From coin tossing to the jacobi polynomials

A simple coin-tossing game leads to the study of real sequences, x and y, with the remarkable property that the products of their differences are majorized by the differences of their product. Such sequences are said to form a double-dipping pair. The following conjecture arises when the game is governed by sampling balls from urns: if A, a, B, b are non-negative integers with A ≥ a and B ≥ b, then x and y form a double-dipping pair, where x k = ( A − K a ) , y k = ( B − K b ) , k = 0 , 1 , 2 , … . The conjecture is proved here under the additional restriction -b ≤ A − B ≤ a. The proof is based, in part, upon the observation that the polynomials, x → ∑ k ( a k ) ( b n − k ) ( 1 − x ) k , n ≤ a + b, have reciprocals, all of whose Taylor coefficients (about x = 0) are non-negative.