In this paper, we define and study free Jacobi processes of parameters λ > 0 and 0 < θ ≤ 1, as the limit of the complex version of the matrix Jacobi process already defined by Y. Doumerc. In the first part, we focus on the stationary case for which we compute the law (that does not depend on time) and derive, for λ ∈]0, 1] and 1/θ ≥ λ + 1 a free SDE analogous to the classical one. In the second...

From the main equation (ax2 + bx + c)y′′ n (x) + (dx + e)y′ n(x)− n((n− 1)a+ d)yn(x) = 0, n∈ Z+, six finite and infinite classes of orthogonal polynomials can be extracted. In this work, first we have a survey on these classes, particularly on finite classes, and their corresponding rational orthogonal polynomials, which are generated by Mobius transform x = pz−1 + q, p = 0, q ∈ R. Some new int...

Methods of numerical integration of ordinary differential equations exploiting the Cayley transform arise in a variety of contexts, ranging from the classical mid-point rule to symplectic and (almost) Poisson integrators, to numerical methods on Lie Groups. In earlier work, the first author investigated the interplay between the Cayley transform and the Jacobi identity in establishing certain e...

The Askey-Wilson function transform is a q-analogue of the Jacobi function transform with kernel given by an explicit non-polynomial eigenfunction of the Askey-Wilson second order q-difference operator. The kernel is called the Askey-Wilson function. In this paper an explicit expansion formula for the Askey-Wilson function in terms of Askey-Wilson polynomials is proven. With this expansion form...

The Fast Fourier Transform (FFT) is an algorithm of paramount importance in signal processing as it allows to apply the Fourier transform in O(n logn) instead of O(n) arithmetic operations. Graph Signal Processing (GSP) is a recent research domain that generalizes classical signal processing tools, such as the Fourier transform, to situations where the signal domain is given by any arbitrary gr...

Integrability of Hamiltonian systems has been a subject of considerable interest for several decades. One way to understand the dynamics of such systems is to find a family of smooth solutions, called generating functions, to the time-independent Hamilton-Jacobi equation. These generating functions define symplectic transformations which transform the given completely integrable Hamiltonian sys...

Formulae expressing explicitly the coefficients of an expansion of double Jacobi polynomials which has been partially differentiated an arbitrary number of times with respect to its variables in terms of the coefficients of the original expansion are stated and proved. Extension to expansion of triple Jacobi polynomials is given. The results for the special cases of double and triple ultraspher...

The recent numerical implementation by Fornberg and collaborators of the so-called unified method to linear elliptic PDEs in polygonal domains involves the computation of the finite Fourier transform of the Legendre polynomials. A variation of this approach, introduced by two of the authors, also involves the same computation. Here, instead of expressing the finite Fourier transform of the Lege...

Despite the ubiquitous use of distance transforms in the shape analysis literature and the popularity of fast marching and fast sweeping methods—essentially HamiltonJacobi solvers, there is very little recent work leveraging the Hamilton-Jacobi to Schrödinger connection for representational and computational purposes. In this work, we exploit the linearity of the Schrödinger equation to (i) des...

We show that requiring diffeomorphic equivalence for one-dimensional stationary states implies that the reduced action S0 satisfies the quantum Hamilton-Jacobi equation with the Planck constant playing the role of a covariantizing parameter. The construction shows the existence of a fundamental initial condition which is strictly related to the Möbius symmetry of the Legendre transform and to i...