Summation with polynomial roots
by fungarwai, May 26, 2020, 12:52 PM
Reciprocal type
Credit to the idea posted by ccmmjj in mathchina forum
Proof
Identity of a conditional symmetric polynomial
Let
Proof
Example
Proof
Suppose is true for
Example
Lagrange polynomial type
Proof
Best Proof Credit to the idea with four variables posted by natmath at #2
Proof Credit to the idea with three variables posted by gasgous at #10
Other Proof
where
Proof
Example
Credit to the idea posted by ccmmjj in mathchina forum
Proof
Identity of a conditional symmetric polynomial
Let
Proof
Distribute balls into boxes
Each box has balls
Let be the event of box contains at least one ball
By Inclusion–exclusion principle
satisfies for the number of any combinations of , therefore
Each box has balls
Let be the event of box contains at least one ball
By Inclusion–exclusion principle
satisfies for the number of any combinations of , therefore
Example
For the combination as example
which is the same as the situation of the coefficient of
Other examples
which is the same as the situation of the coefficient of
Other examples
Proof
Suppose is true for
Example
Lagrange polynomial type
Proof
Best Proof Credit to the idea with four variables posted by natmath at #2
Proof Credit to the idea with three variables posted by gasgous at #10
Other Proof
Let
such that
where
The coefficient of should be 0
i.e.
And then,
Let
As
Credit to the proof posted by natmath at #4
such that
where
The coefficient of should be 0
i.e.
And then,
Let
As
Credit to the proof posted by natmath at #4
https://artofproblemsolving.com/community/c4h2901132
Consider the polynomial
The desired expression is the coefficient of of . We know that (and cyclically) so
But is a quadratic so the coefficient of and should be . This means that So
The coefficient of is .
Consider the polynomial
The desired expression is the coefficient of of . We know that (and cyclically) so
But is a quadratic so the coefficient of and should be . This means that So
The coefficient of is .
where
Proof
When ,
When ,
Suppose
When ,
Suppose
Example
This post has been edited 23 times. Last edited by fungarwai, May 19, 2023, 12:07 PM
by MelonGirl, Sep 11, 2021, 2:51 AM