New complex function space related to both entangled state representation and spin coherent state

Abstract

A new complex function space whose basis is the single-variable Hermite polynomial H2jξ*+τξ2τ is constructed, which is related to both entangled state representation and spin coherent state in Schwinger bosonic realization. New binomial theorem involving two-variable Hermite polynomial is derived, which helps to constitute the new complex function space. We also present a new integration transformation of the basis H2jξ*+τξ2τ with its reciprocal transformation which is useful to deriving some operator identities.

Holomorphic Hermite polynomials in two variables

“…5 Whenever a double infinite summation appears we understand it in the sense [35, Integrating (29) with respect to µ α and interchanging summation with integration (Lebesgue's dominated convergence theorem with the domination Φ makes it possible) gives, after applying orthonormality (24), that the right hand side of (29) reduces to The bisequence (h…”

“…the polynomials (6) are pretty often referred to as Ito's polynomials after [27], but are also appearing in the literature under the names of complex or 2D Hermite polynomials [2,3,8,9,10,13,15,14,16,17,20,22,23,24,25,26,29,40,42,43]. The major object of our investigation are the Hermite polynomials H m,n as defined in (4), as well as consequences of the orthogonality relations which the polynomial functions (z 1 , z 2 ) → H m,n (z 1 , z 2 ) satisfy with respect to some measures which are not product ones.…”