Universita di Modena e Reggio Emilia,

Via Campi 213/A,

41100 Modena,

ITALY

and

Universita di Palermo,

Via Archirafi 36,

I-90123 Palermo,

ITALY

and

3 Magnitohorsky Prov.,

02094 Kyiv,

UKRAINE

E-mail: migliore.agostino@unimore.it, agostino_migliore2002@yahoo.it, vtretynyk@yahoo.com, messina@fisica.unipa.it

**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.