Divergent Products of Real Numbers

$\zeta(\,)$ denotes the Riemann zeta function.
$\gamma = \zeta(1) \approx 0.577215664901532860606512090082402431\ldots$ denotes the Euler-Mascheroni constant.
$A \approx 1.282427129100622636875342568869791727767688\ldots$ denotes the Glaisher-Kinkelin constant.

$c \; (c>0)$ is a constant (real).

HYPERSECTIONS

$\displaystyle \prod_{\substack{n\geq1\\k \text{ shifts}}} c = \underbrace{1.\,\ldots\,.1}_{k \text{ shifts}}\,.\,c\,.\,c\,.\,c\,.\,c\,.\,c\,.\,c\,.\,\ldots = c^{-\frac{1}{2}-k} = \dfrac{1}{c^k\sqrt{c}}$ (unstable)

$\displaystyle \prod_{\substack{n\geq1\\k \text{ shifts}}} c^n = \underbrace{1.\,\ldots\,.1}_{k \text{ shifts}}\,.\,c\,.\,c^2\,.\,c^3\,.\,c^4\,.\,c^5\,.\,c^6\,.\,\ldots = c^{-\frac{1}{12}+\frac{k^2}{2}}$ (unstable)

SUPERSECTIONS

$\displaystyle \prod_{\substack{n\geq1\\ \text{single shift}}} c = 1\,.\,c\,.\,c\,.\,c\,.\,c\,.\,c\,.\,c\,.\,\ldots = c^{-\frac{3}{2}} = \dfrac{1}{c\sqrt{c}}$ (unstable)

$\displaystyle \prod_{\substack{n\geq1\\ \text{double shift}}} c = 1\,.\,1\,.\,c\,.\,c\,.\,c\,.\,c\,.\,c\,.\,c\,.\,\ldots = c^{-\frac{5}{2}} = \dfrac{1}{c^2\sqrt{c}}$ (unstable)

$\displaystyle \prod_{\substack{n\geq1\\ \text{triple shift}}} c = 1\,.\,1\,.\,1\,.\,c\,.\,c\,.\,c\,.\,c\,.\,c\,.\,c\,.\,\ldots = c^{-\frac{7}{2}} = \dfrac{1}{c^3\sqrt{c}}$ (unstable)

$\displaystyle \prod_{\substack{n\geq1\\ \text{single shift}}} c^n = 1\,.\,c\,.\,c^2\,.\,c^3\,.\,c^4\,.\,c^5\,.\,c^6\,.\,\ldots = c^{\frac{5}{12}}$ (unstable)

$\displaystyle \prod_{\substack{n\geq1\\ \text{double shift}}} c^n = 1\,.\,1\,.\,c\,.\,c^2\,.\,c^3\,.\,c^4\,.\,c^5\,.\,c^6\,.\,\ldots = c^{\frac{23}{12}}$ (unstable)

$\displaystyle \prod_{\substack{n\geq1\\ \text{triple shift}}} c^n = 1\,.\,1\,.\,1\,.\,c\,.\,c^2\,.\,c^3\,.\,c^4\,.\,c^5\,.\,c^6\,.\,\ldots = c^{\frac{53}{12}}$ (unstable)

$\displaystyle \prod_{n\geq1} 1.c = 1\,.\,c\,.\,1\,.\,c\,.\,1\,.\,c\,.\,1\,.\,c\,.\,\ldots = c^{-\frac{1}{2}} = \dfrac{1}{\sqrt{c}}$ (unstable)

$\displaystyle \prod_{n\geq1} c.1 = c\,.\,1\,.\,c\,.\,1\,.\,c\,.\,1\,.\,c\,.\,1\,.\,\ldots = 1$ (unstable)

SECTIONS

$\displaystyle \prod_{n\geq 1} c = c\,.\,c\,.\,c\,.\,c\,.\,c\,.\,c\,.\,\ldots = c^{-\frac{1}{2}} = \dfrac{1}{\sqrt{c}}$ (convergent)

$\displaystyle \prod_{n\geq0} c^n = 1\,.\,c\,.\,c^2\,.\,c^3\,.\,c^4\,.\,c^5\,.\,c^6\,.\,\ldots = c^{\frac{5}{12}}$ (unstable)

$\displaystyle \prod_{n\geq1} c^n = c\,.\,c^2\,.\,c^3\,.\,c^4\,.\,c^5\,.\,c^6\,.\,\ldots = c^{-\frac{1}{12}}$ (unstable)

Euler’s reflection formula

$\displaystyle \prod_{n\geq1} \dfrac{n^2}{n^2-s^2} = \dfrac{1}{1-s^2}\cdot\dfrac{4}{4-s^2}\cdot\dfrac{9}{9-s^2}\cdot\dfrac{16}{16-s^2}\cdot\,\cdots \\ = \Gamma(1+s)\,\Gamma(1-s) = \dfrac{\pi s}{\sin(\pi s)}$ (convergent)

