# Identity of a sum

• Jun 17th 2014, 09:37 AM
Peppo
Identity of a sum
Let $x_{1},\dots,x_{N}$ real numbers such that $$\underset{i=1}{\overset{N}{\sum}}\frac{x_{i}}{i+ j}=\frac{1}{2j+1}$$ for each natural number $j\in\left[1,N\right]$. Find (in function of $N$) the value of $$\underset{i=1}{\overset{N}{\sum}}\frac{x_{i}}{2i +1}.$$
• Jun 19th 2014, 11:26 AM
Peppo
Re: Identity of a sum
I know that I have to use the inverse of the Hilbert matrix, but I can understand how.
• Jun 20th 2014, 01:11 AM
Idea
Re: Identity of a sum
Well, yes but...
First

$\displaystyle X = H^{-1} B$

where H is (like) the Hilbert matrix and

B is the matrix with entries $\displaystyle \frac{1}{2j+1}$

Then evaluate
$\displaystyle B^t H^{-1} B=\frac{n(n+1)}{(2n+1)^2}$
• Jun 20th 2014, 11:34 PM
Peppo
Re: Identity of a sum
Quote:

Originally Posted by Idea
Well, yes but...
First

$\displaystyle X = H^{-1} B$

where H is (like) the Hilbert matrix and

B is the matrix with entries $\displaystyle \frac{1}{2j+1}$

Then evaluate
$\displaystyle B^t H^{-1} B=\frac{n(n+1)}{(2n+1)^2}$

Yes I see what you say... but how I can prove that $B^{t}H^{-1}B=\frac{N\left(N+1\right)}{\left(2N+1\right)^{2} }?$ Have I to use the explicit expression of $H^{-1}?$
• Jun 21st 2014, 03:02 AM
Idea
Re: Identity of a sum
To start, let me say that I don't know how to solve this problem.
I arrived at the answer by checking the results for some small values of n and then making a wild guess.

If you have an explicit expression for the i-j entry of the inverse of H then that would be one way.
There is such a formula for the inverse of the Hilbert matrix in terms of binomial coefficients and it is quite complicated.

Another way would be to use induction on the size of H.
• Jul 8th 2014, 02:55 AM
Peppo
Re: Identity of a sum
I'm not able to use the explicit inverse of the Cauchy (Hlibert) matrix. This is a new "hint" for this problem: if you consider the Shifted Legendre polynomials $$P_{n}\left(x\right)=\frac{1}{n!}\frac{d^{n}}{dx^ {n}}\left(x-x^{2}\right)^{n}$$ you have $$\int_{0}^{1} P_{n}\left(x\right)P_{m}\left(x\right)dx=\frac{ \delta_{mn} }{2n+1}$$ where $\delta_{mn}$ is the Kronecker delta. How can use it? Can someone help me?