$(q;l,λ)-$deformed Heisenberg algebra: representations, special functions and quantization

This paper addresses a new characterization of Sudarshan's diagonal representation of the density matrix elements $ρ(z',z)$, derivedfrom $(q;l,λ)-$deformed boson coherent states.The induced $ρ(z',z)$ self-reproducing property with the associated self-reproducing kernel $K(z',z)$ is computed and analyzed. An explicit construction of novel classes of generalized continuous $(q;l,λ)-$Hermite polynomials is provided with the corresponding recursion relations and exact resolution of the moment problems giving their orthogonality weight functions. Besides, the Berezin-Klauder-Toeplitz quantization of classical phase space observables and relevant normal and anti-normal forms are investigated and discussed.

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