An -algebraic number is a number which satisfies
(1)
|
where is the Rogers L-function and are integers not all equal to 0 (Gordon and Mcintosh 1997). Loxton (1991, p. 289) gives a slew of similar identities having rational coefficients
(2)
|
instead of integers.
The only known -algebraic numbers of order 1 are
(3)
| |||
(4)
| |||
(5)
| |||
(6)
| |||
(7)
|
(Loxton 1991, pp. 287 and 289; Bytsko 1999), where .
The only known rational -algebraic numbers are 1/2 and 1/3:
(8)
|
(9)
|
(Lewin 1982, pp. 317-318; Gordon and McIntosh 1997).
There are a number of known quadratic -algebraic numbers. Watson (1937) found
(10)
|
(11)
|
(12)
|
where , , and are the roots of
(13)
|
so that
(14)
| |||
(15)
| |||
(16)
|
(Loxton 1991, pp. 287-288). These are known as Watson's identities.
Higher-order algebraic identities include
(17)
| |
(18)
| |
(19)
| |
(20)
| |
(21)
| |
(22)
| |
(23)
| |
(24)
| |
(25)
|
where
(26)
| |||
(27)
| |||
(28)
| |||
(29)
| |||
(30)
| |||
(31)
| |||
(32)
|
(Gordon and McIntosh 1997).