Papers by José Luis dos Santos Cardoso
Journal of Mathematical Analysis and Applications, 2006
We present a general theory for studying the difference analogues of special functions of hyperge... more We present a general theory for studying the difference analogues of special functions of hypergeometric type on the linear-type lattices, i.e., the solutions of the second order linear difference equation of hypergeometric type on a special kind of lattices: the linear type lattices. In particular, using the integral representation of the solutions we obtain several difference-recurrence relations for such functions. Finally, applications to q-classical polynomials are given.
Journal of Difference Equations and Applications, 2021
Let I ⊆ R be an interval and β : I → I a strictly increasing and continuous function with a uniqu... more Let I ⊆ R be an interval and β : I → I a strictly increasing and continuous function with a unique fixed point s 0 ∈ I that satisfies (s 0 − t)(β(t) − t) ≥ 0 for all t ∈ I, where the equality holds only when t = s 0. The general quantum operator defined by Hamza et al., D β [f ](t) := f β(t) − f (t) β(t) − t if t = s 0 and D β [f ](s 0) := f ′ (s 0) if t = s 0 , generalizes the Jackson q-operator Dq and also the Hahn (quantum derivative) operator, Dq,ω. Regarding a β−Sturm Liouville eigenvalue problem associated with the above operator D β , we construct the β−Lagrange's identity, show that it is self-adjoint in L 2 β ([a, b]), and exhibit some properties for the corresponding eigenvalues and eigenfunctions.
Methods and Applications of Analysis, 2015
Let 0 < q < 1, ω ≥ 0, ω 0 := ω/(1 − q), and I a set of real numbers. Consider the so-called quant... more Let 0 < q < 1, ω ≥ 0, ω 0 := ω/(1 − q), and I a set of real numbers. Consider the so-called quantum derivative operator, Dq,ω, acting on functions f : I → K (where K = R or C) as Dq,ω[f ](x) := f qx + ω − f (x) (q − 1)x + ω , x ∈ I \ {ω 0 } , and Dq,ω[f ](ω 0) := f ′ (ω 0) whenever ω 0 ∈ I and this derivative exists. This operator was introduced by W. Hahn in 1949. Its inverse operator is given in terms of the so-called Jackson-Thomae (q, ω)−integral, also called Jackson-Nörlund (q, ω)−integral. For ω = 0 one obtains the Jackson's q−operator, Dq, whose inverse operator is given in terms of the so-called Jackson q−integral. In this paper we survey in an unified way most of the useful properties of the Jackson's q−integral and then, by establishing links between Dq,ω and Dq, as well as between the q and the (q, ω) integrals, we show how to obtain the properties of Dq,ω and the (q, ω)−integral from the corresponding ones fulfilled by Dq and the q−integral. We also consider (q, ω)−analogues of the Lebesgue spaces, denoted by L p q,ω [a, b] and L p q,ω [a, b], being a, b ∈ R. It is shown that the condition a ≤ ω 0 ≤ b ensures that these are indeed linear spaces over K. Moreover, endowed with an appropriate norm, L p q,ω [a, b] satisfies some expected properties: it is a Banach space if 1 ≤ p ≤ ∞, separable if 1 ≤ p < ∞, and reflexive if 1 < p < ∞.
arXiv (Cornell University), Oct 19, 2017
We study Fourier-Bessel series on a q-linear grid, defined as expansions in complete q-orthogonal... more We study Fourier-Bessel series on a q-linear grid, defined as expansions in complete q-orthogonal systems constructed with the third Jackson q-Bessel function, and obtain sufficient conditions for uniform convergence. The convergence results are illustrated with specific examples of expansions in q-Fourier-Bessel series.
We derive analytic bounds for the zeros of the third Jackson q-Bessel function
The Ramanujan Journal, 2018
We study Fourier-Bessel series on a q-linear grid, defined as expansions in complete q-orthogonal... more We study Fourier-Bessel series on a q-linear grid, defined as expansions in complete q-orthogonal systems constructed with the third Jackson q-Bessel function, and obtain sufficient conditions for uniform convergence. The convergence results are illustrated with specific examples of expansions in q-Fourier-Bessel series.
ALVAREZ-NODARSE Abstract. The functions of hypergeometric type are the solutions y = y (z) of the... more ALVAREZ-NODARSE Abstract. The functions of hypergeometric type are the solutions y = y (z) of the dierential equation (z)y00 + (z)y0 + y = 0, where and are poly- nomials of degrees not higher than 2 and 1, respectively, and is a constant. Here we consider a class of functions of hypergeometric type: those that sat- isfy the condition +
Journal of Physics A: Mathematical and General, 2003
We present a general procedure for finding recurrence relations of the radial wavefunctions for N... more We present a general procedure for finding recurrence relations of the radial wavefunctions for Nth-dimensional isotropic harmonic oscillators and hydrogenlike atoms.
Journal of Fourier Analysis and Applications, 2010
A q-type Hölder condition on a function f is given in order to establish (uniform) convergence of... more A q-type Hölder condition on a function f is given in order to establish (uniform) convergence of the corresponding basic Fourier series S q [f ] to the function itself, on the set of points of the q-linear grid. Furthermore, by adding other conditions, one guarantees the (uniform) convergence of S q [f ] to f on and "outside" the set points of the q-linear grid.
Journal of Difference Equations and Applications, 2005
We present a general procedure for finding linear recurrence relations for the solutions of the s... more We present a general procedure for finding linear recurrence relations for the solutions of the second order difference equation of hypergeometric type. Applications to wave functions of certain discrete system are also given.
International Journal of Mathematics and Mathematical Sciences, 2003
We derive analytic bounds for the zeros of the third Jacksonq-Bessel functionJv(3)(z;q).
The functions of hypergeometric-type are the solutions y = yν (z) of the differential equation σ(... more The functions of hypergeometric-type are the solutions y = yν (z) of the differential equation σ(z)y ′′ + τ (z)y ′ + λy = 0, where σ and τ are polynomials of degrees not higher than 2 and 1, respectively, and λ is a constant. Here we consider a class of functions of hypergeometric type: those that satisfy the condition λ + ντ ′ + 1 2 ν(ν − 1)σ ′′ = 0, where ν is an arbitrary complex (fixed) number. We also assume that the coefficients of the polynomials σ and τ do not depend on ν. To this class of functions belong Gauss, Kummer, and Hermite functions, and also the classical orthogonal polynomials. In this work, using the constructive approach introduced by Nikiforov and Uvarov, several structural properties of the hypergeometric-type functions y = yν (z) are obtained. Applications to hypergeometric functions and classical orthogonal polynomials are also given.
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Papers by José Luis dos Santos Cardoso