## Return to Answer

2 corrected a mathematical typo and typesetting

This is a partial answer...perhaps someone will improve on it!

Historically, most of the q-analog $q$-analog formulae (beginning from Euler) were derived based on the assumption that |q|$|q|< 1 1$ (to ensure series convergence) or $q=p^k$. q=p^k$for a prime$p$. John Baez in one of his weekly finds (week184) discusses the geometric interpretation of q=1$q=1$(counting over$CP^n$), q=-1 \mathbb CP^n$), $q=-1$ (counting over $RP^n$) \mathbb RP^n$) and q=prime$q=$a prime power (counting over PG($F_q.$))PG($\mathbb F_q$)). There is no discussion for other values of q.$q$. However, in Gasper and Rahman's Basic Hypergeometric Series, there is an inversion identity on page 4 which can be used when |q|$|q| > 1. 1$:$(a; q)_n = (a^{-1}; p)_n (-a)^n p^{n/2p^{-n(n-1)/2} $where$p=1/q$. This returns a new expression in base |$\frac{1}{q}$|$|1/q| < 11$. You can see some examples of the identity being applied in Gasper's Lecture Notes on q-series (Exercise 1.1, Exercise 2.3, pg 14) I have no idea how to interpret the result geometrically. Could it be relevant to Buildings, Buekenhout geometry or p-adic$p$-adic geometry?? 1 This is a partial answer...perhaps someone will improve on it! Historically, most of the q-analog formulae (beginning from Euler) were derived based on the assumption that |q|< 1 (to ensure series convergence) or$q=p^k$. John Baez in one of his weekly finds (week184) discusses the geometric interpretation of q=1 (counting over$CP^n$), q=-1 (counting over$RP^n$) and q=prime power (counting over PG($F_q.$)). There is no discussion for other values of q. However, in Gasper and Rahman's Basic Hypergeometric Series, there is an inversion identity on page 4 which can be used when |q| > 1.$(a; q)_n = (a^{-1}; p)_n (-a)^n p^{n/2} $This returns a new expression in base |$\frac{1}{q}\$| < 1. You can see some examples of the identity being applied in Gasper's Lecture Notes on q-series (Exercise 1.1, Exercise 2.3, pg 14)

I have no idea how to interpret the result geometrically. Could it be relevant to Buildings, Buekenhout geometry or p-adic geometry ??