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

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In summary, Ramanujan discovered a relationship between e^pi and the golden ratio, involving an infinite nested arithmetic series of repeating terms. This led to the possibility of presenting an algebraic expression for a transcendental number, which contradicted some definitions of transcendental numbers. There may be subgroups of transcendental numbers that can be expressed as finite algebraic expressions, while others can only be expressed as infinite arithmetic or product series. Ramanujan also found a geometric expression for this relationship, using the "silver ratio" which refers to the "halving" of a non-right angle of a right triangle. This topic is further explored in Gelfond's constant and Gelfond-Schneider theorem.
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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.
 
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cc math said:
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
 
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