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## Basic q-series

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Moderator: Shobhit

### Basic q-series

Wed Jul 24, 2013 12:51 pm
zaidalyafey
Global Moderator

Posts: 354
Prove the following

$\displaystyle \lim_{q \to 1}\frac{(a)_{\infty}}{(aq^x)_{\infty}}=(1-a)^x$
$\displaystyle \sum_{k\geq 0}\frac{f^{(k)} (s)}{(s+1)_k} (-s)^{k}= \int^{1}_0x^{s} f(s x) \left( \frac{x}{s}\right) \, dx$

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### Re: Basic q-series

Wed Jul 31, 2013 3:41 pm
zaidalyafey
Global Moderator

Posts: 354
It can be solved using the q-binomial theorem . If someone is interested I can show it.
$\displaystyle \sum_{k\geq 0}\frac{f^{(k)} (s)}{(s+1)_k} (-s)^{k}= \int^{1}_0x^{s} f(s x) \left( \frac{x}{s}\right) \, dx$

Wanna learn what we discuss , see tutorials

### Re: Basic q-series

Sun May 18, 2014 6:46 pm

Posts: 3
Location: Bournemouth, United Kingdom
zaidalyafey wrote:
It can be solved using the q-binomial theorem . If someone is interested I can show it.

I am interested.

### Re: Basic q-series

Sun May 18, 2014 7:32 pm
zaidalyafey
Global Moderator

Posts: 354
$\displaystyle \sum_{k\geq 0}\frac{f^{(k)} (s)}{(s+1)_k} (-s)^{k}= \int^{1}_0x^{s} f(s x) \left( \frac{x}{s}\right) \, dx$

Wanna learn what we discuss , see tutorials