Formula: T_0 = 1, T_1 = x, T_{n+1}(x) = 2 x T_n(x) - T_{n-1}(x); equivalently T_n(x) = cos(n arccos x)
References
Chebyshev (1853). The original is The recurrence T_{n+1} = 2 x T_n - T_{n-1} with T_0 = 1, T_1 = x and the identity T_n(cos t) = cos(n t) are the standard published definitions.
Examples
V <- c(1, 2, 3, 4, 5, 6, 7, 8)
Chebbasis(V)
#> $basis
#> [,1] [,2] [,3] [,4] [,5] [,6]
#> [1,] 1 1 1 1 1 1
#> [2,] 1 2 7 26 97 362
#> [3,] 1 3 17 99 577 3363
#> [4,] 1 4 31 244 1921 15124
#> [5,] 1 5 49 485 4801 47525
#> [6,] 1 6 71 846 10081 120126
#> [7,] 1 7 97 1351 18817 262087
#> [8,] 1 8 127 2024 32257 514088
#>
#> $degree
#> [1] 5
#>
#> $trig
#> [,1] [,2] [,3] [,4] [,5] [,6]
#> [1,] 1 1 1 1 1 1
#> [2,] NA NA NA NA NA NA
#> [3,] NA NA NA NA NA NA
#> [4,] NA NA NA NA NA NA
#> [5,] NA NA NA NA NA NA
#> [6,] NA NA NA NA NA NA
#> [7,] NA NA NA NA NA NA
#> [8,] NA NA NA NA NA NA
#>
#> $n
#> [1] 8
#>
#> $method
#> [1] "Chebyshev polynomial basis (first kind)"
#>
#> attr(,"class")
#> [1] "morie_rich_result" "list"