A proof request

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# A proof request

Submitted by Zhangaini on Sun, 08/07/2011 - 20:36

Forums:

Dear PMs

The link: http://www.proofwiki.org/wiki/Category:Boubaker_Polynomials

contains a sub-link about looking for a proof !!:

http://www.proofwiki.org/wiki/Generating_Function_for_Boubaker_Polynomials

Can you help on this? (as a math challenge or a simple problem)?

Regards

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## Re: A proof request

That is great!

After having a good rest, you can use other results (with proofs) existing in this link:

http://www.proofwiki.org/wiki/Category:Boubaker_Polynomials

For sure you'll find something!

Regards

## Re: A proof request

Yeah, the sign is wrong, the generating function should be

f(x,t) = (1 + 3t^2) / (1 + t(t-x))

Proof:

f = 1 + xt + (x^2+2)t^2 + sum_{n>=3} (xB_{n-1}t^n - B_{n-2}t^n) =

= 1 + xt + (x^2+2)t^2 + xt sum_{n>=3} B_{n-1}t^{n-1} -

t^2 sum_{n>=3} B_{n-2}t^{n-2} =

= 1 + xt + (x^2+2)t^2 + xt (f - 1 - xt) - t^2 (f - 1).

and then solve for f.

Regards,

csguy

## Re: A proof request

This is very nice. I wanted to solve it via Taylor series. Again, I wanted to bring an artillery against fly. :)

Thanks

joking

## Re: A proof request

Great set of questions. I'll think about these issues and get back to you. Anyone that wishes to create a PM encyclopedia entry on these topics are free to do so ... I AM TOLD

From: Zhangaini

To: milogardner

Date: 2011-08-07 15:46:21

Subject: Re: Some Links two Bourbaki links show up

Message:

No problrm dear milogardner!

Thank you indeed for answer!

For being direct and useful, would I explain to you that the link: http://www.proofwiki.org/wiki/Category:Boubaker_Polynomials

contains a sub-link about looking for a proof !!:

http://www.proofwiki.org/wiki/Generating_Function_for_Boubaker_Polynomials

Can you help on this? (as a math challenge or a simple problem)?

## Re: A proof request

Dear colleagues!

I looked at this problem but unfortunetly I do not think that the ordinary generating function for Boubaker polynomials has it's closed form as it is written in the article.

The function is given by f(x,t)=(1+3*t^2)/(1-t*(t-x)). I tried to calculate it's Taylor series (actually Maclaurin series), but the result is different then B_n(x). For n=0 it is correct, but for n=1 I obtained that

df(x,t)/dt (x,0) = -x

and it should be x (for coefficients to be equal to B_1(x)). It is just a matter of sign, but further calculations of derivatives gives even greater differences between Taylor series of f(x,t) and given ordinary generating function. I'd like to recall that a power series always determines its coefficients uniquely.

So my guess is that there is some kind of a mistake in the definition of f(x,t). Perhaps there's some mistake in a sign? I do not know, I didn't check this (I am after 7h of driving, just a bit tired :)).

Best regards

joking