Related papers: Signal processing, orthogonal polynomials, and Heu…
We introduce the algebraic Heun operator associated to any bispectral pair of operators. We show that these operators are natural generalizations of the ordinary Heun operator. This leads to a simple construction of the operators commuting…
The Heun-Hahn operator on the uniform grid is defined. This operator is shown to map polynomials of degree $n$ to polynomials of degree $n+1$, to be tridiagonal in bases made out of either Pochhammer or Hahn polynomials and to be bilinear…
We introduce a nine-parameter Heun-type differential equation and obtain three classes of its solutions as series of square integrable functions written in terms of the Jacobi polynomial. The expansion coefficients of the series satisfy…
Using an algebraic method for solving the wave equation in quantum mechanics, we encountered a new class of orthogonal polynomials on the real line. It consists of a four-parameter polynomial with continuous spectrum on the whole real line…
This paper addresses a general method of polynomial transformation of hypergeometric equations. Examples of some classical special equations of mathematical physics are generated. Heun's equation and exceptional Jacobi polynomials are also…
Orthogonal polynomials with respect to the hypergeometric distribution on lattices in polyhedral domains in ${\mathbb R}^d$, which include hexagons in ${\mathbb R}^2$ and truncated tetrahedrons in ${\mathbb R}^3$, are defined and studied.…
It is shown that the tridiagonalization of the hypergeometric operator $L$ yields the generic Heun operator $M$. The algebra generated by the operators $L,M$ and $Z=[L,M]$ is quadratic and a one-parameter generalization of the Racah…
Quantum signal processing is a powerful framework in quantum algorithms, playing a central role in Hamiltonian simulation and related applications. The sequence of polynomials implemented at each step of this protocol provides a polynomial…
The time and band limiting operator is introduced to optimize the reconstruction of a signal from only a partial part of its spectrum. In the discrete case, this operator commutes with the so-called algebraic Heun operator which appears in…
Intelligent signal processing for wireless communications is a vital task in modern wireless systems, but it faces new challenges because of network heterogeneity, diverse service requirements, a massive number of connections, and various…
Time and band limiting operators are expressed as functions of the confluent Heun operator arising in the spheroidal wave equation. Explicit formulas are obtained when the bandwidth parameter is either small or large and results on the…
This survey contains the introduction to the subject. Many new results are also included.
We introduce Heun algebras of Lie type. They are obtained from bispectral pairs associated to simple or solvable Lie algebras of dimension three or four. For $\mathfrak{su}(2)$, this leads to the Heun-Krawtchouk algebra. The corresponding…
We discuss several open problems on spectrally bounded operators, some new, some old, adding in a few new insights.
The q-Heun operator of the big q-Jacobi type on the exponential grid is defined. This operator is the most general second order q-difference operator that maps polynomials of degree $n$ to polynomials of degree $n+1$. It is tridiagonal in…
The Riemann-Hilbert problems for multiple orthogonal polynomials of types I and II are used to derive string equations associated to pairs of Lax-Orlov operators. A method for determining the quasiclassical limit of string equations in the…
We study some classes of symmetric operators for the discrete series representations of the quantum algebra U_q(su_{1,1}), which may serve as Hamiltonians of various physical systems. The problem of diagonalization of these operators…
In applications such as social, energy, transportation, sensor, and neuronal networks, high-dimensional data naturally reside on the vertices of weighted graphs. The emerging field of signal processing on graphs merges algebraic and…
The Painlev\'{e} equations arise from the study of Hankel determinants generated by moment matrices, whose weights are expressed as the product of ``classical" weights multiplied by suitable ``deformation factors", usually dependent on a…
We describe some examples of classical and explicit h-transforms as particular cases of a general mechanism, which is related to the existence of symmetric diffusion operators having orthogonal polynomials as spectral decomposition.