Generalised Stirling operators, with Applications

The main objective of the present project are as follows:
(i) for several important classes of infinite-dimensional algebras of creation and annihilation operators, derive the corresponding generalised Stirling operators;
(ii) study combinatorial properties of the generalised Stirling operators;
(iii) consider applications of the generalised Stirling numbers in the theory of random measures, and beyond.

The project will start with a study of the algebra generated by the following operators acting on the space of infinite-dimensional polynomials: the operators of multiplication by a variable (treated as creation operators); and the difference derivatives (treated as annihilation operators). It is expected that the corresponding particle density, i.e., the smeared product of the creation and annihilation operators at a point, will lead us to the gamma random measure and the negative binomial point processes. Another class of the algebras to be considered are those where the difference operators are replaced by the operators of q-differentiation. In that case, the creation and annihilation operators satisfy the infinite-dimensional q-commutation relations. While the corresponding generalised Stirling numbers are well understood in the one-mode case, the q-deformed Stirling operators will present an important and highly non-trivial object of studies.