For a positive integer, expressions of the form , , and can be expressed in terms of and only using the Euler formula and binomial theorem.
For ,
(1)
| |||
(2)
| |||
(3)
| |||
(4)
| |||
(5)
| |||
(6)
|
The first few values are given by
(7)
| |||
(8)
| |||
(9)
| |||
(10)
|
Other related formulas include
(11)
| |||
(12)
|
where is the floor function.
A product formula for is given by
(13)
|
The function can also be expressed as a polynomial in (for odd) or times a polynomial in as
(14)
|
where is a Chebyshev polynomial of the first kind and is a Chebyshev polynomial of the second kind. The first few cases are
(15)
| |||
(16)
| |||
(17)
| |||
(18)
|
Similarly, can be expressed as times a polynomial in as
(19)
|
The first few cases are
(20)
| |||
(21)
| |||
(22)
| |||
(23)
|
Bromwich (1991) gave the formula
(24)
|
where .
For , the multiple-angle formula can be derived as
(25)
| |||
(26)
| |||
(27)
| |||
(28)
| |||
(29)
| |||
(30)
|
The first few values are
(31)
| |||
(32)
| |||
(33)
| |||
(34)
|
Other related formulas include
(35)
| |||
(36)
| |||
(37)
|
The function can also be expressed as a polynomial in (for even) or times a polynomial in as
(38)
|
The first few cases are
(39)
| |||
(40)
| |||
(41)
| |||
(42)
|
Similarly, can be expressed as a polynomial in as
(43)
|
The first few cases are
(44)
| |||
(45)
| |||
(46)
| |||
(47)
|
Bromwich (1991) gave the formula
(48)
|
where .
The first few multiple-angle formulas for are
(49)
| |||
(50)
| |||
(51)
|
are given by Beyer (1987, p. 139) for up to .
Multiple-angle formulas can also be written using the recurrence relations
(52)
| |||
(53)
| |||
(54)
|