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).
