This summation sums to zero. Why?

[EngWiPy]

Hi,

I am reading a paper, and at some point the authors claim that:

$$\sum_{m=1}^{L+1}\frac{\prod_{\substack{l=1\\l\neq m}}^{L+1}\frac{\lambda(m)}{\lambda(m)-\lambda(l)}}{\lambda^r(m)}=0$$

the question is HOW?

Any tiny hint will be highly appreciated.

Thanks

 

[Amir Livne]

In general there is no equality.
It must depend on the definitions of λ, r and L.
Can you provide more details?

 

[EngWiPy]

In general there is no equality.
It must depend on the definitions of λ, r and L.
Can you provide more details?

Lambdas are positive numbers, r is between 1 and L. That is all

 

[Amir Livne]

I don't think that is correct.

Define $\lambda(m)=m$, and pick $L=r=2$. Then
$$\frac{\frac{\lambda(1)}{\left(\lambda(1)-\lambda(2)\right)\left(\lambda(1)-\lambda(3)\right)}}{\lambda(1)^2}+\frac{\frac{\lambda(2)}{\left(\lambda(2)-\lambda(1)\right)\left(\lambda(2)-\lambda(3)\right)}}{\lambda(2)^2}+\frac{\frac{\lambda(3)}{\left(\lambda(3)-\lambda(1)\right)\left(\lambda(3)-\lambda(2)\right)}}{\lambda(3)^2}=\frac{1}{2}-\frac{1}{2}+\frac{1}{6}=\frac{1}{6}$$

 

[CellCoree]

The reason this summation sums to zero is likely due to the fact that it is a telescoping series. This means that many terms in the summation will cancel each other out, leaving only the first and last terms. In this case, the first term is \frac{\prod_{\substack{l=1\\l\neq m}}^{L+1}\frac{\lambda(m)}{\lambda(m)-\lambda(l)}}{\lambda^r(m)} when m=1, and the last term is \frac{\prod_{\substack{l=1\\l\neq m}}^{L+1}\frac{\lambda(m)}{\lambda(m)-\lambda(l)}}{\lambda^r(m)} when m=L+1. These two terms will cancel each other out, resulting in a sum of zero. However, without more context or information about the specific variables and functions involved, it is difficult to provide a more detailed explanation. I recommend consulting with the authors or a mathematics expert for further clarification.