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Proving identities of special functions

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Moderator: Shobhit

Posts: 138
Location: North Londinium, UK
Mighty efficient there, RV (as always)! :mrgreen:

Not my turn, mind, but just out of interest... Have you tried proving the equivalent canonical product for the Barnes' G-Function (Double Gamma Function)? I'm sure you'd find it a doddle...

\(\displaystyle G(z+1)=G(z)\,\Gamma(z)\)

\(\displaystyle G(z+1)=(2\pi)^{z/2}\text{exp}\left(-\frac{z+z^2(1+\gamma)}{2}\right)\, \prod_{k=1}^{\infty}\left(1+\frac{z}{k}\right)\text{exp}\left(\frac{z^2}{2k}-z\right)\)

More on the Barnes' G-Function here...

Shobhit Site Admin
Site Admin

Posts: 850
Location: Jaipur, India

Problem 11

Show that

\(\displaystyle _2\phi_1 \left[\begin{matrix}a,b \\ \frac{aq}{b}\end{matrix}\; ; q;-\frac{q}{b} \right] = \frac{(-q;q)_{\infty}(aq;q^2)_\infty(aq^2/b^2;q^2)_\infty }{(aq/b;q)_\infty (-q/b;q)_\infty}\)

where \(\displaystyle _2\phi_1 \left[\begin{matrix}a,b \\ c \end{matrix}\; ; q;-\frac{q}{b} \right]\) is the Basic Hypergeometric Series.


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