New proofs of interlacing of zeros of Eulerian polynomials. III

Abstract

: Many generating functions of combinatorial systems have palindromic coefficients. A notable example is the n th Eulerian polynomial A n ( x ). It is known that a palindromic polynomial f ( x ) of degree 2 n can be expressed as x n Q ( x + 1 x ) for some polynomial Q ( x ) of degree n . By exploring the real-rootedness of Q ( x ), we are able to infer the corresponding property of f ( x ). By representing A n ( x ) in the said form, we give new proof of the real-rootedness and interlacing property of A n ( x ). This same approach applied to the n th alternating Eulerian polynomial (cid:98) A n ( x ) allows us to infer the interlacing/alternating property of the real and imaginary parts of its non-real zeros. The analogous type B results are also presented.