Is this a new identity? quarter-Gamma ratio

Ideas
1

I was teaching myself about mathematical identities like pi and varpi and the lemniscate and I came across the Euler Reflection formula. After some LLM iterations and Google-fu, I started asking deeper questions about how all geometry related to one another and what the transitions to other shapes like a circle to a square and vice versa. After a while, I started noticing a hard to pin down pattern and I think I identified it specifically. I’ve colloquially named it “G*” in a nod to Gauss and the AGM and the wildcard operator “*”. I was doing this as a fun learning exercise and am by no means a scholarly mathematician, so I’d love your interpretations of the math I’ve been building.

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