A new mathematical identity involving binomial coefficients

Abstract:
Studying the dynamics of a two-level atom quadratically coupled to two bosonic modes we have been faced by very involved expressions containing coupled sums of complicated terms consisting in products of several binomial coefficients. The need of simplifying such an expression has led us to derive a new formula concerning binomial coefficient. This formula allows to drastically reduce the number of sums and the number of binomial factors for each term in our original expression. From a mathematical (and more general) point of view, our result expresses a novel decomposition of a generic binomial coefficient and leads to the definition of a new kind of polynomial functions. It is of relevance that we succeed in establishing relations between such novel functions and well-known polynomials, in particular Jacobi's, Krawtchouk's and Hermite's polynomials. We believe that this formula might result in a powerful tool for simplifying several mathematical expressions arising in the study of several kinds of matter-field multiple interactions.