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Soarer
Frequent VisitorJoined: 18 Jan 2004Posts: 181
Location: Hong Kong

Posted: Mon Oct 25, 2004 7:27 pm Post subject: Recurrence relation



Consider a recurrence satisfying T1=1, and Tn=T1T_(n1)+T2T_(n2)+…+T_(n1)T1 I consider f(x)=T1x+T2x^2+…. (suppose f(x) is convergent) then i get f(x)^2 = f(x)x f(x) = [1sqrt(14x)]/2.
But then how to find out the general term of T_n?





Kenny TM~
Frequent VisitorJoined: 20 Jan 2004Posts: 127

Posted: Tue Oct 26, 2004 7:14 pm Post subject: Re: Recurrence relation



Soarer wrote:

Consider a recurrence satisfying T1=1, and Tn=T1T_(n1)+T2T_(n2)+…+T_(n1)T1 I consider f(x)=T1x+T2x^2+…. (suppose f(x) is convergent) then i get f(x)^2 = f(x)x f(x) = [1sqrt(14x)]/2.
But then how to find out the general term of T_n?

This would be useful: http://www.research.att.com/cgibin/access.cgi/as/njas/sequences/eisA.cgi?Anum=A000108
Edit:
Since f is a g.f. of , we could use the binomial expansion of to get the coefficients of each term.





Soarer
Frequent VisitorJoined: 18 Jan 2004Posts: 181
Location: Hong Kong

Posted: Tue Oct 26, 2004 7:49 pm Post subject:



Thanks





Soarer
Frequent VisitorJoined: 18 Jan 2004Posts: 181
Location: Hong Kong

Posted: Tue Oct 26, 2004 9:17 pm Post subject:



How about a variation?
Tn=T0T_(n1)+T1T_(n2)+…+T_(n1)T0





Soarer
Frequent VisitorJoined: 18 Jan 2004Posts: 181
Location: Hong Kong

Posted: Tue Oct 26, 2004 9:19 pm Post subject: Re: Recurrence relation



Kenny TM~ wrote:

Soarer wrote:

Consider a recurrence satisfying T1=1, and Tn=T1T_(n1)+T2T_(n2)+…+T_(n1)T1 I consider f(x)=T1x+T2x^2+…. (suppose f(x) is convergent) then i get f(x)^2 = f(x)x f(x) = [1sqrt(14x)]/2.
But then how to find out the general term of T_n?

This would be useful: http://www.research.att.com/cgibin/access.cgi/as/njas/sequences/eisA.cgi?Anum=A000108
Edit:
Since f is a g.f. of , we could use the binomial expansion of to get the coefficients of each term.

may i know what does the binonmial theorem with noninteger power look like?





Kenny TM~
Frequent VisitorJoined: 20 Jan 2004Posts: 127





Soarer
Frequent VisitorJoined: 18 Jan 2004Posts: 181
Location: Hong Kong

Posted: Wed Oct 27, 2004 7:08 pm Post subject:



Let’s say T0=1. For taylor’s series, do you mean using this formula,
f(x)=f(0)+f'(0)x+f”(0)x^2+… and so on?





Kenny TM~
Frequent VisitorJoined: 20 Jan 2004Posts: 127

Posted: Thu Oct 28, 2004 7:30 pm Post subject:



Soarer wrote:

Let’s say T0=1. For taylor’s series, do you mean using this formula,
f(x)=f(0)+f'(0)x+f”(0)x^2+… and so on?

It is
—
If we define , then T’ would be same as the original T.
BTW, if (in the original def.) , then . (C(n) is the function described in here)
—
Also, we can define the binomial coefficent for natural k as (not using Gamma function):





Soarer
Frequent VisitorJoined: 18 Jan 2004Posts: 181
Location: Hong Kong

Posted: Thu Oct 28, 2004 9:04 pm Post subject:



ah yes…. i typed taylor’s formula wrongly…. missing out all those factorials in the denominator.. sorry.
Thanks anyway.





Polam
Frequent VisitorJoined: 03 Nov 2003Posts: 333

Posted: Thu Oct 28, 2004 9:50 pm Post subject:



This is very interesting. I was working on the following:
with , , and
with . Is there anything that you can say about them?





