Divergent Products of Integers

$\zeta(\,)$ denotes the Riemann zeta function.

$e \approx 2.7182818284590452353602874713526624977572470\ldots$ denotes the Euler’s number.
$\gamma = \zeta(1) \approx 0.577215664901532860606512090082402431\ldots$ denotes the Euler-Mascheroni constant.
$A \approx 1.282427129100622636875342568869791727767688\ldots$ denotes the Glaisher-Kinkelin constant.

$r$ is the power or order in the infinite product.

GENERA

$\displaystyle \prod_{n\geq1} \, (an+b)^{cn+d} = a^{-\frac{d}{2}-\frac{c}{12}} \left(A\,e^{-\frac{b}{2a}-\frac{1}{12}}\right)^c \left(\dfrac{\Gamma(1\!+\!\frac{b}{a})}{\sqrt{2\pi}}\right)^{\frac{bc}{a}-d} \qquad \textit{(unstable)}$

$\displaystyle \prod_{n\geq1} \, (an\!+\!b)^{cn} = a^{-\frac{c}{12}} \left(A\,e^{-\frac{b}{2a}-\frac{1}{12}}\right)^c \left(\dfrac{\Gamma(1\!+\!\frac{b}{a})}{\sqrt{2\pi}}\right)^{\frac{bc}{a}} \qquad \textit{(unstable)}$

$\displaystyle \prod_{n\geq1} \, (an+b)^{n+d} = a^{-\frac{d}{2}-\frac{1}{12}} \; A\,e^{-\frac{b}{2a}-\frac{1}{12}} \left(\dfrac{\Gamma(1\!+\!\frac{b}{a})}{\sqrt{2\pi}}\right)^{\frac{b}{a}-d} \qquad \textit{(unstable)}$

$\displaystyle \prod_{n\geq1} \, (an\!+\!b)^n = a^{-\frac{1}{12}} \; A\,e^{-\frac{b}{2a}-\frac{1}{12}} \left(\dfrac{\Gamma(1\!+\!\frac{b}{a})}{\sqrt{2\pi}}\right)^{\frac{b}{a}} \qquad \textit{(unstable)}$

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

$\displaystyle \prod_{n\geq1} \, (n\!+\!b)^{cn+d} = \left(A\,e^{-\frac{b}{2}-\frac{1}{12}}\right)^c \left(\dfrac{\Gamma(1\!+\!b)}{\sqrt{2\pi}}\right)^{bc-d} \qquad \textit{(semi-stable)}$

$\displaystyle \prod_{n\geq1} \, (n\!+\!b)^{cn} = \left(A\,e^{-\frac{b}{2}-\frac{1}{12}}\right)^c \left(\dfrac{\Gamma(1\!+\!b)}{\sqrt{2\pi}}\right)^{bc} \qquad \textit{(semi-stable)}$

$\displaystyle \prod_{n\geq1} \, (n\!+\!b)^{n+d} = A\,e^{-\frac{b}{2}-\frac{1}{12}} \left(\dfrac{\Gamma(1\!+\!b)}{\sqrt{2\pi}}\right)^{b-d} \qquad \textit{(semi-stable)}$

$\displaystyle \prod_{n\geq1} \, (n\!+\!b)^n = A\,e^{-\frac{b}{2}-\frac{1}{12}} \left(\dfrac{\Gamma(1\!+\!b)}{\sqrt{2\pi}}\right)^b \qquad \textit{(semi-stable)}$

$\displaystyle \prod_{n\geq1} \, (n\!+\!b)^d = \left(\dfrac{\sqrt{2\pi}}{\Gamma(1\!+\!b)}\right)^d \qquad \textit{(semi-stable)}$

$\displaystyle \prod_{n\geq1} \, (an)^{cn+d} = a^{-\frac{d}{2}-\frac{c}{12}} \left(A\,e^{-\frac{1}{12}}\right)^c (2\pi)^\frac{d}{2} \qquad \textit{(unstable)}$

$\displaystyle \prod_{n\geq1} \, n^{cn+d} = \left(A\,e^{-\frac{1}{12}}\right)^c (2\pi)^\frac{d}{2} \qquad \textit{(semi-stable)}$

$\displaystyle \prod_{n\geq1} \, a^{cn+d} = a^{-\frac{d}{2}-\frac{c}{12}} \qquad \textit{(unstable)}$

$\displaystyle \prod_{n\geq1} n^r = 1\,.\,2^{r}.\,3^{r}.\,4^{r}.\,5^{r}.\,6^{r}.\ldots = (2\pi)^\frac{r}{2} \qquad \textit{(semi-stable)}$

$\displaystyle \prod_{n\geq1} (2n)^r = 2^{r}.\,4^{r}.\,6^{r}.\,8^{r}.\,10^{r}.\,12^{r}.\ldots = \pi^\frac{r}{2} \qquad \textit{(unstable)}$

$\displaystyle \prod_{n\geq1} (2n\!-\!1)^r = 1\,.\,3^{r}.\,5^{r}.\,7^{r}.\,9^{r}.\,11^{r}.\ldots =1 \qquad \textit{(unstable)}$

$\displaystyle \prod_{n\geq1} \, n^{2r} = \prod_{n\geq1} \, (n^r)^2 = \prod_{n\geq1} \, (n^2)^r = 1\,.\,4^{r}.\,9^{r}.\,16^{r}.\,25^{r}.\,36^{r}.\ldots = \nolinebreak (2\pi)^r \quad \textit{(semi-stable)}$

SUBGENERA

$\displaystyle \prod_{n\geq2} \, n^{2r} = \prod_{n\geq2} \, (n^r)^2 = \prod_{n\geq2} \, (n^2)^r = 4^{r}.\,9^{r}.\,16^{r}.\,25^{r}.\ldots = (2\pi)^r \qquad \textit{(unstable)}$

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

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

$\displaystyle \prod_{n\geq2} (n+1)^{n+r} = 3^{2+r}.\,4^{3+r}.\,5^{4+r}.\,6^{5+r}.\,7^{6+r}.\,8^{7+r}.\ldots = A \, e^{-\frac{13}{12}} \left(\dfrac{\pi}{2}\right)^\frac{r-1}{2} \quad \textit{(unstable)}$

$\displaystyle \prod_{n\geq2} (n^2)^{n+r} = 4^{2+r}.\,9^{3+r}.\,16^{4+r}.\,25^{5+r}.\,36^{6+r}.\,49^{7+r}.\ldots \\ = \left(A\,e^{-\frac{7}{12}}\right)^2 (2\pi)^r = A^2 \, e^{-\frac{7}{6}} \, (2\pi)^r \qquad \textit{(unstable)}$

$\displaystyle \prod_{n\geq2} (n^2-1)^{n+r} = 3^{2+r}.\,8^{3+r}.\,15^{4+r}.\,24^{5+r}.\,35^{6+r}.\,48^{7+r}.\ldots \\ = \left(A\,\sqrt{2}\,e^{-\frac{7}{12}}\right)^2 \pi^r = 2A^2 \, e^{-\frac{7}{6}} \, \pi^r \qquad \textit{(unstable)}$

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