Discrete Math And Its Applications Ucf Pdf
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Dr. Hongwei Long
We give complex weight functions with respect to which the Jacobi, Laguerre, little q-Jacobi and Askey-Wilson polynomials are orthogonal. The complex functions obtained are weight functions in a wider range of parameters than the real The complex functions obtained are weight functions in a wider range of parameters than the real weight functions.
They also provide an alternative to the recent distributional weight functions of Morton and Krall, and the more recent hyperfunction weight functions of Kim. Publication Date: Pure Mathematics. Diagonalization of certain integral operators II more. We establish an integral representation of a right inverse of the Askey-Wilson finite difference operator in an L2 space weighted by the weight function of the continuous q-Jacobi polynomials. We characterize the eigenvalues of this We characterize the eigenvalues of this integral operator and prove a q-analog of the expansion of eixy in Jacobi polynomials of argument x.
We also outline a general procedure of finding. The associated Askey - Wilson polynomials more. Pure Mathematics and American Mathematical Society. Small eigenvalues of large Hankel matrices:The indeterminate case more.
In this paper we characterise the indeterminate case by the eigenvalues of the Hankel matrices being bounded below by a strictly positive constant. An explicit lower bound is given in terms of the orthonormal polynomials and we find An explicit lower bound is given in terms of the orthonormal polynomials and we find expresions for this lower bound in a number of indeterminate moment problems.
Lower Bound. Some basic bilateral sums and integrals more. Pure Mathematics and Boolean Satisfiability. We use generating functions to express orthogonality relations in the form of q-beta. The integrand of such a q-beta. This method Pure Mathematics and Generating Function.
Small eigenvalues of large Hankel matrices: the indeterminate case more. Abstract: In this paper we characterise the indeterminate case by the eigenvalues of the Hankel matrices being bounded below by a strictly positive constant.
An explicit lower bound is given in terms of the orthonormal polynomials and we Publisher: arxiv. Approximation Theory , Orthogonal polynomials , and Pure Mathematics. Three routes to the exact asymptotics for the one-dimensional quantum walk more. We demonstrate an alternative method,for calculating the asymptotic behaviour of the discrete one-coin quantum walk on the infinite line, via the Jacobi polynomials that arise in the path integral representation. We calculate the We calculate the asymtotics using a method,that is significantly easier to use than the Darboux method.
It also provides a single integral representation for the wavefunction that works over. Editorial more. Approximation Theory. Jacobi polynomials from compatibility conditions more. On Asymptotics of Jacobi Polynomials more. The combinatorics of q-Hermite polynomials and the Askey-Wilson integral more. The combinatorics of q-Hermite polynomials and the Askey-Wilson integral.
Dennis Stanton, University of Minnesota, Minneapolis. Gerard Viennot, University of Bordeaux, Talence, Deformed Complex Hermite Polynomials more. We study a class of bivariate deformed Hermite polynomials and some of their properties using classical analytic techniques and the Wigner map. We also prove the positivity of certain determinants formed by the deformed polynomials. Along the way we also work out some additional properties of the undeformed complex Hermite polynomials and their relationships to the standard Hermite polynomials of a single real variable.
Linear q-Difference Equations more. It is proved that such an equation has a fundamental set of n-linearly independent solutions and a q-type Wronskian is derived. Fundamental systems of solutions are Fundamental systems of solutions are constructed for the n-th order linear q-difference equation with constant coefficients. Examples illustrating the results are presented.
Applied Mathematics and Pure Mathematics. Generalizations of Chebyshev polynomials and polynomial mappings more. Dedication more. Hypergeometric origins of Diophantine properties associated with the Askey scheme more. Here the Diophantine property refers to integer valued zeros. It turns out that the same procedure can also be applied to polynomials arising from the basic hypergeometric series.
Journal Name: Proc. Soc Publication Date: Some orthogonal polynomials arising from coherent states more. Ladder operators and differential equations for orthogonal polynomials more. Under some integrability conditions we derive raising and lowering differential recurrence relations for polynomials orthogonal with respect to a weight function supported in the real line.
We also derive a second-order We also derive a second-order differential equation satisfied by these polynomials. We discuss the Lie algebra generated by the generalized creation and annihilation operators. From the differential equations, Plancherel—Rotach type asymptotics are derived.
Under certain conditions, stated in the text, an Airy function emerges. Ladder operators for q-orthogonal polynomials more. Nonlinear equations for the recurrence coefficients of discrete orthogonal polynomials more.
Simeonov and Mourad Ismail. Difference Operators for Orthogonal Polynomials more. Applications of q-Taylor theorems more. We establish two new q-analogues of a Taylor series expansion for polynomials using special Askey—Wilson polynomial bases. Combining these expansions with an earlier expansion theorem we derive inverse relations and evaluate certain Combining these expansions with an earlier expansion theorem we derive inverse relations and evaluate certain linearization coefficients.
Byproducts include new summation theorems, new results on a q-exponential function, and quadratic transformations for q-series. Asymptotics of basic Bessel functions and q-Laguerre polynomials more. Asymptotics of extreme zeros of the Meixner-Pollaczek polynomials more. This solves the problem of Our technique is constructive and gives an explicit representation of the sought entire function. Applications to q-series identities are given. Lattice Paths and Positive Trigonometric Sums more.
A trigonometric polynomial generalization to the positivity of an alternating sum of binomial coefficients is given. The proof uses lattice paths, and identifies the trigonometric sum as a polynomial with positive integer The proof uses lattice paths, and identifies the trigonometric sum as a polynomial with positive integer coefficients.
Some special cases of the q -analogue conjectured by Bressoud are established, and new conjectures are given. Monotonicity Properties of Determinants of Special Functions more. Our results recover and Our results recover and generalize some known determinantal inequalities.
We also show that a certain determinant formed by the Fibonacci numbers are nonnegative while determinants involving Hermite polynomials of imaginary arguments are shown to be completely monotonic functions.
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