From special operators (1-Dx )n+α to Laguerre Polynomials

Laguerre polynomials Ln α (x) are shown to be the transforms of monomials by the special operators (1-Dx )n+α . From this their current properties such as Rodrigues formula, Lucas symbolic formula, orthogonality, generating functions, etc… are systematically obtained. This success opens the way for the study of special functions from special operators by the powerful operator calculus.

Exact Density Matrix Elements for a Driven Dissipative Bosonic Mode in a Kerr-Like Medium

For a prototype Hamiltonian describing a driven, dissipative bosonic mode in a Kerr-like medium, the exact matrix elements of the reduced density matrix are obtained from a generating function in terms of the normal characteristic functions. The approach is based on the Heisenberg equations and operator calculus. The special and limiting cases are discussed.

A Family of Banach Spaces Over ℝ∞

In T. L. Gill and W. W. Zachary, Functional Analysis and the Feynman Operator Calculus (Springer, New York, 2016), the topology of [Formula: see text] was replaced with a new topology and denoted by [Formula: see text]. This space was then used to construct Lebesgue measure on [Formula: see text] in a manner that is no more difficult than the same construction on [Formula: see text]. More important for us, a new class of separable Banach spaces [Formula: see text], [Formula: see text], for the HK-integrable functions, was introduced. These spaces also contain the [Formula: see text] spaces and the Schwartz space as continuous dense embeddings. This paper extends the work in T. L. Gill and W. W. Zachary, Functional Analysis and the Feynman Operator Calculus (Springer, New York, 2016) from [Formula: see text] to [Formula: see text].

Lagrange problem for fractional ordinary elliptic system via Dubovitskii–Milyutin method

In the paper, we investigate a weak maximum principle for Lagrange problem described by a fractional ordinary elliptic system with Dirichlet boundary conditions. The Dubovitskii–Milyutin approach is used to find the necessary conditions. The fractional Laplacian is understood in the sense of Stone–von Neumann operator calculus.

Krawtchouk transforms and convolutions

We present an operator calculus based on Krawtchouk polynomials, including Krawtchouk transforms and corresponding convolution structure which provides an inherently discrete alternative to Fourier analysis. This approach is well suited for applications such as digital image processing. This paper includes the theoretical aspects and some basic examples.