Power series expansion of a Hankel determinant

Hankel continued fraction and its applications

A regular continued C-fraction is associated with a power series. The coefficients of this fraction are determined via either Hankel determinants or inverse derivatives. We prove the equivalence of these approaches to the construction of regular continued C-fractions.

In this paper, we investigated spectrum estimation for vowels using power series expansion. In general, sound signals can not keep their characteristics when the spectral peaks and dips are missing. Consequently, the missing spectral peaks and dips are estimated by using power series expansion. First, vowels are divided into cycle by cycle. Second, the one‐cycle vowels are converted into frequency by FFT. Third, damaged spectra were made for demonstrating spectrum estimation. Fourth, we estimated missing spectral peaks and dips using power series expansion. In our spectrum estimation, the differential coefficients for power series expansion were calculated from nearby spectrum of missing spectrum. In this experiment, the estimated spectrum was almost the same compared with the original spectrum in the case of approximation by power series expansion on the rank of 6. The spectrum is restored about the range of 180 Hz using nearby 10‐Hz samples. Consequently, power series expansion can estimate spectral peaks and dips for recovering spectrum.

The main objective of this contribution is a constructive generalization of the holomorphic power and Laurent series expansions in ℂ to dimension 3 using the framework of hypercomplex function theory. This first article on hand deals with generalized Fourier and Taylor series expansions in the space of square integrable quaternion-valued functions which possess peculiar properties regarding the hypercomplex derivative and primitive. In analogy to the complex one-dimensional case, both series expansions are orthogonal series with respect to the unit ball in ℝ3 and their series coefficients can be explicitly (one-to-one) linked with each other. Finally, very compact and efficient representation formulae (recurrence, closed-form) for the elements of the orthogonal bases are presented.

We completely characterize the appearance of Schur functions corresponding to partitions of the form $\nu = (1^a, b)$ (hook shapes) in the Schur function expansion of the plethysm of two Schur functions, $$s_\lambda[s_\mu] = \sum_{\nu} a_{\lambda, \mu, \nu} s_\nu.$$ Specifically, we show that no Schur functions corresponding to hook shapes occur unless $\lambda$ and $\mu$ are both hook shapes and give a new proof of a result of Carbonara, Remmel and Yang that a single hook shape occurs in the expansion of the plethysm $s_{(1^c, d)}[s_{(1^a, b)}]$. We also consider the problem of adding a row or column so that $\nu$ is of the form $(1^a,b,c)$ or $(1^a, 2^b, c)$. This proves considerably more difficult than the hook case and we discuss these difficulties while deriving explicit formulas for a special case.

Fast computation of the Gauss hypergeometric function with all its parameters complex with application to the Pöschl–Teller–Ginocchio potential wave functions

Various authors have presented the aberration function of an optical system as a power series expansion with respect to the ray coordinates in the exit pupil and the coordinates of the intersection point with the image field of the optical system. In practical applications, for reasons of efficiency and accuracy, an expansion with the aid of orthogonal polynomials is preferred for which, since the 1980s, orthogonal Zernike polynomials have become the reference. In the literature, some conversion schemes of power series coefficients to coefficients for the corresponding Zernike polynomial expansion have been given. In this paper we present an analytic solution for the conversion problem from a power series expansion in three or four dimensions to a double Zernike polynomial expansion. The solution pertains to a general optical system with four independent pupil and field coordinates and to a system with rotational symmetry in which case three independent coordinate combinations have to be considered. The conversion of the coefficients is analytically in closed form and the result is independent of a specific sampling scheme or sampling density as this is the case for the commonly used least squares fitting techniques. Computation schemes are given that allow the evaluation of coefficients of arbitrarily high order in pupil and field coordinates.

This paper presents a method to solve the American option pricing problem in the Black Scholes framework that generalizes the Barone-Adesi, Whaley method [1]. An auxiliary parameter is introduced in the American option pricing problem. Power series expansions in this parameter of the option price and of the corresponding free boundary are derived. These series expansions have the Baroni-Adesi, Whaley solution of the American option pricing problem as zero-th order term. The coefficients of the option price series are explicit formulae. The partial sums of the free boundary series are determined solving numerically nonlinear equations that depend from the time variable as a parameter. Numerical experiments suggest that the series expansions derived are convergent. The evaluation of the truncated series expansions on a grid of values of the independent variables is easily parallelizable. The cost of computing the n-th order truncated series expansions is approximately proportional to n as n goes to infinity. The results obtained on a set of test problems with the first and second order approximations deduced from the previous series expansions outperform in accuracy and/or in computational cost the results obtained with several alternative methods to solve the American option pricing problem [1]-[3]. For example when we consider options with maturity time between three and ten years and positive cost of carrying parameter (i.e. when the continuous dividend yield is smaller than the risk free interest rate) the second order approximation of the free boundary obtained truncating the series expansions improves substantially the Barone-Adesi, Whaley free boundary [1]. The website: http://www.econ.univpm.it/recchioni/finance/w20 contains material including animations, an interactive application and an app that helps the understanding of the paper. A general reference to the work of the authors and of their coauthors in mathematical finance is the website: http://www.econ.univpm.it/recchioni/finance.

Domain-partition power series in vibration analysis of variable-cross-section rods

Discrete orthogonal polynomials and difference equations of several variables

We study discrete semiclassical orthogonal polynomials of class s = 1. By considering particular solutions of the Pearson equation, we obtain five canonical families of such polynomials. We also consider limit relations between these and other families of orthogonal polynomials.

where |y1P #1. These were first discussed by Schur [6]1 with |7pl <1 in connection with functions bounded in the unit circle. In ?2 it is shown that to an arbitrary power series there correspond continued fractions (1.1), and, conversely, to every such continued fraction there corresponds a power series which agrees with the power series for the 2pth and (2p+1)th approximants of the continued fraction up to and including the terms involving zP and zP-1, respectively. An important new feature which is given in ?3 is the simple algorithm for the expansion of a power series into a continued fraction (1.1), and, conversely, for the computation of the corresponding power series from the continued fraction. Heretofore this computation has been exceedingly cumbersome. In ?4 conditions under which these continued fractions converge to the corresponding power series are considered, in particular, if the generating function is meromorphic. In ?5 convergence regions for the expansions (1.1) are found, and in ?6 convergence regions for the continued fraction whose approximants are the even and odd approximants of (1.1). In ?7 certain transformations of (1.1) are discussed.

Many Hankel determinant computations arising in combinatorial analysis can be done using results from the theory of standard orthogonal polynomials. Here, we will emphasize special sequences which require the inclusion of discrete Sobolov orthogonality to find their closed form.

Orthogonal polynomials, Hankel determinants and small eigenvalues associated with a deformed octic Freud weight

The six-vertex model with domain wall boundary conditions, or squareice, is considered for particular values of its parameters, corresponding to1-,2-,and 3-enumerations of alternating sign matrices (ASMs). Using Hankel determinant representationsfor the partition function and the boundary correlator of homogeneous square ice, it is shownhow the ordinary and refined enumerations can be derived in a very simple and straightforwardway. The derivation is based on the standard relationship between Hankel determinantsand orthogonal polynomials. For the particular sets of parameters corresponding to1-,2-,and 3-enumerations of ASMs, the Hankel determinant can be naturally related to ContinuousHahn, Meixner–Pollaczek, and Continuous Dual Hahn polynomials, respectively. Thisobservation allows for a unified and simplified treatment of ASM enumerations.In particular, along the lines of the proposed approach, we provide a completesolution to the long-standing problem of the refined 3-enumeration of ASMs.