Prove the following

\(\displaystyle \lim_{q \to 1}\frac{(a)_{\infty}}{(aq^x)_{\infty}}=(1-a)^x\)

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## Basic q-series

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I am interested.

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**Moderator:** Shobhit

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Prove the following

\(\displaystyle \lim_{q \to 1}\frac{(a)_{\infty}}{(aq^x)_{\infty}}=(1-a)^x\)

\(\displaystyle \lim_{q \to 1}\frac{(a)_{\infty}}{(aq^x)_{\infty}}=(1-a)^x\)

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

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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

zaidalyafey wrote:

It can be solved using the q-binomial theorem . If someone is interested I can show it.

I am interested.

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

Wanna learn what we discuss , see tutorials

4 posts
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