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This paper addresses a new characterization of Sudarshan's diagonalrepresentation of the density matrix elements $\rho(z',z)$, derivedfrom$(q;l,\lambda)-$deformed boson coherent states.The induced $\rho(z',z)$self-reproducing property with the associated self-reproducing kernel $K(z',z)$is computed and analyzed. An explicit construction of novel classes ofgeneralized continuous $(q;l,\lambda)-$Hermite polynomials is provided with thecorresponding recursion relations and exact resolution of the moment problemsgiving their orthogonality weight functions. Besides, theBerezin-Klauder-Toeplitz quantization of classical phase space observables andrelevant normal and anti-normal forms are investigated and discussed.

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