Abstract: In this paper we review a special set of orthonormal functions, namely Zernike polynomials which are widely used in representing the aberrations of optical systems. We give the recurrence relations, relationship to other special functions, as well as scaling and other properties of these important polynomials. Mathematica code for certain operations are given in the Appendix.

Abstract: This paper presents a computational technique based on the collocation method and Muntz polynomials for the solution of fractional differential equations. An appropriate representation of the solution via the Muntz polynomials reduces its numerical treatment to the solution of a system of algebraic equations. The main advantage of the present method is its superior accuracy and exponential convergence. Consequently, one can obtain good results even by using a small number of collocation points. The accuracy and performance of the proposed method are examined by means of some numerical experiments.

Abstract: This paper provides an effective solution for the simulation of cables and interconnects with the inclusion of the effects of parameter uncertainties. The problem formulation is based on the telegraphers equations with stochastic coefficients, whose solution requires an expansion of the unknown parameters in terms of orthogonal polynomials of random variables. The proposed method offers accuracy and improved efficiency in computing the parameter variability effects on system responses with respect to the conventional Monte Carlo approach. The approach is validated against results available in the literature, and applied to the stochastic analysis of a commercial multiconductor flat cable.

Abstract: In this paper, we give relations involving values of q-Bernoulli, q-Euler, and Bernstein polynomials. Using these relations, we obtain some interesting identities on the q-Bernoulli, q-Euler, and Bernstein polynomials.

Abstract: This paper focuses on the derivation of an enhanced transmission-line model allowing the stochastic analysis of a realistic multiconductor interconnect. The proposed model, which is based on the expansion of the well-known telegraph equations in terms of orthogonal polynomials, includes the variability of geometrical or material properties of the interconnect due to uncertainties like fabrication process or temperature. A real application example involving the frequency-domain analysis of a coupled microstrip and the computation of the parameters variability effects on the transmission-line response concludes this paper.

Abstract: This is a survey of results constituting the foundations of the modern convergence theory of Pade approximants. Bibliography: 204 titles.

Abstract: We consider weighted 𝑞-Genocchi numbers and polynomials. We investigated some interesting properties of the weighted 𝑞-Genocchi numbers related to weighted 𝑞-Bernstein polynomials by using fermionic 𝑝-adic integrals on ℤ𝑝.

Abstract: A comprehensive review of the discrete quantum mechanics with the pure imaginary shifts and the real shifts is presented in parallel with the corresponding results in the ordinary quantum mechanics. The main subjects to be covered are the factorized Hamiltonians, the general structure of the solution spaces of the Schr?dinger equation (Crum's theorem and its modification), the shape invariance, the exact solvability in the Schr?dinger picture as well as in the Heisenberg picture, the creation/annihilation operators and the dynamical symmetry algebras, the unified theory of exact and quasi-exact solvability based on the sinusoidal coordinates, and the infinite families of new orthogonal (the exceptional) polynomials. Two new infinite families of orthogonal polynomials, the X? Meixner?Pollaczek and the X? Meixner polynomials, are reported.

Abstract: Pade approximation and Rational interpolation.- Integral approximants for functions of higher monodromic dimension.- Asymptotics of Hermite-Pade Polynomials and related convergence results.- Rational approximation.- On the behavior of zeros and poles of best uniform polynomial and rational approximants.- Once again: the Adamjan-Arov-Krein approximation theory.- Diagonal Pade approximants, rational Chebyshev approximants and poles of functions.- On the use of the Caratheodory-Fejer method for investigating'1/9' and similar constants.- Multidimensional and Multivariate problems.- Simultaneous rational approximation to some q-hypergeometric functions.- Minimal Pade-sense matrix approximations around s = 0 and s = ?.- (Pade)y of (Pade)x approximants of F(x,y).- Different techniques for the construction of multivariate rational interpolants.- Rational approximants of hypergeometric series in ?n.- Orthogonal polynomials and the Moment problem.- Some orthogonal systems of p+1Fp-type Laurent polynomials.- The moment problem on equipotential curves.- Difference equations, continued fractions, Jacobi matrices and orthogonal polynomials.- Multipoint Pade approximation and orthogonal rational functions.- L-Polynomials orthogonal on the unit circle.- Continued fractions.- Schur's algorithm extended and Schur continued fractions.- Some recent results in the analytic theory of continued fractions.- Best a posteriori truncation error estimates for continued fractions if (an/1) with twin element regions.- Convergence acceleration for Miller's algorithm.- Convergence acceleration.- A new approach to convergence acceleration methods.- Applications.- General T-fraction solutions to Riccati differential equations.- A simple alternative principle for rational ?-method approximation.- Evaluation of Fermi-Dirac integral.- An application of operator Pade approximants to multireggeon processes.

