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q-series

ZaidAlyafey

Well-known member
MHB Math Helper
Jan 17, 2013
1,667

Opalg

MHB Oldtimer
Staff member
Feb 7, 2012
2,725

ZaidAlyafey

Well-known member
MHB Math Helper
Jan 17, 2013
1,667
Great !

I am looking for a proof for the following

\(\displaystyle \frac{(-b)_{\infty}}{(a)_{\infty}} = \sum_{k \geq 0} \frac{\left(-b/a\right)_k} {(q)_k} a^k \)
 

Opalg

MHB Oldtimer
Staff member
Feb 7, 2012
2,725

ZaidAlyafey

Well-known member
MHB Math Helper
Jan 17, 2013
1,667
Thanks for the link , I really really appreciate it. I thought that q-series are interesting it reminds me of the famous relation between prime numbers and zeta function due to Euler . I was reading the NoteBook by Ramnujan and I was amazed by the vast results related to Hypergeometric functions which are a special case of q-series . I was hoping to read about the Jacobi theta function but I thought it would be better to start by q-series.
 

ZaidAlyafey

Well-known member
MHB Math Helper
Jan 17, 2013
1,667
By the way Opalg , do you think it is a hard thing to deal with because I was depressed to see how Ramanujan worked with these stuff and I was like what is that !
 

Opalg

MHB Oldtimer
Staff member
Feb 7, 2012
2,725
By the way Opalg , do you think it is a hard thing to deal with because I was depressed to see how Ramanujan worked with these stuff and I was like what is that !
Please don't expect to understand everything that Ramanujan could do! That way madness lies. (Tmi)

There are results that Ramanujan somehow apprehended by intuition, that even today cannot be proved although they appear to be true, and nobody knows how he arrived at them. He must have had some quite unique insight that probably even he could not have explained.
 

ZaidAlyafey

Well-known member
MHB Math Helper
Jan 17, 2013
1,667
Hey , I am confused about the notations !

\(\displaystyle (a)_k = a(a+1)(a+2) \cdots (a+k-1)\)

\(\displaystyle (a)_k =(a;q)_k = \prod_{n=0}^{k-1} (1-aq^n)\)

The latter defines a base $q$ .
 

Opalg

MHB Oldtimer
Staff member
Feb 7, 2012
2,725
Hey , I am confused about the notations !

\(\displaystyle (a)_k = a(a+1)(a+2) \cdots (a+k-1)\)

\(\displaystyle (a)_k =(a;q)_k = \prod_{n=0}^{k-1} (1-aq^n)\)

The latter defines a base $q$ .
I was assuming that $(a)_k$ was an abbreviation for $(a;q)_k$ (as here, for example), with the $q$ not explicitly mentioned. But I could well be wrong.
 

ZaidAlyafey

Well-known member
MHB Math Helper
Jan 17, 2013
1,667

ZaidAlyafey

Well-known member
MHB Math Helper
Jan 17, 2013
1,667
How to prove the following

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

It is kind of easy to prove it for $x\in \mathbb{Z}^+$
 

ZaidAlyafey

Well-known member
MHB Math Helper
Jan 17, 2013
1,667
To simplify for those how don't understand the notations

\(\displaystyle (a;q)_{\infty}= \prod_{k=0}(1-aq^k)\)

Similarilry we have

\(\displaystyle (aq^x;q)_{\infty}= \prod_{k=0}(1-aq^{k+x})\)

So we have to prove that

\(\displaystyle \lim_{q \to 1}\prod_{k=0} \frac{1-aq^k}{1-aq^{k+x}} = (1-a)^ x\)

Any clue ?
 

ZaidAlyafey

Well-known member
MHB Math Helper
Jan 17, 2013
1,667
How to prove the following

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

It is kind of easy to prove it for $x\in \mathbb{Z}^+$
Ok , I think I got it , this is a simple consequence of the q-binomial theorem

Consider the following

\(\displaystyle {}_1\phi_0 (a;- ;q,z) = \sum_{k\geq 0}\frac{(a;q)_k}{(q;q)_k}z^k=\frac{(az;q)_{\infty}}{(z;q)_{\infty}}\) (1)

In (1) let $a = q^{x}$ and $z = a$

\(\displaystyle {}_1\phi_0 (q^x;- ;q,a) = \sum_{k\geq 0}\frac{(q^x;q)_k}{(q;q)_k}a^k=\frac{(aq^{x};q)_ {\infty} }{(a;q)_{\infty}}\)

Hence we have

\(\displaystyle \frac{(aq^{x};q)_{\infty}}{(a;q)_{\infty}}= \sum_{k\geq 0}\frac{(q^x;q)_k}{(q;q)_k}a^k\)

Now consider the limit

\(\displaystyle \lim_{q \to 1}\frac{(aq^{x};q)_{\infty}}{(a;q)_{\infty}}= \lim_{q \to 1} \sum_{k\geq 0}\frac{(q^x;q)_k}{(q;q)_k}a^k\)(2)

Suppose that \(\displaystyle |a|<1\) and \(\displaystyle |q|<1\) so the sum is uniformly convergent on any sub-disk . So we have to approach $1$ from the left to stay in the disk !

The idea is use the L'Hospitale rule

\(\displaystyle \lim_{q \to 1^-}\frac{(q^x;q)_k}{(q;q)_k} = \lim_{q \to 1^-} \frac{(1-q^x)\cdot (1-q^{x+1}) \cdot(1-q^{x+2}) \cdots (1-q^{x+k-1}) }{(1-q)\cdot(1-q^2)\cdot(1-q^3) \cdots(1-q^k)}\)

which can be written as

\(\displaystyle \lim_{q \to 1^-}\frac{(q^x;q)_k}{(q;q)_k} = \lim_{q \to 1^-} \frac{(1-q^x)}{1-q}\cdot \lim_{q \to 1^-}\frac{(1-q^{x+1})}{1-q^2} \cdot \lim_{q \to 1} \frac{(1-q^{x+2})}{1-q^3} \cdots \lim_{q \to 1^-} \frac{(1-q^{x+k-1}) }{(1-q^k)}\)

\(\displaystyle \lim_{q \to 1^-}\frac{(q^x;q)_k}{(q;q)_k} = \frac{x (x+1)(x+2)\cdots (x+k-1)}{1\cdot 2 \cdot 3 \cdots k} = \frac{(x)_k}{k!} \)

Substitute in (2)

\(\displaystyle \lim_{q \to 1^-}\frac{(aq^{x};q)_{\infty}}{(a;q)_{\infty}}= \sum_{k\geq 0}\frac{(x)_k}{k!}a^k\)

The sum on the right is well-know $(1-x)^{-a}$

\(\displaystyle \lim_{q \to 1^-}\frac{(aq^{x};q)_{\infty}}{(a;q)_{\infty}}= (1-x)^{-a} \) (3)

From (3) we conclude that

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