Interesting sum identity

[ognjenmi]

Hi,

I am trying to prove following identity. I obviously can't see something here.

[tex]\sum_{k=1}^{n}(\frac{\prod_{1\leq r\leq n-1}(y_k-z_r)}{\prod_{1\leq r\leq n, r\neq k}(y_k-y_r)})=1[/tex]

Where [tex]y_1,y_2,...,y_n[/tex] are all different.

Any hints?

Regards,
Ognjen

[Guest]

What about [tex]z_r[/tex]

[ognjenmi]

[tex]z_r, 1\le r \le n-1[/tex] is any number.

[HallsofIvy]

Then your "identity" is obviously NOT true since changing [tex]z_r[/tex] will change the left side!

[GeradHum]

This looks like a form of Lagrange interpolation property. Try checking how the sum of Lagrange basis polynomials equals 1, since each term resembles those basis fractions. That’s likely the key to proving the identity.