Divergent Products of Integers

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

HYPERSECTIONS

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

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

$\displaystyle \prod_{\substack{n\geq1\\k \text{ shifts}}} \! an = \underbrace{1.\,\ldots\,.1}_{k \text{ shifts}}.\,a\,.\,2a\,.\,3a\,.\,4a\,.\,5a\,.\,6a\,.\ldots \\ = \prod_{\substack{n\geq1\\k \text{ shifts}}} \! a \;.\! \prod_{\substack{n\geq1\\k \text{ shifts}}} \! n = \dfrac{1}{a^{k+\frac{1}{2}}} \,.\, \sqrt{2\pi} = \underbrace{\dfrac{\sqrt{2\pi}}{\sqrt{a}}}_{\text{seed}} \, \underbrace{\dfrac{1}{a^k}}_{\substack{\text{shift} \\ \text{factor}}} \qquad \textit{(unstable)}$

$\displaystyle \prod_{\substack{n\geq0\\k \text{ shifts}}} \! (an\!+\!1) = \underbrace{1.\,\ldots\,.1}_{k \text{ shifts}}.\,1\,.\,(a\!+\!1)\,.\,(2a\!+\!1)\,.\,(3a\!+\!1)\,.\,(4a\!+\!1)\,.\ldots \\ = \underbrace{\dfrac{\sqrt{2\pi}}{\sqrt{a}\;\Gamma(\frac{1}{a})}}_{\text{seed}} \, \underbrace{\dfrac{1}{a^k}}_{\substack{\text{shift} \\ \text{factor}}} \qquad \textit{(unstable)}$

$\displaystyle \prod_{\substack{n\geq1\\k \text{ shifts}}} \! (an\!+\!1) = \underbrace{1.\,\ldots\,.1}_{k \text{ shifts}}.\,(a\!+\!1)\,.\,(2a\!+\!1)\,.\,(3a\!+\!1)\,.\,(4a\!+\!1)\,.\ldots \\ = \underbrace{\dfrac{\sqrt{2\pi a}}{\Gamma(\frac{1}{a})}}_{\text{seed}} \, \underbrace{\dfrac{1}{a^k}}_{\substack{\text{shift} \\ \text{factor}}} \qquad \textit{(unstable)}$

$\displaystyle \prod_{\substack{n\geq0\\k \text{ shifts}}} \! (n\!+\!b) = \underbrace{1.\,\ldots\,.1}_{k \text{ shifts}}.\,b\,.\,\,(1\!+\!b)\,.\,(2\!+\!b)\,.\,(3\!+\!b)\,.\,(4\!+\!b)\,.\ldots \\ = \underbrace{\dfrac{\sqrt{2\pi}}{\Gamma(b)}}_{\text{seed}} \qquad \textit{(semi-stable)}$

$\displaystyle \prod_{\substack{n\geq1\\k \text{ shifts}}} \! (n\!+\!b) = \underbrace{1.\,\ldots\,.1}_{k \text{ shifts}}.\,(1\!+\!b)\,.\,(2\!+\!b)\,.\,(3\!+\!b)\,.\,(4\!+\!b)\,.\ldots \\ = \underbrace{\dfrac{\sqrt{2\pi}}{\Gamma(1\!+\!b)}}_{\text{seed}} \qquad \textit{(semi-stable)}$

$\displaystyle \prod_{\substack{n\geq0\\k \text{ shifts}}} \! \left(n\!+\!\frac{b}{a}\right) = \underbrace{1.\,\ldots\,.1}_{k \text{ shifts}}\,.\,\frac{b}{a}\,.\left(1\!+\!\frac{b}{a}\right).\left(2\!+\!\frac{b}{a}\right).\left(3\!+\!\frac{b}{a}\right).\left(4\!+\!\frac{b}{a}\right).\ldots \\ = \underbrace{\dfrac{\sqrt{2\pi}}{\Gamma(\frac{b}{a})}}_{\text{seed}} \qquad \textit{(semi-stable)}$

$\displaystyle \prod_{\substack{n\geq1\\k \text{ shifts}}} \! \left(n\!+\!\frac{b}{a}\right) = \underbrace{1.\,\ldots\,.1}_{k \text{ shifts}}.\left(1\!+\!\frac{b}{a}\right).\left(2\!+\!\frac{b}{a}\right).\left(3\!+\!\frac{b}{a}\right).\left(4\!+\!\frac{b}{a}\right).\ldots \\ = \underbrace{\dfrac{\sqrt{2\pi}}{\Gamma(1\!+\!\frac{b}{a})}}_{\text{seed}} \qquad \textit{(semi-stable)}$

