Generalised Stirling operators, with Applications

In his seminal note "Combinatorial aspects of boson algebra", Katriel proved that Stirling numbers of the second kind appear in the representation of powers of the particle density operator in the boson algebra as a sum of Wick ordered terms. This result led to numerous important generalisations of Stirling numbers, when one replaces the boson algebra with another algebra. An example of such a generalisation is the case of a pair of creation and annihilation operators satisfying the q-commutation relation. However, all the available generalisations of Stirling numbers have only be carried out in the one-mode case, i.e., when the algebra under consideration is generated by a single pair of creation and annihilation operators. In a recent paper by the supervisor of the student with co-authors (to appear in Journal of Functional Analysis) , a spatial counterpart of the combinatorial theory of Stirling numbers was developed. In this theory, the natural numbers (interpreted as the size of a population) are replaced by a population distributed in space. Mathematically, this means that one replaces the set of natural numbers by a configuration space. In this framework, the spatial counterpart of Stirling numbers are Stirling operators acting on spaces of symmetric multivariate functions. It was proved that Stirling operators naturally appear in the representation of powers of the particle density operators in the infinite-dimensional boson algebra as a sum of Wick-ordered terms.

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.