First discovery

Previously, it was mentioned that setting ALPHA = 1, GAMMA = 1, BETA = 10 and omitting the d = d * b operation (OP#1) (as for the calculation of e), and using the exclusive or operation, yields

1.543080634815243778477905620757061682601529112365863704737402...

which happens to be cosh(1).

Hence we state the interesting identity:

where ⊕ means exclusive or, and we replace 1 ⊕ 1 = 0 with 1 (to avoid dividing by zero).

The product in the denominator gives 1, 3, 6, 30, 120, 840, 5040... which is A265376 in the The On-Line Encyclopedia of Integer Sequences (OEIS), Published electronically at http://oeis.org. The proof that the above identity is true is straightforward: even terms of the denominator equal (2n-1)! + (2n)! and odd terms equal (2n-1)!. The former sum to 1/e and the latter to sinh(1), and we recall cosh(1) = 1/e + sinh(1).

And there's plenty more of this type of thing, for example (i.e. setting ALPHA = 4, GAMMA = 4, BETA = 10) gives the following:

Although there might seem to be a preponderance for the exclusive or operation to yield values writable in terms of hyperbolic functions, this is not always the case, for example: