A Sequence of Unimodal Polynomials

A finite sequence of real numbers {d0, d1, · · · , dm} is said to be unimodal if there exists an index 0 ≤ j ≤ m such that d0 ≤ d1 ≤ · · · ≤ dj and dj ≥ dj+1 ≥ · · · ≥ dm. A polynomial is said to be unimodal if its sequence of coefficients is unimodal. The sequence {d0, d1, · · · , dm} with dj ≥ 0 is said to be logarithmically concave (or log concave for short) if dj+1dj−1 ≤ dj for 1 ≤ j ≤ m − 1. It is easy to see that if a sequence is log concave then it is unimodal [16]. Unimodal polynomials arise often in combinatorics, geometry, and algebra, and have been the subject of considerable research. The reader is referred to [11, 6] for surveys of the diverse techniques employed to prove that specific families of polynomials are unimodal. In this paper we prove the unimodalityof a specific class of Jacobi polynomials. The general Jacobi polynomials P (α,β) m (z) can be defined by