uru mai: indistinguishable points of different q-deformations

Since the negative of the derivative of the Gamma function evaluated at 1 is the Euler-Mascheroni constant -\Gamma'(1) = \gamma, and the,er, most obvious way of q-deforming this is the q-digamma function, here is a phase portrait of (one possible q deformed Euler Mascheroni Constant:


But the Euler-Mascheorni constant \gamma can also be expressed as an infinite series involving the Riemann zeta function:
\gamma=\sum_{m=2}^{\infty} (-1)^m\frac{\zeta(m)}{m}
and poking around google seems to indicate there are a few different ways of q-deforming the Riemann zeta function. Now, this suggests the question… “where do these different q-deformations agree”. I’m using the Maori expression uru mai to denote points where two different q-deformations are indistinguishable. For instance, there are three Jackson q-Bessel functions, and I can certainly make phase portraits of their differences and look for zeros — points uru mai with respect to a number of q-defromations.

(‘uru mai’ is Maori for consistent, just because the word ‘consistent’ has an overuse problem)


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