* Bill Gosper <billgosper@gmail.com> [Aug 29. 2011 16:54]:

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Unfortunately sin^2+cos^2 /=1 in q-land.

Joerg> One can regularize to get that one

(via abs( qexp(I*x) ) == 1 for some nice qexp()). The paper Jan L.\ Cie\'{s}li\'{n}ski: {Improved $q$-exponential and $q$-trigonometric functions}, arXiv:1006.5652v1 [math.CA], (29-June-2010). URL: \url{http://arxiv.org/abs/1006.5652}.} does it for the version where limit_{q-->1} gives the usual functions (exp(), cos(), sin()).

Caution to the casual: In this paper, superscripts of e, E and Script-E are *not* exponents. In fact, I don't even see a relation of the form Script-Exp_q1(x)^2 ~ Script-Exp_q2(2x), so I don't see much motivation, beyond nice q-difference and nice Pythagoras. Can we have q, Pythagoras, *and* identities? The paper claims application to quantum mechanics and "quantum calculus", so if there is an application it would be another case of "q" being a fortuitous name. (Recall that q-Vandermonde sum for the expected rank of a random bitmatrix with bit probablities p and q.)

(Implementations are given in http://www.jjj.de/pari/qtrig2r.gpi the nonregularized ones are in http://www.jjj.de/pari/qtrig2.gpi )

I recently did essentially the same with the products prod(n>=0, 1 - x*q^n) and prod(n>=0, 1 + x*q^n)

(See http://www.jjj.de/pari/qtrig1r.gpi the nonregularized ones are in http://www.jjj.de/pari/qtrig1.gpi ) [the majority of lines in both files are relations given as comments; no need to understand pari syntax]

Are the latter ones easily related to your q-trigs?

Not that I can see--I've never found a convincing q-exponential underpinning. (But I wouldn't swear there isn't one.) Likewise, I haven't found a convincing tan, cot, etc. I should check if qsin(i x) is anything like a sinh. Then we might have qexp:=qcosh+qsinh. I'm unsanguine. --rwg I was hoping to test qsin(7 pi/22) for exact expressibility as a quintic solution, but Julian's result says even lowly q-cos pi/5 is quintic for q>1, only quadratic for q=1.

For those in qtrig2r.gpi one can happily rediscover all kinds of modular equations. I was slightly excited to find a two-parameter version of Theta4(q^2)^2 = Theta4(q) * Theta3(q) which, expressed in my q-exp() is E(+q^2, -q^2)^2 = E(+q, -q) * E(-q, +q) The equation in question is E(+q^2, -q*x)^2 = E(+q, -x) * E(-q, +x)

I stopped investigating, however, after looking into Whittaker/Watson chapter.21 and suspecting my finding is really a specialization of stuff given there.

Btw. your paper stops at the prime p=5. I could, with some effort, get one modular equation for p=7 (indeed Somos' q56_18_240a) but none for p>7.

The _only_ thing I rely on is that my program expressing things as eta-products is working (and it's hard to imagine things going wrong there), the rest comes from (the analogs of) most simple trigonometric identities such as tan() * cot() == 1. (That is why I still much like my q-trigs...)

cheers, jj

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