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Andreas Dieckmann science forum beginner
Joined: 25 Jul 2005
Posts: 18

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



Quote:  Hi,
In the book by Slater on 'Generalized Hypergeometric
Functions' I found the
solution to the sum_k x^k /(1x^k) on page 91 to be
f(x)=x/(1x) * 2PHI1(x,x;x^2;x,x) ,
where 2PHI1 is a qhypergeometric 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[1a*q^k,{k,0,n1}]*Product[1b*q^k,{k,0,n1}])*x^n/
(Product[1c*q^k,{k,0,n1}]*Product[1q*q^k,{k,0,n1}]),{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

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



Hi,
In the book by Slater on 'Generalized Hypergeometric Functions' I found the
solution to the sum_k x^k /(1x^k) on page 91 to be
f(x)=x/(1x) * 2PHI1(x,x;x^2;x,x) ,
where 2PHI1 is a qhypergeometric 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

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



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

Write x^p/(1x^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

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



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) / (1x^(2k+1))).
Do you happen to know a closed form for the
similar sum Sum (x^k / (1x^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

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



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

Posted: Tue Jul 11, 2006 12:11 pm Post subject:
Re: Need Help with Limit



Andreas Dieckmann wrote:
Quote:  Hi,
the expression is Sum (x^(2k+1) / (1x^(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/(1x)+1/(1x^3)+1/(1x^5)+... (a)
as 0<x<1, 1 divide denominator is larger than 1. as k increases x^k
approaches 0.
So, 1/(1x)+1/(1x^3)+1/(1x^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

Posted: Tue Jul 11, 2006 11:59 am Post subject:
Re: Need Help with Limit



On Tue, 11 Jul 2006 06:24:48 EDT, Andreas Dieckmann
<adieckmann@aol.com> wrote:
Quote:  Hi,
the expression is Sum (x^(2k+1) / (1x^(2k+1))), k=0..inf, x<1
Any ideas ?

That depends. What's the question?
************************
David C. Ullrich 

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Andreas Dieckmann science forum beginner
Joined: 25 Jul 2005
Posts: 18

Posted: Tue Jul 11, 2006 10:24 am Post subject:
Need Help with Limit



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

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