Double dipping: The case of the missing binomial coefficient identities
The author considers sequences \(\{x_ n\}\) and \(\{y_ n\}\) of real numbers with the remarkable property that the products of their differences are majorized by the difference of their coordinatewise product \(\{x_ n y_ n\}\). Such sequences are said to represent a double-dipping pair. The simplest example, \(x_ n=(1-p)^ n\), \(y_ n=(1-q)^ n\) with \(0\leq p\), \(q\leq 1\) \((n=0,1,\dots)\) arises when playing a coin-tossing game [see the author, Discrete Math. 84, No. 2, 111-118 (1990; Zbl 0707.90109)]. In this case the double-dipping property leads to the theorem that the sum of any \(N\) terms from the set \(\{p^ m q^ n;\;m,n=0,1,\dots\}\) does not exceed \(1+(p+q-pq)+\cdots+(p+q- pq)^{N- 1}\) \((N\geq 1)\). The main purpose of this paper is to prove the theorems for the games based on sampling balls from urns, with various replacement schemes. It is found that real sequences \(\{x_ n\}\) and \(\{y_ n\}\) form a double- dipping pair whenever \(x_ n={B-n\choose b}\) and \(y_ n={C-n\choose c}\), where \(B\), \(b\), \(C\), \(c\) are non-negative integers with \(B\geq b\) and \(C\geq c\). The results are rephrased in the language of the theory of majorization, which enables to increase the possible scope of the theorems by suggesting additional problems.
scientific article; zbMATH DE number 501197
scientific article