cos(nq) is a polynomial
function of cos(q).
The following relation defines a polynomial of degree n
known as the Chebyshev polynomial of degree n:
cos (nq) =
Tn (cos q)
The symbol T comes from careful Russian transliterations like
Tchebyshev,
Tchebychef (French) or
Tschebyschow
(German).
Alternate spellings include
Tchebychev (French)
and "Chebychev".
The trigonometric formula
cos (n+2)q =
2 cos q cos (n+1)q -
cos nq
translates into a simple recurrence relation which makes Chebyshev polynomials
very easy to tabulate, namely:
T0 (x) | = | 1 |
Tn+2 (x) =
2 x Tn+1 (x) - Tn (x)
|
T1 (x) | = |
| x |
|

Pafnuty Chebyshev |
T2 (x) | = |
-1 | + | 2 x2 | |
T3 (x) | = |
| -3 x | + | 4 x3 | |
T4 (x) | = |
1 | - | 8 x2 | + | 8 x4 | |
T5 (x) | = |
| 5 x | - | 20 x3 | + | 16 x5 |
T6 (x) | = |
-1 | + | 18 x2 | - | 48 x4 |
+ | 32 x6 |
T7 (x) | = |
| -7 x | + | 56 x3 | - | 112 x5 |
+ | 64 x7 |
T8 (x) | = |
1 | - | 32 x2 | + | 160 x4 |
- | 256 x6 | + | 128 x8 |
Knowing only the highest term of Tn and its obvious n distinct real zeroes,
we obtain immediately Tn as a product of n factors:
If n > 0, then Tn (x) = 2 n-1 |
n-1 |
( x - cos (k+½)
p/n ) |
Õ |
k=0 |
The case Tn (0) = (-1)n tells
something nice about a product of cosines.
Inverse formulas :
x0 | = | T0 |
|
x1 | = |
| T1 |
| |
2 x2 | = |
T0 | + | T2 | |
4 x3 | = |
| 3 T1 | + | T3 | |
8 x4 | = |
3 T0 | + | 4 T2 | + | T4 | |
16 x5 | = |
| 10 T1 | + | 5 T3 | + | T5 |
32 x6 | = |
10 T0 | + | 15 T2 | + | 6 T4 |
+ | T6 |
64 x7 | = |
| 35 T1 | + | 21 T3 | + | 7 T5 |
+ | T7 |
128 x8 | = |
35 T0 | + | 56 T2 | + | 28 T4 |
+ | 8 T6 | + | T8 |
(2014-07-26) A solution looking for a problem :
Chebyshev polynomials verify Tm(Tn(x)) =
Tmn(x). This unique property makes it possible to define pairs of
closely related functions from any pair of arithmetic functions u and v
(with subexponential growth) that are
Dirichlet inverses of each other,
using the following symmetrical relations:
|
¥ |
g ( x ) = |
å |
u(n) f ( Tn(x) ) |
|
n = 1 |
|
¥ |
f ( x ) = |
å |
v(n) g ( Tn(x) ) |
|
n = 1 |
If f (0) = 0,
those series are usually absolutely convergent,
because Tn(x) decreases exponentially with n,
for any fixed x in ]-1,+1[.
Proof :
Expand the latter right-hand-side using the definition of g :
å m
å n
u(n) v(m) f ( Tmn (x) )
=
å k
[
å d|k u(d) v(k/d)
]
f ( Tk (x) )
u and v being Dirichlet inverses,
the bracket is either 1 (if k = 1) or 0.
This applies, in particular, when u is a totally multiplicative
arithmetic function [i.e., such that u(mn) = u(m) u(n)
for any m & n ]
in which case its Dirichlet inverse can be expressed using the
Möbius function (m) :
v(n) = m(n) u(n)
Using Tn(x) = x1/n instead of Chebyshev polynomials,
this pattern was used in 1859 by Riemann to link
his (normalized) prime-counting function
f = p
with the celebrated jump function
g = J he obtained with u(n) = 1/n.
Bienaymé-Chebyshev inequality
|
Chebyshev economization
|
Pafnuty Chebyshev (1821-1894)