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Question Number 3198 by Filup last updated on 07/Dec/15

Π_(i=a) ^n i=? for ∣i∣<1, a∈R

$$\underset{{i}={a}} {\overset{{n}} {\prod}}{i}=?\:\:\:\:\:\mathrm{for}\:\mid{i}\mid<\mathrm{1},\:\:{a}\in\mathbb{R} \\ $$

Commented byFilup last updated on 07/Dec/15

=(1/a)×(1/(a+1))×...×(1/n) =(1/((((n!)/((a−1)!))))) =(((a−1)!)/(n!)) correct?

$$=\frac{\mathrm{1}}{{a}}×\frac{\mathrm{1}}{{a}+\mathrm{1}}×...×\frac{\mathrm{1}}{{n}} \\ $$ $$=\frac{\mathrm{1}}{\left(\frac{{n}!}{\left({a}−\mathrm{1}\right)!}\right)} \\ $$ $$=\frac{\left({a}−\mathrm{1}\right)!}{{n}!} \\ $$ $$ \\ $$ $$\mathrm{correct}? \\ $$

Commented byprakash jain last updated on 07/Dec/15

I don′t think factorial function is defined for real numbers.

$$\mathrm{I}\:\mathrm{don}'\mathrm{t}\:\mathrm{think}\:\mathrm{factorial}\:\mathrm{function}\:\mathrm{is} \\ $$ $$\mathrm{defined}\:\mathrm{for}\:\mathrm{real}\:\mathrm{numbers}. \\ $$

Commented byFilup last updated on 07/Dec/15

What about in Γ notation?

$$\mathrm{W}{hat}\:\mathrm{about}\:\mathrm{in}\:\Gamma\:\mathrm{notation}? \\ $$

Commented byprakash jain last updated on 07/Dec/15

Let us say n=i+k k∈Z^+ Π_(i=1) ^n i=a(a+1)(a+2)...(a+k)

$$\mathrm{Let}\:\mathrm{us}\:\mathrm{say}\:{n}={i}+{k}\:{k}\in\mathbb{Z}^{+} \\ $$ $$\underset{{i}=\mathrm{1}} {\overset{{n}} {\prod}}{i}={a}\left({a}+\mathrm{1}\right)\left({a}+\mathrm{2}\right)...\left({a}+{k}\right) \\ $$

Commented byprakash jain last updated on 07/Dec/15

If n=i+k, k∈Z^+ then Γ(n) and Γ(i) be derived with Γ(x+1)=xΓ(x)

$$\mathrm{If}\:{n}={i}+{k},\:{k}\in\mathbb{Z}^{+} \\ $$ $$\mathrm{then}\:\Gamma\left({n}\right)\:{and}\:\Gamma\left({i}\right)\:\mathrm{be}\:\mathrm{derived}\:\mathrm{with} \\ $$ $$\Gamma\left({x}+\mathrm{1}\right)={x}\Gamma\left({x}\right) \\ $$

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