Is-there-any-pi-product-notation-rules-I-discovered-some-such-as-k-a-b-k-b-a-1-k-a-b-c-c-b-a-1-k-a-b-c-k-k-a-b-c-k-a-b-k-k-a-b-k-c-k-a-c-b

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 [\dots] ?

Commented by MJS last updated on 10/Nov/19

\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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