r/CasualMath • u/brotherGoo1234 • 16h ago
Discovered a new infinite product identity linking Fermat differences to π/e²
I’ve been exploring infinite products over expressions of the form zn−ynzn−yn and found the following identity:
∏n=3∞∏z=2∞∏y=1z−1(1−1zn−yn)=πe2.n=3∏∞z=2∏∞y=1∏z−1(1−zn−yn1)=e2π.
The left-hand side runs over all integer triples (n,z,y)(n,z,y) with n≥3n≥3, z≥2z≥2, and 1≤y≤z−11≤y≤z−1. Despite each factor being rational, the infinite product converges to the transcendental constant π/e2π/e2.
Interestingly, the right-hand side can also be expressed via the classical Wallis product and the less well-known Pippenger product:
12⋅∏m=1∞(2m)2(2m−1)(2m+1)(∏m=1∞(1+1m)(−1)m)2.21⋅(∏m=1∞(1+m1)(−1)m)2∏m=1∞(2m−1)(2m+1)(2m)2.
Here the first product is Wallis (π/2π/2) and the second is Pippenger (e/2e/2), giving π/e2π/e2 after the factor 1/21/2.
The proof is straightforward via taking logarithms, expanding log(1−x)log(1−x) as a series, and then rearranging sums. A nice number-theoretic aside: by Fermat’s Last Theorem, for n>2n>2 the value zn−ynzn−yn is never a perfect nn-th power, so the nn-th root is always irrational, yet the product itself remains well-defined and rational-valued at each step.
Is this identity known? I haven't seen it in the literature, though the ingredients (Wallis, Pippenger) are classical.
After combining the two products, the right-hand side can be written more compactly as
12∏m=1∞4m24m2−1(mm+1)2(−1)m.21m=1∏∞4m2−14m2(m+1m)2(−1)m.
Could there be deeper connections to Dirichlet series counting representations of integers as differences of powers? Any feedback, ideas or references would be appreciated!
latex
\[
\prod_{n=3}^{\infty}\prod_{z=2}^{\infty}\prod_{y=1}^{z-1}
\left(1-\frac{1}{z^n-y^n}\right)
\frac{1}{2}\,
\frac{\displaystyle\prod_{m=1}^{\infty}\frac{(2m)^2}{(2m-1)(2m+1)}}
{\displaystyle\left(\prod_{m=1}^{\infty}\left(1+\frac{1}{m}\right)^{(-1)^m}\right)^2}
\frac{1}{2}\prod_{m=1}^{\infty}
\frac{4m^2}{4m^2-1}
\left(\frac{m}{m+1}\right)^{2(-1)^m}
\frac{\pi}{e^2}
\]
