Some Finite Inversion Formulae

Abstract

In a great many problems in probability, one is given one set of quantities Xj expressed in terms of a second set of quantities Yk, and one needs an expression for the Yk in terms of the Xj. Basically, of course, this is just a problem in solving equations, but the results can be expressed in a particularly elegant form when the coefficients involve the binomial quantities . This situation is one that occurs very frequently, and the resulting formulae simplify the derivations of many identities. The present expository account is intended to collect together various results in a convenient form. We shall need to recall briefly a few of the main properties of the binomial coefficients.