Is there any pi/product notation rules I discovered some such as: Π_(k=a) ^b [k]=((b!)/((a−1)!)) Π_(k=a) ^b [c]=c^(b−a+1) Π_(k=a) ^b [c∙k]=Π_(k=a) ^b [c]∙Π_(k=a) ^b [k] Π_(k=a) ^b [k+c]=Π_(k=a+c) ^(b+c) [k] But what about this Π


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Question Number 73340 by Raxreedoroid last updated on 10/Nov/19

$Is there any pi / product notation rules$ $I discovered some such as :$ $\underset{k = a}{\prod^{b}} [k] = \frac{b!}{(a - 1)!}$ $\underset{k = a}{\prod^{b}} [c] = c^{b - a + 1}$ $\underset{k = a}{\prod^{b}} [c \cdot k] = \underset{k = a}{\prod^{b}} [c] \cdot \underset{k = a}{\prod^{b}} [k]$ $\underset{k = a}{\prod^{b}} [k + c] = \underset{k = a + c}{\prod^{b + c}} [k]$ $But what about this$ $\underset{k = a}{\prod^{b}} [c \cdot k + d]$
Commented by mathmax by abdo last updated on 10/Nov/19

$w h a t m e a n s [. . .] ?$
Commented by MJS last updated on 10/Nov/19
$Π_(k=a) ^b [c×k+d] let d=c×e Π_(k=a) ^b [c×(k+e)]=(((b+e)!)/((a+e−1)!))c^(1−a+b) = =Π_(k=a) ^b [c]×Π_(k=a) ^b [k+e] if (d/c)=e∈Q\Z Π_(k=a) ^b [k+(d/c)]=(a+(d/c))((Γ(b+(d/c)+1))/(Γ(a+(d/c)+1)))$
$\underset{k = a}{\prod^{b}} [c \times k + d]$ $let d = c \times e$ $\underset{k = a}{\prod^{b}} [c \times (k + e)] = \frac{(b + e)!}{(a + e - 1)!} c^{1 - a + b} =$ $= \underset{k = a}{\prod^{b}} [c] \times \underset{k = a}{\prod^{b}} [k + e]$ $if \frac{d}{c} = e \in Q ∖ Z$ $\underset{k = a}{\prod^{b}} [k + \frac{d}{c}] = (a + \frac{d}{c}) \frac{Γ (b + \frac{d}{c} + 1)}{Γ (a + \frac{d}{c} + 1)}$
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