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Roots of a polynomial defined by a sequence are real

Source: Iran TST 2013: TST 1, Day 1, Problem 3

April 17, 2013

algebrapolynomialinductionalgebra proposed

Problem Statement

For nonnegative integers

m

and

n

, define the sequence

a(m,n)

of real numbers as follows. Set

a(0,0)=2

and for every natural number

n

, set

a(0,n)=1

and

a(n,0)=2

. Then for

m,n\geq1

, define

a(m,n)=a(m-1,n)+a(m,n-1).

Prove that for every natural number

k

, all the roots of the polynomial

P_{k}(x)=\sum_{i=0}^{k}a(i,2k+1-2i)x^{i}

are real.