solving boundary value problems integral and integro differential equations using gegenbauer integration matrices we introduce a hybrid gegenbauer ultraspherical integration method hgim for solving boundary value problems bvps integral and integro differential equations the proposed approach recasts the original problems into their integral formulations which are then discretized into linear systems of algebraic equations using gegenbauer integration matrices gims the resulting linear systems are well conditioned and can be easily solved using standard linear system solvers a study on the error bounds of the proposed method is presented and the spectral convergence is proven for two point bvps tpbvps comparisons with other competitive methods in the recent literature are included the proposed method results in an efficient algorithm and spectral accuracy is verified using eight test examples addressing the aforementioned classes of problems the proposed method can be applied on a broad range of mathematical problems while producing highly accurate results the developed numerical scheme provides a viable alternative to other solution methods when high order approximations are required using only a relatively small number of solution nodes

cox s regression model for counting processes a large sample study the cox regression model for censored survival data specifies that covariates have a proportional effect on the hazard function of the life time distribution of an individual in this paper we discuss how this model can be extended to a model where covariate processes have a proportional effect on the intensity process of a multivariate counting process this permits a statistical regression analysis of the intensity of a recurrent event allowing for complicated censoring patterns and time dependent covariates furthermore this formulation gives rise to proofs with very simple structure using martingale techniques for the asymptotic properties of the estimators from such a model finally an example of a statistical analysis is included

on the coefficients of integrated expansions of bessel polynomials a new formula expressing explicitly the integrals of bessel polynomials of any degree and for any order in terms of the bessel polynomials themselves is proved another new explicit formula relating the bessel coefficients of an expansion for infinitely differentiable function that has been integrated an arbitrary number of times in terms of the coefficients of the original expansion of the function is also established an application of these formulae for solving ordinary differential equations with varying coefficients is discussed

connection coefficients between boas buck polynomial sets abstract in this paper a general method to express explicitly connection coefficients between two boas buck polynomial sets is presented as application we consider some generalized hypergeometric polynomials from which we derive some well known results including duplication and inversion formulas

evaluation of cox s model and logistic regression for matched case control data with time dependent covariates a simulation study case control studies are typically analysed using the conventional logistic model which does not directly account for changes in the covariate values over time yet many exposures may vary over time the most natural alternative to handle such exposures would be to use the cox model with time dependent covariates however its application to case control data opens the question of how to manipulate the risk sets through a simulation study we investigate how the accuracy of the estimates of cox s model depends on the operational definition of risk sets and or on some aspects of the time varying exposure we also assess the estimates obtained from conventional logistic regression the lifetime experience of a hypothetical population is first generated and a matched case control study is then simulated from this population we control the frequency the age at initiation and the total duration of exposure as well as the strengths of their effects all models considered include a fixed in time covariate and one or two time dependent covariate s the indicator of current exposure and or the exposure duration simulation results show that none of the models always performs well the discrepancies between the odds ratios yielded by logistic regression and the true hazard ratio depend on both the type of the covariate and the strength of its effect in addition it seems that logistic regression has difficulty separating the effects of inter correlated time dependent covariates by contrast each of the two versions of cox s model systematically induces either a serious under estimation or a moderate over estimation bias the magnitude of the latter bias is proportional to the true effect suggesting that an improved manipulation of the risk sets may eliminate or at least reduce the bias

on the coefficients of integrated expansions and integrals of ultraspherical polynomials and their applications for solving differential equations an analytical formula expressing the ultraspherical coefficients of an expansion for an infinitely differentiable function that has been integrated an arbitrary number of times in terms of the coefficients of the original expansion of the function is stated in a more compact form and proved in a simpler way than the formula suggested by phillips and karageorghis 27 1990 823 a new formula expressing explicitly the integrals of ultraspherical polynomials of any degree that has been integrated an arbitrary number of times of ultraspherical polynomials is given the tensor product of ultraspherical polynomials is used to approximate a function of more than one variable formulae expressing the coefficients of differentiated expansions of double and triple ultraspherical polynomials in terms of the original expansion are stated and proved some applications of how to use ultraspherical polynomials for solving ordinary and partial differential equations are described

