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Need Help with Limit
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Andreas Dieckmann
science forum beginner


Joined: 25 Jul 2005
Posts: 18

PostPosted: Wed Jul 12, 2006 7:24 pm    Post subject: Re: Need Help with Limit Reply with quote

Quote:
Hi,
In the book by Slater on 'Generalized Hypergeometric
Functions' I found the
solution to the sum_k x^k /(1-x^k) on page 91 to be
f(x)=x/(1-x) * 2PHI1(x,x;x^2;x,x) ,
where 2PHI1 is a q-hypergeometric function,
which doesn't seem to be implemented for instance in
Mathematica.
Thanks for all answers.
Andreas

the function 2PHI1(a,b;c;q,x) can be defined in Mathematica as
qHyp2F1[a_,b_,c_,q_,x_]:=
Sum[(Product[1-a*q^k,{k,0,n-1}]*Product[1-b*q^k,{k,0,n-1}])*x^n/
(Product[1-c*q^k,{k,0,n-1}]*Product[1-q*q^k,{k,0,n-1}]),{n,0,hiNumber}]
where hiNumber is an integer as 100 or 1000, chosen so large
(-> infinity), that the result no longer changes.

Andreas
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Andreas Dieckmann
science forum beginner


Joined: 25 Jul 2005
Posts: 18

PostPosted: Wed Jul 12, 2006 10:08 am    Post subject: Re: Need Help with Limit Reply with quote

Hi,
In the book by Slater on 'Generalized Hypergeometric Functions' I found the
solution to the sum_k x^k /(1-x^k) on page 91 to be
f(x)=x/(1-x) * 2PHI1(x,x;x^2;x,x) ,
where 2PHI1 is a q-hypergeometric function,
which doesn't seem to be implemented for instance in Mathematica.
Thanks for all answers.
Andreas
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Robert B. Israel
science forum Guru


Joined: 24 Mar 2005
Posts: 2151

PostPosted: Tue Jul 11, 2006 5:54 pm    Post subject: Re: Need Help with Limit Reply with quote

In article <19525980.1152613518622.JavaMail.jakarta@nitrogen.mathforum.org>,
Andreas Dieckmann <adieckmann@aol.com> wrote:
Quote:
Hi,
the expression is Sum (x^(2k+1) / (1-x^(2k+1))), k=0..inf, x<1
Any ideas ?
Andreas

Write x^p/(1-x^p) = sum_{j=1}^infty x^{pj}
so your sum is
sum_{n=1}^infty g(n) x^n
where g(n) is the number of odd divisors of n.
I don't think there's a closed form. See A001227 in the
Encyclopedia of Integer Sequences
<http://www.research.att.com/~njas/sequences/A001227>

Robert Israel israel@math.ubc.ca
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia Vancouver, BC, Canada
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David C. Ullrich
science forum Guru


Joined: 28 Apr 2005
Posts: 2250

PostPosted: Tue Jul 11, 2006 2:07 pm    Post subject: Re: Need Help with Limit Reply with quote

On Tue, 11 Jul 2006 08:53:30 EDT, Andreas Dieckmann
<adieckmann@aol.com> wrote:

Quote:
David:
I wanted to know whether there is a 'closed expression' that gives the same
result as function of x as the quoted sum...

What sum? You shouldn't make people chase around several posts
to try to answer a question, you should _quote_ the relevant
parts.

Ok, the sum is Sum (x^(2k+1) / (1-x^(2k+1))).

Do you happen to know a closed form for the
similar sum Sum (x^k / (1-x^k)) ? If that sum
is f(x) then your sum is just f(x) - f(x^2).

Quote:
cyclomethane:
the sum converges for Abs(x)<1

Andreas


************************

David C. Ullrich
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Andreas Dieckmann
science forum beginner


Joined: 25 Jul 2005
Posts: 18

PostPosted: Tue Jul 11, 2006 12:53 pm    Post subject: Re: Need Help with Limit Reply with quote

David:
I wanted to know whether there is a 'closed expression' that gives the same
result as function of x as the quoted sum...

cyclomethane:
the sum converges for Abs(x)<1

Andreas
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cyclomethane@gmail.com
science forum beginner


Joined: 02 Jul 2006
Posts: 5

PostPosted: Tue Jul 11, 2006 12:11 pm    Post subject: Re: Need Help with Limit Reply with quote

Andreas Dieckmann wrote:
Quote:
Hi,
the expression is Sum (x^(2k+1) / (1-x^(2k+1))), k=0..inf, x<1
Any ideas ?
Andreas

If you expand the 'Sum Blah',
Sum_k (Sum_p x^( (2K+1)*p ) ), k=0..inf, * 0<x<1, p=1..inf

and expand more like this.
1/(1-x)+1/(1-x^3)+1/(1-x^5)+... -(a)
as 0<x<1, 1 divide denominator is larger than 1. as k increases x^k
approaches 0.

So, 1/(1-x)+1/(1-x^3)+1/(1-x^5)+... >1+1+1+...

This is not that formal but I hope this help.

For x<=0, I didnt solved.
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David C. Ullrich
science forum Guru


Joined: 28 Apr 2005
Posts: 2250

PostPosted: Tue Jul 11, 2006 11:59 am    Post subject: Re: Need Help with Limit Reply with quote

On Tue, 11 Jul 2006 06:24:48 EDT, Andreas Dieckmann
<adieckmann@aol.com> wrote:

Quote:
Hi,
the expression is Sum (x^(2k+1) / (1-x^(2k+1))), k=0..inf, x<1
Any ideas ?

That depends. What's the question?

Quote:
Andreas


************************

David C. Ullrich
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Andreas Dieckmann
science forum beginner


Joined: 25 Jul 2005
Posts: 18

PostPosted: Tue Jul 11, 2006 10:24 am    Post subject: Need Help with Limit Reply with quote

Hi,
the expression is Sum (x^(2k+1) / (1-x^(2k+1))), k=0..inf, x<1
Any ideas ?
Andreas
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