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The Chebyshev polynomials, named after Pafnuty Chebyshev,[1] are sequences of polynomials (of orthogonal polynomials) which are related to de Moivre's formula and which are easily defined recursively, like Fibonacci or Lucas numbers. One usually distinguishes between Chebyshev polynomials of the first kind, which are denoted , and Chebyshev polynomials of the second kind, which are denoted . The letter T is used because of the alternative transliterations of the name Chebyshev as Tchebychev (French) or Tschebyschow (German.)
Chebyshev polynomials of the first kind
The Chebyshev polynomials of the first kind[2] are defined by the recurrence relation
The first few Chebyshev polynomials of the first kind are
-
A closed-form formula (would be a Binet formula of the first type, but for the factor ) (Cf. Lucas numbers#Binet's closed-form formula) giving the Chebyshev polynomials of the first kind is
where and are the roots of the quadratic polynomial in terms of
-
The ordinary generating function for is
The exponential generating function for is
Triangle of coefficients of Chebyshev polynomials of the first kind
Chebyshev polynomials of the first kind
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The triangle of coefficients of Chebyshev polynomials of the first kind gives the infinite sequence of finite sequences
- {{1}, {1}, {-1, 2}, {-3, 4}, {1, -8, 8}, {5, -20, 16}, {-1, 18, -48, 32}, {-7, 56, -112, 64}, {1, -32, 160, -256, 128}, {9, -120, 432, -576, 256}, {-1, 50, -400, 1120, -1280, 512}, {-11, 220, -1232, 2816, -2816, 1024}, {1, -72, 840, -3584, 6912, -6144, 2048}, {13, -364, 2912, -9984, 16640, -13312, 4096}, ...}
Triangle of coefficients of Chebyshev polynomials of the first kind . (Cf. A008310)
- {1, 1, -1, 2, -3, 4, 1, -8, 8, 5, -20, 16, -1, 18, -48, 32, -7, 56, -112, 64, 1, -32, 160, -256, 128, 9, -120, 432, -576, 256, -1, 50, -400, 1120, -1280, 512, -11, 220, -1232, 2816, -2816, 1024, 1, -72, 840, -3584, 6912, -6144, 2048, 13, -364, 2912, -9984, 16640, -13312, 4096, ...}
Rows of coefficients of Chebyshev polynomials of the first kind
Row sums of coefficients of Chebyshev polynomials of the first kind
Columns of coefficients of Chebyshev polynomials of the first kind
Columns of absolute values of coefficients of Chebyshev polynomials of the first kind
Compare with the (2,1)-Pascal triangle columns.
Rising diagonals of coefficients of Chebyshev polynomials of the first kind
(...)
Rising diagonal sums of coefficients of Chebyshev polynomials of the first kind
Except for the 0 th rising diagonal, which sums to 1, the rising diagonal sums are all 0. Thus the j th rising diagonal sums to .
Falling diagonals of coefficients of Chebyshev polynomials of the first kind
(...)
Chebyshev polynomials of the second kind
The Chebyshev polynomials of the second kind[3] are defined by the recurrence relation
The first few Chebyshev polynomials of the second kind are
A closed-form formula (would be a Binet formula of the second type, except that the exponents are instead of ) (Cf. Fibonacci numbers#Binet's closed-form formula) giving the Chebyshev polynomials of the second kind is
where and are the roots of the quadratic polynomial in terms of
-
The ordinary generating function for is
The exponential generating function for is
Triangle of coefficients of Chebyshev polynomials of the second kind
Chebyshev polynomials of the second kind
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The triangle of coefficients of Chebyshev polynomials gives the infinite sequence of finite sequences
- {{1}, {2}, {-1, 4}, {-4, 8}, {1, -12, 16}, {6, -32, 32}, {-1, 24, -80, 64}, {-8, 80, -192, 128}, {1, -40, 240, -448, 256}, {10, -160, 672, -1024, 512}, {-1, 60, -560, 1792, -2304, 1024}, {-12, 280, -1792, 4608, -5120, 2048}, {1, -84, 1120, -5376, 11520, -11264, 4096}, ...}
Triangle of coefficients of Chebyshev polynomials . (Cf. A008312)
- {1, 2, -1, 4, -4, 8, 1, -12, 16, 6, -32, 32, -1, 24, -80, 64, -8, 80, -192, 128, 1, -40, 240, -448, 256, 10, -160, 672, -1024, 512, -1, 60, -560, 1792, -2304, 1024, -12, 280, -1792, 4608, -5120, 2048, 1, -84, 1120, -5376, 11520, -11264, 4096, ...}
Rows of coefficients of Chebyshev polynomials of the second kind
Row sums of coefficients of Chebyshev polynomials of the second kind
Columns of coefficients of Chebyshev polynomials of the second kind
Columns of absolute values of coefficients of Chebyshev polynomials of the second kind
Compare with the (1,1)-Pascal triangle columns.
Rising diagonals of coefficients of Chebyshev polynomials of the second kind
(...)
Rising diagonal sums of coefficients of Chebyshev polynomials of the second kind
The rising diagonal sums are alternatively 1 and 0.
Falling diagonals of coefficients of Chebyshev polynomials of the second kind
(...)
See also
Notes