orthogonal polynomials associated polynomials and functions of the second kind abstract a survey is given of the interaction between orthogonal polynomials associated polynomials and functions of the second kind with an emphasis on asymptotic results various formulas are presented in a unified way in terms of wronskians of solutions of linear recurrence relations some of these formulas are classical and go back to the previous century but usually they are hard to locate in the literature some new formulas are also given in particular a formula expressing the derivative of an orthogonal polynomial in terms of the orthogonal polynomials and the associated polynomials

characterization theorem for laguerre hahn orthogonal polynomials on non uniform lattices abstract a characterization theorem for laguerre hahn orthogonal polynomials on non uniform lattices is stated and proved this theorem proves the equivalence between the riccati equation for the formal stieltjes function linear first order difference relations for the orthogonal polynomials as well as for the associated polynomials of the first kind and linear first order difference relations for the functions of the second kind

on associated polynomials and decay rates for birth death processes abstract we consider sequences of orthogonal polynomials and pursue the question of how partial knowledge of the orthogonalizing measure for the associated polynomials can lead to information about the orthogonalizing measure for the original polynomials in particular we relate the supports of the two measures and their moments as an application we analyse the relation between two decay rates connected with a birth death process

chinchilla middle ear transmission matrix model and middle ear flexibility the function of the middle ear me in transforming me acoustic inputs and outputs sound pressures and volume velocities can be described with an acoustic two port transmission matrix this description is independent of the load on the me cochlea or ear canal and holds in either direction forward from ear canal to cochlea or reverse from cochlea to ear canal a transmission matrix describing me function in chinchilla an animal commonly used in auditory research is presented computed from measurements of forward me function input admittance ytm me pressure gain gmep me velocity transfer function hv and cochlear input admittance yc in the same set of ears ravicz and rosowski 2012b j acoust soc am 132 2437 2454 2013a j acoust soc am 133 2208 2223 2013b j acoust soc am 134 2852 2865 unlike previous estimates these computations require no assumptions about the state of the inner ear effectiveness of me manipulations or measurements of sound transmission in the reverse direction these element values are generally consistent with physical constraints and the anatomical me transformer ratio differences from a previous estimate in chinchilla songer and rosowski 2007 j acoust soc am 122 932 942 may be due to a difference in me flexibility between the two subject groups

accurate simple zeros of polynomials in floating point arithmetic in the paper we examine the local behavior of newton s method in floating point arithmetic for the computation of a simple zero of a polynomial assuming that an good initial approximation is available we allow an extended precision twice the working precision in the computation of the residual we prove that for a sufficient number of iterations the zero is as accurate as if computed in twice the working precision we provide numerical experiments confirming this

stieltjes polynomials and functions of the second kind abstract in this paper we discuss known and new facts on the asymptotic behaviour of the stieltjes polynomials and the close relation to the asymptotic behaviour of the boundary values of the functions of the second kind on 1 1 most of the results are derived by new methods of proof where mainly the theory of real orthogonal polynomials is used

accurate floating point summation part i faithful rounding given a vector of floating point numbers with exact sum s we present an algorithm for calculating a faithful rounding of s i e the result is one of the immediate floating point neighbors of s if the sum s is a floating point number we prove that this is the result of our algorithm the algorithm adapts to the condition number of the sum i e it is fast for mildly conditioned sums with slowly increasing computing time proportional to the logarithm of the condition number all statements are also true in the presence of underflow the algorithm does not depend on the exponent range our algorithm is fast in terms of measured computing time because it allows good instruction level parallelism it neither requires special operations such as access to mantissa or exponent it contains no branch in the inner loop nor does it require some extra precision the only operations used are standard floating point addition subtraction and multiplication in one working precision for example double precision certain constants used in the algorithm are proved to be optimal

finite perturbations of orthogonal polynomials in this paper we study orthogonal polynomials pn which arise from a given system of orthogonal polynomials pn by a finite perturbation of the recurrence coefficients i e the recurrence coefficients n n 1 of pn respectively n n 1 of pn are related to each other by n l n m and n 1 l n 1 m for n n where l and m are fixed nonnegative integers closed expressions for the polynomial pn in terms of pn and its associated polynomial pn 1 1 as well as for the orthogonality measure of pn in terms of the orthogonality measure of pn are given in fact the results are presented for the most general case when is an arbitrary not necessarily positive measure