Abstract: Exactly solvable rationally-extended radial oscillator potentials, whose wave functions can be expressed in terms of Laguerre-type exceptional orthogonal polynomials, are constructed in the framework of kth-order supersymmetric quantum mechanics, with special emphasis on k = 2. It is shown that for μ = 1, 2, and 3, there exist exactly μ distinct potentials of μth type and associated families of exceptional orthogonal polynomials, where μ denotes the degree of the polynomial gμ arising in the denominator of the potentials.

Abstract: The main object of this paper is to introduce and investigate a new generalization of the family of Euler polynomials by means of a suitable generating function We establish several interesting properties of these general polynomials and derive explicit representations for them in terms of a certain generalized Hurwitz-Lerch Zeta function and in series involving the familiar Gaussian hypergeometric function Finally, we propose an analogous generalization of the closely-related Genocchi polynomials and show how we can fruifully exploit some potentially useful linear connections of all these three important families of generalized Bernoulli, Euler and Genocchi polynomials with one another

Abstract: A comprehensive review of the discrete quantum mechanics with the pure imaginary shifts and the real shifts is presented in parallel with the corresponding results in the ordinary quantum mechanics. The main subjects to be covered are the factorised Hamiltonians, the general structure of the solution spaces of the Schroedinger equation (Crum's theorem and its modification), the shape invariance, the exact solvability in the Schroedinger picture as well as in the Heisenberg picture, the creation/annihilation operators and the dynamical symmetry algebras, the unified theory of exact and quasi-exact solvability based on the sinusoidal coordinates, the infinite families of new orthogonal (the exceptional) polynomials. Two new infinite families of orthogonal polynomials, the X_\ell Meixner-Pollaczek and the X_\ell Meixner polynomials are reported.

Abstract: In this paper, we present a unified family of polynomials including not only the Apostol-Bernoulli, Euler and Genocchi polynomials, but also a general family of polynomials suggested by Ozden et al. [H. Ozden, Y. Simsek, H.M. Srivastava, A unified presentation of the generating functions of the generalized Bernoulli, Euler and Genocchi polynomials. Comput. Math. Appl. 60 (10) (2010) 2779-2787]. We obtain the explicit representation of this unified family, in terms of a Gaussian hypergeometric function. Some symmetry identities and multiplication formula are also given.

Abstract: Sigma-delta modulation is a popular method for analog-to-digital conversion of bandlimited signals that employs coarse quantization coupled with oversampling. The standard mathematical model for the error analysis of the method measures the performance of a given scheme by the rate at which the associated reconstruction error decays as a function of the oversampling ratio λ. It was recently shown that exponential accuracy of the form O(2−rλ) can be achieved by appropriate one-bit sigma-delta modulation schemes. By general information-entropy arguments, r must be less than 1. The current best-known value for r is approximately 0:088. The schemes that were designed to achieve this accuracy employ the “greedy” quantization rule coupled with feedback filters that fall into a class we call “minimally supported.” In this paper, we study the discrete minimization problem that corresponds to optimizing the error decay rate for this class of feedback filters. We solve a relaxed version of this problem exactly and provide explicit asymptotics of the solutions. From these relaxed solutions, we find asymptotically optimal solutions of the original problem, which improve the best-known exponential error decay rate to r ≈ 0.102. Our method draws from the theory of orthogonal polynomials; in particular, it relates the optimal filters to the zero sets of Chebyshev polynomials of the second kind. © 2011 Wiley Periodicals, Inc.