$\displaystyle \prod_{\substack{n\geq0\\k \text{ shifts}}} \! (an\!+\!b) = \underbrace{1.\,\ldots\,.1}_{k \text{ shifts}}.\,b\,.\,(a\!+\!b)\,.\,(2a\!+\!b)\,.\,(3a\!+\!b)\,.\,(4a\!+\!b)\,.\ldots \\ = \underbrace{\dfrac{\sqrt{2\pi}}{\sqrt{a}\;\Gamma(\frac{b}{a})}}_{\text{seed}} \, \underbrace{\dfrac{1}{a^k}}_{\substack{\text{shift} \\ \text{factor}}} \qquad \textit{(unstable)}$

$\displaystyle \prod_{\substack{n\geq1\\k \text{ shifts}}} \! (an\!+\!b) = \underbrace{1.\,\ldots\,.1}_{k \text{ shifts}}.\,(a\!+\!b)\,.\,(2a\!+\!b)\,.\,(3a\!+\!b)\,.\,(4a\!+\!b)\,.\ldots \\ = \underbrace{\dfrac{\sqrt{2\pi}}{\sqrt{a}\;\Gamma(1\!+\!\frac{b}{a})}}_{\text{seed}} \, \underbrace{\dfrac{1}{a^k}}_{\substack{\text{shift} \\ \text{factor}}} \qquad \textit{(unstable)}$

SUPERSECTIONS

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

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

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

$\displaystyle \prod_{\substack{n\geq1\\ \text{single shift}}} c^n = \prod_{\substack{n\geq0\\ \text{no shift}}} c^n = 1\,.\,c\,.\,c^2\,.\,c^3\,.\,c^4\,.\,c^5\,.\,c^6\,.\,\ldots = c^{\frac{5}{12}} \qquad \textit{(unstable)}$

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

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

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

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

$\displaystyle \prod_{n\geq1} n.c = 1\,.\,c\,.\,2\,.\,c\,.\,3\,.\,c\,.\,4\,.\,c\,.\,\ldots = \dfrac{\sqrt{\pi}}{\sqrt{c}} \qquad \textit{(unstable)}$

$\displaystyle \prod_{n\geq1} c.n = c\,.\,1\,.\,c\,.\,2\,.\,c\,.\,3\,.\,c\,.\,4\,.\,\ldots = \sqrt{2\pi} \qquad \textit{(unstable)}$

$\displaystyle \prod_{n\geq1} (an\!+\!b)^{1-(-1)^n} \!=\! (a\!+\!b)^2\,.\,1\,.\,(3a\!+\!b)^2\,.\,1\,.\,(5a\!+\!b)^2\,.\,1\,.\,\ldots \\ = 2\sqrt{\pi}\,\dfrac{\Gamma(\frac{b}{2a})}{\Gamma(\frac{b}{a}+1)\,\Gamma(\frac{b}{2a}-\frac{1}{2})} \qquad \textit{(unstable)}$

$\displaystyle \prod_{n\geq1} (an\!+\!b)^\frac{1-(-1)^n}{2} \!=\! (a\!+\!b)\,.\,1\,.\,(3a\!+\!b)\,.\,1\,.\,(5a\!+\!b)\,.\,1\,.\,\ldots \\ = \sqrt{2\sqrt{\pi}\,\dfrac{\Gamma(\frac{b}{2a})}{\Gamma(\frac{b}{a}+1)\,\Gamma(\frac{b}{2a}-\frac{1}{2})}} \qquad \textit{(unstable)}$

$\displaystyle \prod_{n\geq1} (an\!+\!b)^{1+(-1)^n} \!=\! 1\,.\,(2a\!+\!b)^2\,.\,1\,.\,(4a\!+\!b)^2\,.\,1\,.\,(6a\!+\!b)^2\,.\,1\,.\,\ldots \\ = \dfrac{\sqrt{\pi}}{a}\,\dfrac{\Gamma(\frac{b}{2a}-\frac{1}{2})}{\Gamma(\frac{b}{a}+1)\,\Gamma(\frac{b}{2a})} \qquad \textit{(unstable)}$

$\displaystyle \prod_{n\geq1} (an\!+\!b)^\frac{1+(-1)^n}{2} \!=\! 1\,.\,(2a\!+\!b)\,.\,1\,.\,(4a\!+\!b)\,.\,1\,.\,(6a\!+\!b)\,.\,1\,.\,\ldots \\ = \sqrt{\dfrac{\sqrt{\pi}}{a}\,\dfrac{\Gamma(\frac{b}{2a}-\frac{1}{2})}{\Gamma(\frac{b}{a}+1)\,\Gamma(\frac{b}{2a})}} \qquad \textit{(unstable)}$