rounding error bounds for the clenshaw and forsythe algorithms for the evaluation of orthogonal polynomial series rounding error bounds of the forsythe and the clenshaw smith algorithm for the evaluation of finite series of orthogonal polynomials are presented the forward error is studied by means of a direct approach that permits to obtain sharp bounds the bounds are illustrated with several numerical tests

accurate sum and dot product algorithms for summation and dot product of floating point numbers are presented which are fast in terms of measured computing time we show that the computed results are as accurate as if computed in twice or k fold working precision k ge 3 for twice the working precision our algorithms for summation and dot product are some 40 faster than the corresponding xblas routines while sharing similar error estimates our algorithms are widely applicable because they require only addition subtraction and multiplication of floating point numbers in the same working precision as the given data higher precision is unnecessary algorithms are straight loops without branch and no access to mantissa or exponent is necessary

an isohazard function for impulse noise existing criteria for safe exposure to impulse noise do not consider the frequency spectrum of an impulse as a variable in the evaluation of the hazards to the auditory system this study was designed to determine the relative potential that impulsive energy concentrated at different frequencies has in causing auditory system trauma one hundred and thirty 130 chinchillas divided into 22 groups of 5 to 7 animals were used pre and postexposure audiograms were measured on each animal using avoidance conditioning procedures quantitative histology was used to determine the extent and pattern of the sensory cell damage cochleograms noise exposure employed seven different computer generated narrow band impulses approximately 400 hz bandwidth having center frequencies located at 0 260 0 775 1 025 1 350 2 075 2 450 and 3 550 khz presented at two to four different intensities an isohazard weighting function derived from the audiometric and histological data demonstrates that equivalent amounts

fast algorithms for discrete polynomial transforms consider the vandermonde like matrix p p k cos j n j k 0 n where the polynomials p k satisfy a three term recurrence relation if p k are the chebyshev polynomials t k then p coincides with c n 1 cos jk n j k 0 n this paper presents a new fast algorithm for the computation of the matrix vector product pa in o n log 2 n arithmetical operations the algorithm divides into a fast transform which replaces pa with c n 1 and a subsequent fast cosine transform the first and central part of the algorithm is realized by a straightforward cascade summation based on properties of associated polynomials and by fast polynomial multiplications numerical tests demonstrate that our fast polynomial transform realizes pa with almost the same precision as the clenshaw algorithm but is much faster for n 128

middle ear function in the chinchilla circuit models and comparison with other mammalian species the middle ear efficiently transmits sound from the ear canal into the inner ear through a broad range of frequencies thus understanding middle ear transmission characteristics is essential in the study of hearing mechanics two models of the chinchilla middle ear are presented in the first model the middle ear is modeled as a lumped parameter system with elements that represent the ossicular chain and the middle ear cavity parameters of this model are fit using available experimental data of two port transmission matrix parameters in an effort to improve agreement between model simulations and the phase of published experimental measurements for the forward pressure transfer function at high frequencies a second model in which a lossless transmission line model of the tympanic membrane is appended to the original model is proposed two port transmission matrix parameter results from this second model were compared with results from previously developed models of the guinea pig cat and human middle ears model results and published experimental data for the two port transmission matrix parameters are found to be qualitatively similar between species quantitative differences in the two port transmission matrix parameters suggest that the ossicular chains of chinchillas cats and guinea pigs are less flexible than in humans

theory of recursive generation of systems of orthogonal polynomials an illustrative example abstract pending its publication in full detail the theory of recursive generation of systems of orthogonal polynomials is verbally described and illustrated by means of a typical example of practical application for this purpose the theory is applied to the polynomials of legendre chosen as initially given orthogonal sequence

middle ear phenomenology the view from the three windows to provide a common ground for the comparison between theory and experiment this paper presents a framework for the phenomenological description of middle ear mechanics the framework defines those measurements sufficient to characterize the transduction properties of the middle ear and its components phenomenological equations are represented in the form of an equivalent electrical circuit that can be used to deduce testable relations among measurable quantities two applications are then discussed first the classical concept of the middle ear transformer ratio is generalized to include any effects of eardrum flexion or nonrotational ossicular motion middle ear models predict that the resulting transformer ratios vary considerably with frequency second the conditions under which the topology of existing circuit analogs satisfactorily approximates middle ear mechanics are given most middle ear models cannot be used to correctly predict the absolute pressures in the cochlea