Abstract: This paper is concerned with Hermite-Pade rational approximants of analytic functions and their connection with multiple orthogonal polynomial ensembles of random matrices. Results on the analytic theory of such approximants are discussed, namely, convergence and the distribution of the poles of the rational approximants, and a survey is given of results on the distribution of the eigenvalues of the corresponding random matrices and on various regimes of such distributions. An important notion used to describe and to prove these kinds of results is the equilibrium of vector potentials with interaction matrices. This notion was introduced by A.A. Gonchar and E.A. Rakhmanov in 1981. Bibliography: 91 titles.

Abstract: A new family of hierarchical vector bases is proposed for triangles and tetrahedra These functions span the curl-conforming reduced-gradient spaces of Nedelec The bases are constructed from orthogonal scalar polynomials to enhance their linear independence, which is a simpler process than an orthogonalization applied to the final vector functions Specific functions are tabulated to order 65 Preliminary results confirm that the new bases produce reasonably well-conditioned matrices

Abstract: The nonlinear steepest descent method for rank-two systems relies on the notion of g-function. The applicability of the method ranges from orthogonal polynomials (and generalizations) to Painleve transcendents, and integrable wave equations (KdV, NonLinear Schrodinger, etc.). For the case of asymptotics of generalized orthogonal polynomials with respect to varying complex weights we can recast the requirements for the Cauchy-transform of the equilibrium measure into a problem of algebraic geometry and harmonic analysis and completely solve the existence and uniqueness issue without relying on the minimization of a functional. This addresses and solves also the issue of the “free boundary problem", determining implicitly the curves where the zeroes of the orthogonal polynomials accumulate in the limit of large degrees and the support of the measure. The relevance to the quasi-linear Stokes phenomenon for Painleve equations is indicated. A numerical algorithm to find these curves in some cases is also explained.

Abstract: We show how all the quantal systems related to the exceptional Laguerre and Jacobi polynomials can be constructed in a direct and systematic way, without the need of shape invariance and Darboux-Crum transformation. Furthermore, the prepotential need not be assumed a priori. The prepotential, the deforming function, the potential, the eigenfunctions and eigenvalues are all derived within the same framework. The exceptional polynomials are expressible as a bilinear combination of a deformation function and its derivative. Subject Index: 010, 064

Abstract: We consider the random matrix model with external source in the case where the potential V(x) is an even polynomial and the external source has two eigenvalues ±a of equal multiplicity. We show that the limiting mean eigenvalue distribution of this model can be characterized as the first component of a pair of measures (μ 1 , μ 2 ) that solve a constrained vector equilibrium problem. The proof is based on the steepest-descent analysis of the associated Riemann-Hilbert problem for multiple orthogonal polynomials. We illustrate our results in detail for the case of a quartic double-well potential V(x) = 1/4x 4 - t/2x 2 . We are able to determine the precise location of the phase transitions in the ta-plane, where either the constraint becomes active or the two intervals in the support come together (or both). © 2010 Wiley Periodicals, Inc.

Abstract: This paper presents two main results. The first result pertains to uniform approximation with Bernstein polynomials. We show that, given a power-form polynomial g, we can obtain a Bernstein polynomial of degree m with coefficients that are as close as desired to the corresponding values of g evaluated at the points 0,1m,2m,...,1, provided that m is sufficiently large. The second result pertains to a subset of Bernstein polynomials: those with coefficients that are all in the unit interval. We show that polynomials in this subset map the open interval (0,1) into the open interval (0,1) and map the points 0 and 1 into the closed interval [0,1]. The motivation for this work is our research on probabilistic computation with digital circuits. Our design methodology, called stochastic logic, is based on Bernstein polynomials with coefficients that correspond to probability values; accordingly, the coefficients must be values in the unit interval. The mathematics presented here provides a necessary and sufficient test for deciding whether polynomial operations can be implemented with stochastic logic.