On Mon, Feb 25, 2013 at 10:18 AM, Bill Gosper <billgosper@gmail.com> wrote:
On Sun, Feb 24, 2013 at 11:21 PM, Bill Gosper <billgosper@gmail.com>wrote:
These bilateral products invariant in the unit disk may be common: Product[1 + (-1/2 + (I*Sqrt[7])/2)*z^2^n + z^(3*2^n) + (-1/2 - (I*Sqrt[7])/2)*z^2^(1 + n), {n, -oo,oo}] == I/Sqrt[7]
Product[((I + z^2^k)*(-I + z^2^(1 + k)))/(1 + z^2^k), {k, -oo, oo}] == -((-1)^(3/4)/(2*Sqrt[2])),
Product[1 + z^2^k*(-1 + z^2^k)*((-1)^(1/3) + z^2^k),{k, -oo, oo}] == -(-1)^(2/3)/3
Product[1 + I*Sqrt[2]*z^3^k - z^(2*3^k) + z^(4*3^k) - I*Sqrt[2]*z^3^(1 + k), {k, -oo, oo}] == I/Sqrt[2]
|z|<1.
There seem(s) to be about a dozen of these ^3^k flavor. I've been so amused at the z-invariance that I've neglected to give the underlying telescoping forms. Here's this latest: Product[2*Cos[(2*t)/3^k] + 2*Sqrt[2]*Sin[t/3^k] - 1, {k, 1, Infinity}] == 1 + Sqrt[2]*Sin[t] (∀t) rwg
[...]
Multiplying the identities gives the curious, bilateral product Out[331]=Product[1 - x^2^k + x^2^(1 + k), {k, -Infinity, Infinity}] == 1/3 when |x|<1.
--rwg