Onq-extended eigenvectors of the integral and finite Fourier transforms

Abstract

Mehta has shown that eigenvectors of the N × N finite Fourier transform can be written in terms of the standard Hermite eigenfunctions of the quantum harmonic oscillator (1987 J. Math. Phys. 28 781). Here, we construct a oneparameter family of q-extensions of these eigenvectors, based on the continuous q-Hermite polynomials of Rogers. In the limit when q → 1 these q-extensions coincide with Mehta's eigenvectors, and in the continuum limit as N → ∞ they give rise to q-extensions of eigenfunctions of the Fourier integral transform.

q-Extension of Mehta's eigenvectors of the finite Fourier transform forq, a root of unity

“…First, let 0 < q < 1 and take λ = κ as in (3.3). Then (4.6) holds (see [9] and [11] By similar reasoning as in Remark 3.4 we see that (1.4) is a limit of (4.6) in these two cases (as q ↑ 1 in the first case and as M → ∞ in the second case). These limits are formal because we have to take termwise limits for f Let ǫ = ±1.…”

“…Another feature of the present paper, compared with [9], [8] and [11], is that we emphasize a more conceptual approach by using Theorem 4.1 due to Dahlquist [16] and Matveev [21] and the (trivial) Lemma 5.1, rather than repeating a technical argument in each special situation.…”

On Fourier integral transforms forq-Fibonacci andq-Lucas polynomials

“…In particular, we have proved that these families of q-polynomials exhibit a simple transformation behavior (4.7) and (4.13) under the classical Fourier integral transform. Let us emphasize that one actually may use the Mehta-Dahlquist-Matveev techniques (see [20,10,19,3,4]) in order to show that the Fourier integral transformation formulas (4.7) and (4.13) in fact entail a similar behavior of the q-Fibonacci and q-Lucas polynomials under the discrete (finite) Fourier transform as well.…”

q-deformed harmonic and Clifford analysis and theq-Hermite and Laguerre polynomials

“…We also need a good q-deformed Leibniz rule based on (2). We deform commutation relation (13) in stead of (12) because x 2 is scalar. Therefore we can elegantly extend ∂ q t t 2 = q 2 t 2 ∂ q t + (q + 1)t to…”

“…Because all the different types of the q-Hermite polynomials satisfy many properties that are analogues of properties of the Hermite polynomials, see e.g. [11,12,13], they are interesting objects of study themselves. In this study we define a theory of q-deformed derivatives in higher dimensions and a q-deformed Laplace operator acting on functions with commuting variables.…”