SUBSECTIONS

$\displaystyle \prod_{n\geq 2} c = \prod_{n\geq1} c = c\,.\,c\,.\,c\,.\,c\,.\,c\,.\,c\,.\,\ldots = c^{-\frac{1}{2}} = \dfrac{1}{\sqrt{c}}$ (convergent)

$\displaystyle \prod_{n\geq 3} c = \prod_{n\geq1} c = c\,.\,c\,.\,c\,.\,c\,.\,c\,.\,c\,.\,\ldots = c^{-\frac{1}{2}} = \dfrac{1}{\sqrt{c}}$ (convergent)

$\displaystyle \prod_{n\geq2} c^n = \prod_{n\geq1} c^{n+1} = c^2\,.\,c^3\,.\,c^4\,.\,c^5\,.\,c^6\,.\,c^7\,.\,\ldots = c^{-\frac{7}{12}}$ (unstable)

$\displaystyle \prod_{n\geq3} c^n = \prod_{n\geq1} c^{n+2} = c^3\,.\,c^4\,.\,c^5\,.\,c^6\,.\,c^7\,.\,c^8\,.\,\ldots = c^{-\frac{13}{12}}$ (unstable)

Euler’s truncated reflection formula

$\displaystyle \prod_{n\geq2} \dfrac{n^2}{n^2-s^2} = \prod_{n\geq1} \dfrac{(n+1)^2}{(n+1)^2-s^2} = \dfrac{4}{4-s^2}\cdot\dfrac{9}{9-s^2}\cdot\dfrac{16}{16-s^2}\cdot\,\cdots \\ = \Gamma(2+s)\,\Gamma(2-s) = \dfrac{\pi s}{\sin(\pi s)}\cdot(1-s^2)$ (convergent)

$\displaystyle \prod_{n\geq3} \dfrac{n^2}{n^2-s^2} = \prod_{n\geq1} \dfrac{(n+2)^2}{(n+2)^2-s^2} = \dfrac{9}{9-s^2}\cdot\dfrac{16}{16-s^2}\cdot\dfrac{25}{25-s^2}\cdot\,\cdots \\ = \dfrac{\Gamma(3+s)\,\Gamma(3-s)}{4} = \dfrac{\pi s}{\sin(\pi s)}\cdot\dfrac{(1-s^2)\,(4-s^2)}{4}$ (convergent)

$\displaystyle \prod_{n\geq4} \dfrac{n^2}{n^2-s^2} = \prod_{n\geq1} \dfrac{(n+3)^2}{(n+3)^2-s^2} = \dfrac{16}{16-s^2}\cdot\dfrac{25}{25-s^2}\cdot\dfrac{36}{36-s^2}\cdot\,\cdots \\ = \dfrac{\Gamma(4+s)\,\Gamma(4-s)}{36} = \dfrac{\pi s}{\sin(\pi s)}\cdot\dfrac{(1-s^2)\,(4-s^2)\,(9-s^2)}{36}$ (convergent)

INFRASECTIONS

$\displaystyle \prod_{n\geq m+1} c = \prod_{n\geq1} c = c\,.\,c\,.\,c\,.\,c\,.\,c\,.\,c\,.\,\ldots = c^{-\frac{1}{2}} = \dfrac{1}{\sqrt{c}}$ (unstable)

$\displaystyle \prod_{n\geq m+1} c.c = \prod_{n\geq1} c.c = c\,.\,c\,.\,c\,.\,c\,.\,c\,.\,c\,.\ldots = c^{-\frac{1}{2}} = \dfrac{1}{\sqrt{c}}$ (unstable)

$\displaystyle \prod_{n\geq m+1} c^n = \prod_{n\geq1} c^{n+m} = c^{m+1}\,.\,c^{m+2}\,.\,c^{m+3}\,.\,c^{m+4}\,.\,\ldots = c^{-\frac{1}{12}-\frac{m}{2}}$ (unstable)

Euler’s truncated reflection formula (generalized)

$\displaystyle \prod_{n\geq m+1} \dfrac{n^2}{n^2-s^2} = \prod_{n\geq1} \dfrac{(n+m)^2}{(n+m)^2-s^2} \\ = \dfrac{(m+1)^2}{(m+1)^2-s^2}\cdot\dfrac{(m+2)^2}{(m+2)^2-s^2}\cdot\dfrac{(m+3)^2}{(m+3)^2-s^2}\cdot\,\cdots \\ = \dfrac{\Gamma(m+1+s)\,\Gamma(m+1-s)}{\prod_{k=1}^m k^2} = \dfrac{\pi s}{\sin(\pi s)}\cdot\prod_{k=1}^m\left(1-\left(\frac{s}{k}\right)^2\right)$ (convergent)

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