"But Mathematica should be able to average π_q for us over the unit disk." Whoa, harder than I expected. It requires termwise integrating the *square* of a 𝜗 series. So far, all I have is Integrate[π_q,{q,0,1}] == 32/65 + 4*Sum[ Sum[1/(17 + 4*k*(3 + k) - 20*n - 8*k*n + 8*n^2), {n, ∞}] - Sum[1/(25 + 4*k*(3 + k) - 20*n - 8*k*n + 8*n^2), {n, ∞}] - Sum[1/(5 + 4*k^2 - 4*n + 8*k*n + 8*n^2), {n, ∞}] + Sum[1/(13 + 4*k^2 - 4*n + 8*k*n + 8*n^2), {n, ∞}], {k, ∞}] ~ 1.19153301072378 (accuracy unknown). Compare with the Dedekind 𝜂(q) and (q,q)_∞ averages over [0,1] at https://www.wolframcloud.com/obj/3ed3a0a9-14f1-42e1-8c6a-6f7fd0bbf483 . (which might rather be averaged over [-1, 1] or even the unit disk.) Can anyone do this double sum? —rwg Mathematica caution: I inadvisedly assigned a value (involving QFactorial[-1/2,q]^2) to Subscript[π,q], and now I can't figure out how to undo it!-{ (I'm chicken to Remove[π].) On Fri, Dec 6, 2019 at 10:04 PM Bill Gosper <billgosper@gmail.com> wrote:

My idea for q-Trigonometry was simply to substitute QFactorial for Factorial in the identity z! (-z)! == π z/Sin[π z]. (z = -1/2 is where we get (-1/2)! = √π.) This determines the "q-extension" of π to be

q^(1/4) QFactorial[-1/2,q^2]^2== π*(1+(q-1)+1/6 (q-1)^2-1/180 (q-1)^4+1/180 (q-1)^5-(37 (q-1)^6)/7560+4/945 (q-1)^7-(119 (q-1)^8)/32400+(9 (q-1)^9)/2800+O[q-1]^10)

(Arguably, q^(1/8) QFactorial[-1/2,q]^2).

This expansion (which should be in powers of 1-q) dates from an ancient Macsyma. I was unable to drag it out of Mathematica, which is woefully ignorant of expanding *q*-things near the all-important q➝1 limit. Likewise, other points on the unit circle. But Mathematica should be able to average π_q for us over the unit disk.

For every triplet of distinct positive integers a,b,c, there is a polynomial identity relating π_q^a, π_q^b, and π_q^c, because π_q==(1-q^2) 𝜂(q^2)^4)/𝜂(q)^2, where 𝜂(q) := q^(1/24) Product[1-q^n,{n,∞}] = DedekindEta[Log@q/2/I/π]

and 𝜂(q) satisfies an equivalent infinitude of three term identities.

The simplest π_q identity is Out[95]= 4==-(1+q^4)^2 (π_q^2)^2/(π_q^4)^2+(1+q^2)^2 (1+q^4) (π_q)^2/π_q^2/π_q^4 (Converted to 𝜂 functions, this is Jacobi's Æquatio Identica Satis Abstrusa. But the *least* abstruse, it turns out!).

Testing with our π_q expansion near 1, π_q:>π*(q+1/6 (-1+q)^2-1/180 (-1+q)^4+1/180 (-1+q)^5-(37 (-1+q)^6)/7560+4/945 (-1+q)^7-(119 (-1+q)^8)/32400+(9 (-1+q)^9)/2800)

% /. q -> q^2; % /. q -> q^2;

(My need to special case q^2 and q^4 affirms that the design of Series is seriously deficient.)

%95 /. {%, %%, %%%} <Big mess> In[117]:= Series[Activate[%],{q,1,9}] Out[117]= 4==4+O[q-1]^10

Quod erat orandum. Why idolize a mere number? —rwg