SECTIONS

$\displaystyle \prod_{n\geq1} c = \prod_{n\geq0} c = c\,.\,c\,.\,c\,.\,c\,.\,c\,.\,c\,.\,\ldots = c^{-\frac{1}{2}} = \dfrac{1}{\sqrt{c}} \qquad \textit{(unstable)}$

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

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

$\displaystyle \prod_{n\geq1} c^\frac{1}{n} = c^\gamma \qquad \textit{(unstable)}$

$\displaystyle \prod_{n\geq0} \, (n+b) = b.(1+b).(2+b).(3+b).\ldots = \dfrac{\sqrt{2\pi}}{\Gamma(b)} \qquad \textit{(semi-stable)}$

$\displaystyle \prod_{n\geq1} \, (n+b) = (1+b).(2+b).(3+b).\ldots = \dfrac{\sqrt{2\pi}}{\Gamma(1\!+\!b)} \qquad \textit{(semi-stable)}$

$\displaystyle \prod_{n\geq0} \left(n\!+\!\frac{b}{a}\right) = \frac{b}{a}\,.\left(1\!+\!\frac{b}{a}\right).\left(2\!+\!\frac{b}{a}\right).\left(3\!+\!\frac{b}{a}\right).\ldots = \dfrac{\sqrt{2\pi}}{\Gamma(\frac{b}{a})} \qquad \textit{(semi-stable)}$

$\displaystyle \prod_{n\geq1} \left(n\!+\!\frac{b}{a}\right) = \left(1\!+\!\frac{b}{a}\right).\left(2\!+\!\frac{b}{a}\right).\left(3\!+\!\frac{b}{a}\right).\ldots = \dfrac{\sqrt{2\pi}}{\Gamma(1\!+\!\frac{b}{a})} \qquad \textit{(semi-stable)}$

$\displaystyle \prod_{n\geq0} \, (an+b) = b\,.\,(a+b)\,.\,(2a+b)\,.\,(3a+b)\,.\,\ldots = \dfrac{\sqrt{2\pi}}{\sqrt{a}\,\Gamma(\frac{b}{a})} \qquad \textit{(unstable)}$

$\displaystyle \prod_{n\geq1} \, (an+b) = (a+b)\,.\,(2a+b)\,.\,(3a+b)\,.\,\ldots = \dfrac{\sqrt{2\pi}}{\sqrt{a}\,\Gamma(1\!+\!\frac{b}{a})} \qquad \textit{(unstable)}$

SUBSECTIONS

$\displaystyle \prod_{n\geq3} c = \prod_{n\geq2} c = \prod_{n\geq1} c = c\,.\,c\,.\,c\,.\,c\,.\,c\,.\,c\,.\,\ldots = c^{-\frac{1}{2}} = \dfrac{1}{\sqrt{c}} \qquad \textit{(unstable)}$

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

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

$\displaystyle \prod_{n\geq4} c^n = \prod_{n\geq1} c^{n+3} = c^4\,.\,c^5\,.\,c^6\,.\,c^7\,.\,c^8\,.\,c^9\,.\,\ldots = c^{-\frac{19}{12}} \qquad \textit{(unstable)}$

$\displaystyle \prod_{n\geq2} \, (n\!+\!b) = (2\!+\!b)\,.\,(3\!+\!b)\,.\,(4\!+\!b)\,.\,\ldots = \dfrac{\sqrt{2\pi}}{\Gamma(2\!+\!b)} \qquad \textit{(semi-stable)}$

$\displaystyle \prod_{n\geq3} \, (n\!+\!b) = (3\!+\!b)\,.\,(4\!+\!b)\,.\,(5\!+\!b)\,.\,\ldots = \dfrac{\sqrt{2\pi}}{\Gamma(3\!+\!b)} \qquad \textit{(semi-stable)}$

$\displaystyle \prod_{n\geq4} \, (n\!+\!b) = (4\!+\!b)\,.\,(5\!+\!b)\,.\,(6\!+\!b)\,.\,\ldots = \dfrac{\sqrt{2\pi}}{\Gamma(4\!+\!b)} \qquad \textit{(semi-stable)}$

$\displaystyle \prod_{n\geq2} \left(n\!+\!\frac{b}{a}\right) = \left(2\!+\!\frac{b}{a}\right).\left(3\!+\!\frac{b}{a}\right).\left(4\!+\!\frac{b}{a}\right).\ldots = \dfrac{\sqrt{2\pi}}{\Gamma(2\!+\!\frac{b}{a})} \qquad \textit{(semi-stable)}$

$\displaystyle \prod_{n\geq3} \left(n\!+\!\frac{b}{a}\right) = \left(3\!+\!\frac{b}{a}\right).\left(4\!+\!\frac{b}{a}\right).\left(5\!+\!\frac{b}{a}\right).\ldots = \dfrac{\sqrt{2\pi}}{\Gamma(3\!+\!\frac{b}{a})} \qquad \textit{(semi-stable)}$

