prove-that-n-0-m-0-n-1-2-m-1-2-n-1-m-1-2-3-n-1-2-m-n-m-1-2-3-2-2-G-2-2-2-2-1-4-0-0-1-2-1-2-3-pi-

Question Number 98434 by M±th+et+s last updated on 14/Jun/20

p r o v e t h a t

Ω = \underset{n = 0}{\sum^{+ \infty}} \underset{m = 0}{\sum^{+ \infty}} \frac{Γ (n + \frac{1}{2}) . Γ (m + \frac{1}{2})}{Γ (n + 1) . Γ (m + 1)} . \frac{{(\frac{2}{3})}^{n} . {(\frac{1}{2})}^{m}}{(n + m + \frac{1}{2})}

= \frac{\sqrt{3}}{2 \sqrt{2}} . G_{2, 2}^{2, 2} (\frac{1}{4} ∣_{0, 0}^{\frac{1}{2}, \frac{1}{2}})

= \frac{\sqrt{3} π}{\sqrt{2}} K (\frac{3}{4}) = \frac{\sqrt{3} π^{2}}{2 \sqrt{2} A G M (1, \frac{1}{2})}

Commented by M±th+et+s last updated on 13/Jun/20

G_{2, 2}^{2, 2} (Z ∣ ∙) \equiv m e i j e r G - f u n c t i o n \equiv (s p e c i a l h i g h f u n c t i o n)

K (m) i s t h e c o m p l e t e e l l i p t i c i n t e g r a l o f f i r s t k i n d

w i t h p a r a m e t e r m = k^{2}

Γ (s) i s g a m m a f u n c t i o n

A G M (x, y) i s t h e A r i t h m e t i c - g e o m e t r i c m e a n o f x a n d y .

Commented by maths mind last updated on 14/Jun/20

Ω = {(\sum_{n ⩾ 0} \frac{Γ (n + \frac{1}{2})}{Γ (n + 1)})}^{2}

\frac{Γ (n + \frac{1}{2})}{Γ (n + 1)} ⩾ \frac{Γ (n)}{Γ (n + 1)} = \frac{1}{n}

\Rightarrow \sum_{n ⩾ 0} \frac{Γ (n + \frac{1}{2})}{Γ (n + 1)} ⩾ \sum_{n ⩾ 1} \frac{1}{n} \dots S u m / D i v e r g e

c h e k i t s i r

Commented by M±th+et+s last updated on 14/Jun/20

t h a n k y o u s i r .

y o u a r e r i g h s i r i c h e c k e d i t t h e r e w e r e

s o m e t y p o s

* * i a m s o r r y * *

Commented by maths mind last updated on 14/Jun/20

t h a n k y o u f o r t h i s n i c e Q u a t i o n

Commented by M±th+et+s last updated on 14/Jun/20

i a m g l a d y o u l i k e d i t