fast and stable algorithms for discrete spherical fourier transforms abstract in this paper we propose an algorithm for the stable and efficient computation of fourier expansions of square integrable functions on the unit sphere s r 3 as well as for the evaluation of these fourier expansions at special knots the heart of the algorithm is an efficient realization of discrete legendre function transforms based on a modified and stabilized version of the driscoll healy algorithm

accurate evaluation of a polynomial and its derivative in bernstein form in computer aided geometric design a polynomial is usually represented in bernstein form this paper presents two compensated algorithms to accurately evaluate a polynomial and its derivative in bernstein form with floating point coefficients the principle is to apply error free transformation to improve traditional de casteljau algorithm forward error analysis and numerical experiments illustrate the accuracy of our algorithms

accurate floating point product and exponentiation several different techniques and softwares intend to improve the accuracy of results computed in a fixed finite precision here we focus on a method to improve the accuracy of the product of floating point numbers we show that the computed result is as accurate as if computed in twice the working precision the algorithm is simple since it only requires addition subtraction and multiplication of floating point numbers in the same working precision as the given data such an algorithm can be useful for example to compute the determinant of a triangular matrix and to evaluate a polynomial when represented by the root product form it can also be used to compute the integer power of a floating point number

fourth order differential equation for the co modified of semi classical orthogonal polynomials abstract we give a new derivation of the fourth order differential equation satisfied by the co modified orthogonal polynomials of any semi classical family of orthogonal polynomials this procedure generalizes a technique already used in the co recursive and co dilated cases for the classical cases we give explicitly the corresponding differential equation in a determinantal form

integrals of the ising class from an experimental mathematical perspective we analyze ising class integrals our experimental results involvedextreme precision multidimensional quadrature on intricate integrands thus highly parallel computation was required

associated laguerre and hermite polynomials explicit orthogonality relations are found for the associated laguerre and hermite polynomials one consequence is the construction of the n 1 n pade approximation to a 1 b x a b x where a b x is the second solution to the confluent hypergeometric differential equation that does not grow rapidly at infinity

generalized associated polynomials and functions of second kind let pv v n0 pv v v 1 be a sequence of polynomials generated by a three term recurrence relation shifting the recurrence coefficients of the elements of pv v n0 we get a sequence of so called associated polynomials which play an important role in the theory of orthogonal polynomials we generalize this concept of associating for arbitrary polynomials vn n especially if vn is expanded in terms of pv v 0 n their associated polynomials are clenshaw polynomials which are used in numerical mathematics as a consequence of it we present some results from the viewpoint of associated polynomials and from the viewpoint of clenshaw polynomials analogously as for orthogonal polynomials we define functions of second kind for vn we prove some properties of them which depend on the generalized associated polynomials and functions of second kind for orthogonal polynomials

on co polynomials on the real line abstract in this paper we study new algebraic and analytic aspects of orthogonal polynomials on the real line when finite modifications of the recurrence coefficients the so called co polynomials on the real line are considered we investigate the behavior of their zeros mainly interlacing and monotonicity properties furthermore using a transfer matrix approach we obtain new structural relations combining theoretical and computational advantages finally a connection with the theory of orthogonal polynomials on the unit circle is pointed out

cortical responses to the 2f1 f2 combination tone measured indirectly using magnetoencephalography the simultaneous presentation of two tones with frequencies f1 and f2 causes the perception of several combination tones in addition to the original tones the most prominent of these are at frequencies f2 f1 and 2f1 f2 this study measured human physiological responses to the 2f1 f2 combination tone at 500hz caused by tones of 750 and 1000hz with intensities of 65 and 55db spl respectively responses were measured from the cochlea using the distortion product otoacoustic emission dpoae and from the auditory cortex using the 40 hz steady state magnetoencephalographic meg response the perceptual response was assessed by having the participant adjust a probe tone to cause maximal beating best beats with the perceived combination tone the cortical response to the combination tone was evaluated in two ways first by presenting a probe tone with a frequency of 460hz at the perceptual best beats level resulting in a 40 hz response because of interaction with the combination tone at 500hz and secon

on the associated orthogonal polynomials abstract by using the second order recurrence relation this paper gives some new results on associated orthogonal polynomials without referring to the continued fractions tool some results are very useful for obtaining the second order differential equation satisfied by the semi classical orthogonal polynomials hendriksen and van rossum 1985 maroni 1987 cf section 3 also the main formula derives from proposition 2 6 by which the fourth order differential equation satisfied by some laguerre hahn polynomials magnus 1984 is obtained cf belmehdi and ronveaux 1989 dini 1989 ronveaux 1990

