"But Mathematica should be able to average Ο_q for us over the unit disk." Whoa, harder than I expected. It requires termwise integrating the *square* of a π series. So far, all I have is Integrate[Ο_q,{q,0,1}] == 32/65 + 4*Sum[ Sum[1/(17 + 4*k*(3 + k) - 20*n - 8*k*n + 8*n^2), {n, β}] - Sum[1/(25 + 4*k*(3 + k) - 20*n - 8*k*n + 8*n^2), {n, β}] - Sum[1/(5 + 4*k^2 - 4*n + 8*k*n + 8*n^2), {n, β}] + Sum[1/(13 + 4*k^2 - 4*n + 8*k*n + 8*n^2), {n, β}], {k, β}] ~ 1.19153301072378 (accuracy unknown). Compare with the Dedekind π(q) and (q,q)_β 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? βrwg Mathematica caution: I inadvisedly assigned a value (involving QFactorial[-1/2,q]^2) to Subscript[Ο,q], and now I can't figure out how to undo it!-{ (I'm chicken to Remove[Ο].) 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)! == Ο z/Sin[Ο z]. (z = -1/2 is where we get (-1/2)! = βΟ.) This determines the "q-extension" of Ο to be
q^(1/4) QFactorial[-1/2,q^2]^2== Ο*(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 qβ1 limit. Likewise, other points on the unit circle. But Mathematica should be able to average Ο_q for us over the unit disk.
For every triplet of distinct positive integers a,b,c, there is a polynomial identity relating Ο_q^a, Ο_q^b, and Ο_q^c, because Ο_q==(1-q^2) π(q^2)^4)/π(q)^2, where π(q) := q^(1/24) Product[1-q^n,{n,β}] = DedekindEta[Log@q/2/I/Ο]
and π(q) satisfies an equivalent infinitude of three term identities.
The simplest Ο_q identity is Out[95]= 4==-(1+q^4)^2 (Ο_q^2)^2/(Ο_q^4)^2+(1+q^2)^2 (1+q^4) (Ο_q)^2/Ο_q^2/Ο_q^4 (Converted to π functions, this is Jacobi's Γquatio Identica Satis Abstrusa. But the *least* abstruse, it turns out!).
Testing with our Ο_q expansion near 1, Ο_q:>Ο*(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? βrwg