$\displaystyle \prod_{n\geq4} \left(n\!+\!\frac{b}{a}\right) = \left(4\!+\!\frac{b}{a}\right).\left(5\!+\!\frac{b}{a}\right).\left(6\!+\!\frac{b}{a}\right).\ldots = \dfrac{\sqrt{2\pi}}{\Gamma(4\!+\!\frac{b}{a})} \qquad \textit{(semi-stable)}$

$\displaystyle \prod_{n\geq2} \, (an+b) = (2a+b)\,.\,(3a+b)\,.\,(4a+b)\,.\,\ldots = \dfrac{\sqrt{2\pi}}{\sqrt{a}\,\Gamma(2\!+\!\frac{b}{a})} \qquad \textit{(unstable)}$

$\displaystyle \prod_{n\geq3} \, (an+b) = (3a+b)\,.\,(4a+b)\,.\,(5a+b)\,.\,\ldots = \dfrac{\sqrt{2\pi}}{\sqrt{a}\,\Gamma(3\!+\!\frac{b}{a})} \qquad \textit{(unstable)}$

$\displaystyle \prod_{n\geq4} \, (an+b) = (4a+b)\,.\,(5a+b)\,.\,(6a+b)\,.\,\ldots = \dfrac{\sqrt{2\pi}}{\sqrt{a}\,\Gamma(4\!+\!\frac{b}{a})} \qquad \textit{(unstable)}$

INFRASECTIONS

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

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

$\displaystyle \prod_{n\geq m+1} c^n = \prod_{n\geq1} c^{n+m} = c^{m+1}\,.\,c^{m+2}\,.\,c^{m+3}\,.\,c^{m+4}\,.\,\ldots = \underbrace{c^{-\frac{1}{12}}}_{\text{seed}} \underbrace{c^\frac{m}{2}}_{\substack{\text{truncation} \\ \text{factor}}} \qquad \textit{(unstable)}$

$\displaystyle \prod_{n\geq m+1} (n+b) = (m+1+b).(m+2+b).(m+3+b).\ldots \\ = \dfrac{\sqrt{2\pi}}{\Gamma(m\!+\!1\!+\!b)} = \dfrac{\sqrt{2\pi}}{\frac{1}{b}\,\Gamma(1\!+\!b)} \; \dfrac{1}{\prod_{k=0}^m (k\!+\!b)} = \underbrace{\dfrac{\sqrt{2\pi}}{\Gamma(1\!+\!b)}}_{\text{seed}} \; \underbrace{\dfrac{b}{(b)^{\overline{m+1}}}}_{\substack{\text{truncation} \\ \text{factor}}} \qquad \textit{(semi-stable)}$

$\displaystyle \prod_{n\geq m+1} \left(n\!+\!\dfrac{b}{a}\right) = \left(m\!+\!1\!+\!\frac{b}{a}\right).\left(m\!+\!2\!+\!\frac{b}{a}\right).\left(m\!+\!3\!+\!\frac{b}{a}\right).\ldots \\ = \dfrac{\sqrt{2\pi}}{\Gamma(m\!+\!1\!+\!\frac{b}{a})} = \dfrac{\sqrt{2\pi}}{\frac{a}{b}\,\Gamma(1\!+\!\frac{b}{a})} \; \dfrac{1}{\prod_{k=0}^m (k\!+\!\frac{b}{a})} = \underbrace{\dfrac{\sqrt{2\pi}}{\Gamma(1\!+\!\frac{b}{a})}}_{\text{seed}} \; \underbrace{\dfrac{1}{\frac{a}{b}\,(\frac{b}{a})^{\overline{m+1}}}}_{\substack{\text{truncation} \\ \text{factor}}} \qquad \textit{(semi-stable)}$

$\displaystyle \prod_{n\geq m+1} (an+b) = (am+a+b).(am+2a+b).(am+3a+b).\ldots \qquad \textit{(unstable)} \\ = \dfrac{\sqrt{2\pi}}{\sqrt{a}\,\Gamma(m\!+\!1\!+\!\frac{b}{a})} = \dfrac{\sqrt{2\pi}}{\frac{a\sqrt{a}}{b}\,\Gamma(1\!+\!\frac{b}{a})} \; \dfrac{1}{\prod_{k=0}^m (k\!+\!\frac{b}{a})} = \underbrace{\dfrac{\sqrt{2\pi}}{\sqrt{a}\,\Gamma(1\!+\!\frac{b}{a})}}_{\text{seed}} \; \underbrace{\dfrac{1}{\frac{a}{b}\,(\frac{b}{a})^{\overline{m+1}}}}_{\substack{\text{truncation} \\ \text{factor}}}$

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