on semiclassical orthogonal polynomials via polynomial mappings abstract we consider orthogonal polynomials via polynomial mappings in the framework of the semiclassical class we prove that this class is stable under polynomial transformations several consequences of this fact are deduced as an application we analyze in detail cubic transformations for semiclassical orthogonal polynomials of class at most 2 recovering and extending some results proved recently for class 1 and producing new examples of semiclassical orthogonal polynomials of class 2 in particular we show how to obtain integral representations for the regular functionals with respect to which these new semiclassical families are orthogonal

fast evaluation of quadrature formulae on the sphere recently a fast approximate algorithm for the evaluation of expansions in terms of standard l 2 s 2 orthonormal spherical harmonics at arbitrary nodes on the sphere s 2 has been proposed in s kunis and d potts fast spherical fourier algorithms j comput appl math 161 75 98 2003 the aim of this paper is to develop a new fast algorithm for the adjoint problem which can be used to compute expansion coefficients from sampled data by means of quadrature rules we give a formulation in matrix vector notation and an explicit factorisation of the spherical fourier matrix based on the former algorithm starting from this we obtain the corresponding factorisation of the adjoint spherical fourier matrix and are able to describe the associated algorithm for the adjoint transformation which can be employed to evaluate quadrature rules for arbitrary weights and nodes on the sphere we provide results of numerical tests showing the stability of the obtained algorithm using as examples classical gaus legendre and clenshaw curtis quadrature rules as well as the healpix pixelation scheme and an equidistribution

the evaluation of integrals of bessel functions via g function identities a few transformations are presented for reducing certain cases of meijer s g function to a g function of lower order their applications to the integration of a product of bessel functions are given the algorithm has been implemented within mathematica 3 0

on the efficient parallel computation of legendre transforms in this article we discuss a parallel implementation of efficient algorithms for computation of legendre polynomial transforms and other orthogonal polynomial transforms we develop an approach to the driscoll healy algorithm using polynomial arithmetic and present experimental results on the accuracy efficiency and scalability of our implementation the algorithms were implemented in ansi c using the bsplib communications library we also present a new algorithm for computing the cosine transform of two vectors at the same time

the weight functions generating functions and miscellaneous properties of the sequences of orthogonal polynomials of the second kind associated with the jacobi and the gegenbauer polynomials abstract in the introduction the main object of the paper namely the calculation and study of the weight functions and orthogonality relations corresponding to the sequences of polynomials of the second kind associated with the jacoby and the gegenbauer polynomials is briefly discussed section 2 contains the underlying theory restricted to the citation of a few general formulas regarding the orthogonality property of associated polynomials of the second kind section 3 which deals with the application to the case of the jacobi polynomials p n z n n starts with the construction of the orthogonality relation and the weight function 1 of the associated polynomials for arbitrary gt 1 the denominator of the fraction representing 1 comprises the principal value of an integral which is studied in detail in order to bring it in an explicit form most suited for practical use attention is also paid to the associated functions of the second kind and some of their properties next in a subsection 1 is expressed solely in terms of elementary functions in a number of special cases involving subsets of pairs when 1 1 n gt 1 1 1 gt0 and 1 0 gt 1 in another subsection the generating function for the associated polynomials is calculated and the analysis of the result comprises some related formulas in the final subsection the moments of the weight function 1 with respect to 0 1 and 1 are expressed in terms of the corresponding moments of the weight function of the jacobi polynomials both in finite determinantal form and by means of a recurrence relation in section 4 which concerns the application to the case of the gegenbauer polynomials c v n z n n the advantages of a new definition of the associated polynomials of the second kind are at first expounded then follow the orthogonality relation and the weight function 1 for arbitrary v gt 1 2 next 1 is calculated in terms of elementary functions for all integer and half odd integer values of v exceeding 1 2 in the former case a remarkable simplification occurs in the denominator of the fraction representing 1 after that follow three subsections devoted respectively to the functions of the second kind associated with the gegenbauer polynomials two different generating functions for the associated polynomials and related formulas and the moments of the weight function 1 the paper ends with two appendices in which certain types of sums are reduced to a one term expression

polynomial evaluation and associated polynomials we show a connection between the clenshaw algorithm for evaluating a polynomial q n expanded in terms of a system of orthogonal polynomials and special linear combinations of associated polynomials these results enable us to get the derivatives of q n z analogously to the horner algorithm for evaluating polynomials in monomial representations furthermore we show how a polynomial hat q n z given in monomial representation can be evaluated for z in bbb c using the clenshaw algorithm without complex arithmetic from this we get a connection between zeros of polynomials expanded in terms of chebyshev polynomials and the corresponding polynomials in monomial representation with the same coefficients

