoek and Swarttouw 1998).
Letting allows the Chebyshev polynomials of the second kind to be written as
(22)
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The second linearly dependent solution to the transformed differential equation is then given by
(23)
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which can also be written
(24)
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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).