I Is e^pi a unique transcendental, or perhaps not transcendental?

[cc math]

Ramanujan discovered a relationship between e^pi and the golden ratio which involved an infinite nested arithmetic series of repeating term.
Argument: Since such a series is the equivalent of a simple finite algebraic and geometric expression, there can be presented an algebraic expression of a transcendental number which is equal to an algebraic number.
Some, such as Euler, have stated that this should not be possible per their definition of transcendental.
So my question is: Are their subgroups of transcendental numbers, such that those that can be calculated/presented as a finite algebraic expression, and those that can only be presented as an infinite arithmetic or product series, or neither, so as to distinguish them from each other?

Notes:
Ramanujan found a general class equation for e^pi at integer intervals involving infinite product series. But as far as I know, there is only one nested arithmetic relation (Please correct me if I am wrong).

Essentially:
The ‘silver ratio’ of the ‘golden ratio’ is equal to e^(2/5*pi) divided by the ‘silver ratio’ of e^(2*pi)/2.

The geometrical expression of this relation is more straightforward.

By ‘silver ratio’, I refer to the ‘halving’ of a non-right angle of a right triangle, or simply, the sum of the long leg with the hypotenuse of a right triangle, with the short leg as 1.

Thanks for any thoughts on this.

 

[fresh_42]

Ramanujan discovered a relationship between e^pi and the golden ratio which involved an infinite nested arithmetic series of repeating term.
Argument: Since such a series is the equivalent of a simple finite algebraic and geometric expression, there can be presented an algebraic expression of a transcendental number which is equal to an algebraic number.
Some, such as Euler, have stated that this should not be possible per their definition of transcendental.
So my question is: Are their subgroups of transcendental numbers, such that those that can be calculated/presented as a finite algebraic expression, and those that can only be presented as an infinite arithmetic or product series, or neither, so as to distinguish them from each other?

Notes:
Ramanujan found a general class equation for e^pi at integer intervals involving infinite product series. But as far as I know, there is only one nested arithmetic relation (Please correct me if I am wrong).

Essentially:
The ‘silver ratio’ of the ‘golden ratio’ is equal to e^(2/5*pi) divided by the ‘silver ratio’ of e^(2*pi)/2.

The geometrical expression of this relation is more straightforward.

By ‘silver ratio’, I refer to the ‘halving’ of a non-right angle of a right triangle, or simply, the sum of the long leg with the hypotenuse of a right triangle, with the short leg as 1.

Thanks for any thoughts on this.

See
https://en.wikipedia.org/wiki/Gelfond's_constant
and
https://en.wikipedia.org/wiki/Gelfond–Schneider_theorem#Corollaries