szegő transformations and nth order associated polynomials on the unit circle in this paper we analyze the stieltjes functions defined by the szego inverse transformation of a nontrivial probability measure supported on the unit circle such that the corresponding sequence of orthogonal polynomials is defined by either backward or forward shifts in their verblunsky parameters such polynomials are called anti associated respectively associated orthogonal polynomials thus rational spectral transformations appear in a natural way

outage exponent a unified performance metric for parallel fading channels the parallel fading channel which consists of finite number of subchannels is very important because it can be used to formulate many practical communication systems the outage probability on the other hand is widely used to analyze the relationship among the communication efficiency reliability signal to noise ratio snr and channel fading to the best of our knowledge the previous works only studied the asymptotic outage performance of the parallel fading channels which are only valid for a large number of subchannels or high snrs in this paper a unified performance metric which we shall refer to as the outage exponent will be proposed our approach is mainly based on the large deviations theory and meijer s g function it is shown that the proposed outage exponent is not only an accurate estimation of the outage probability for any number of subchannels any snr and any target transmission rate but also provides an easy way to compute the outage capacity finite snr diversity multiplexing tradeoff and snr gain the asymptotic performance metrics such as the delay limited capacity ergodic capacity and diversity multiplexing tradeoff can be directly obtained by letting the number of subchannels or snr tend to infinity similar to gallager s error exponent a reliable function for parallel fading channels which illustrates a fundamental relationship between the transmission reliability and efficiency can also be defined from the outage exponent therefore the proposed outage exponent provides a complete and comprehensive performance measure for parallel fading channels

accurate floating point summation part ii sign k fold faithful and rounding to nearest in part ii of this paper we first refine the analysis of error free vector transformations presented in part i based on that we present an algorithm for calculating the rounded to nearest result of s sum p i for a given vector of floating point numbers p i as well as algorithms for directed rounding a special algorithm for computing the sign of s is given also working for huge dimensions assume a floating point working precision with relative rounding error unit eps we define and investigate a k fold faithful rounding of a real number r basically the result is stored in a vector mathtt res nu of k nonoverlapping floating point numbers such that sum mathtt res nu approximates r with relative accuracy mathtt eps k and replacing mathtt res k by its floating point neighbors in sum mathtt res nu forms a lower and upper bound for r for a given vector of floating point numbers with exact sum s we present an algorithm for calculating a k fold faithful rounding of s using solely the working precision furthermore an algorithm for calculating a faithfully rounded result of the sum of a vector of huge dimension is presented our algorithms are fast in terms of measured computing time because they allow good instruction level parallelism they neither require special operations such as access to mantissa or exponent they contain no branch in the inner loop nor do they require some extra precision the only operations used are standard floating point addition subtraction and multiplication in one working precision for example double precision certain constants used in the algorithms are proved to be optimal

complex jacobi matrices complex jacobi matrices play an important role in the study of asymptotics and zero distribution of formal orthogonal polynomials fops the latter are essential tools in several fields of numerical analysis for instance in the context of iterative methods for solving large systems of linear equations or in the study of pade approximation and jacobi continued fractions in this paper we present some known and some new results on fops in terms of spectral properties of the underlying infinite jacobi matrix with a special emphasis to unbounded recurrence coefficients here we recover several classical results for real jacobi matrices the inverse problem of characterizing properties of the jacobi operator in terms of fops and other solutions of a given three term recurrence is also investigated this enables us to give results on the approximation of the resolvent by inverses of finite sections with applications to the convergence of pade approximants

szegő transformations and rational spectral transformations for associated polynomials in this contribution we analyze rational spectral transformations related to associated polynomials with respect to probability measures supported on the interval 1 1 the connection with rational spectral transformations of measures supported on the unit circle using the szego transformation is presented

accurate evaluation of a polynomial in chebyshev form this paper presents a compensated algorithm to accurately evaluate a polynomial expressed in chebyshev basis of the first and second kind with floating point coefficients the principle is to apply error free transformations to improve the traditional clenshaw algorithm the new algorithm is as accurate as the clenshaw algorithm performed in twice the working precision forward error analysis and numerical experiments illustrate the accuracy and properties of the proposed algorithm