an efficient spectral method for ordinary differential equations with rational function coefficients the authors present some relations that allow the efficient approximate inversion of linear differential operators with rational function coefficients they employ expansions in terms of a large class of orthogonal polynomial families including all the classical orthogonal polynomials these families obey a simple 3 term recurrence relation for differentiation which implies that on an appropriately restricted domain the differentiation operator has a unique banded inverse the inverse is an integration operator for the family and it is simply the tridiagonal coefficient matrix for the recurrence since in these families convolution operators i e matrix representations of multiplication by a function are banded for polynomials the authors are able to obtain a banded representation for linear differential operators with rational coefficients this leads to a method of solution of initial or boundary value problems that besides having an operation count that scales linearly with the order of truncation n is computationally well conditioned among the applications considered is the use of rational maps for the resolution of sharp interior layers 19 refs 2 figs 3 tabs

an accurate solution of parabolic equations by expansion in ultraspherical polynomials abstract an ultraspherical expansion technique is applied to numerically obtain the solution of the third boundary value problem for a linear partial differential equation in a one space variable the method depends basically on the fact that an expansion in a series of ultraspherical polynomials c n for the function and its space derivatives occuring in the partial differential equation is assumed the coefficients of expansion are then determined by reducing the partial differential equation with its boundary and initial conditions to a system of ordinary differential equations for these coefficients this system may be solved analytically or numerically in a step by step manner the expansion for the function enables one to get the sought for approximation for any possible value of the parameter 1 2 numerical experiments show that more accurate results may be obtained by taking to be small and negative the method in its present form may be considered as a generalization of that of dew and scraton the extension of the method to the polar type equations u t p 2 u x 2 k u x x is also considered

the coefficients of differentiated expansions and derivatives of ultraspherical polynomials abstract a formula expressing the ultraspherical coefficients of the general order derivative of an infinitely differentiable function in terms of its original ultraspherical coefficients is stated in a more compact form and proved in a simpler way than the formula suggested by karageorghis and phillips in their recent report 5 formulas expressing explicitly the derivatives of ultraspherical polynomials of any degree and for any order in terms of the ultraspherical polynomials are given the special cases of chebyshev polynomials of the first and second kinds and of legendre polynomials are considered an application of how to use ultraspherical polynomials for solving ordinary and partial differential equations is described

polynomial interpolation results in sobolev spaces abstract in this paper we derive new results related to the lagrange polynomial interpolation on the nodes of gauss type quadrature formulas in particular we obtain asymptotic error estimates of the same order as the distance to the best fit in the polynomial space

connection problems via lowering operators the lowering operator associated with a polynomial set pn n 0 is an operator not depending on n and satisfying the relation pn npn 1 in this paper we express explicitly the connection coefficients between two polynomial sets using their corresponding lowering operators we obtain some well known results as particular cases including some duplication and addition formulas

a chebyshev expansion method for solving nonlinear optimal control problems this paper presents a numerical technique for solving the controlled duffing oscillator the control and state variables are approximated by chebyshev series a technique is provided for approximating the system dynamics boundary conditions and performance index this technique is based on using an explicit formula for the chebyshev polynomials in terms of arbitrary order of their derivatives the system dynamics and the performance index are converted into some algebraic equations then the optimal control problem is reduced to constrained optimization problem results and comparisons are given at the end of this paper

classical orthogonal polynomials dependence on parameters most of the classical orthogonal polynomials continuous discrete and their q analogues can be considered as functions of several parameters ci a systematic study of the variation innitesimal and nite of these polynomials pn x ci with respect to the parameters ci is proposed a method to get recurrence relations for connection coecients linking r c r i pn x ci t opn x ci is given and in some situations explicit expressions are obtained this allows us to compute new integrals or sums of classical orthogonal polynomials using the digamma function a basic theorem on the zeros of ci pn x ci is also proved c 2000 elsevier science b v all rights reserved msc 33c25 42c05 33b15

on the connection coefficients and recurrence relations arising from expansions in series of laguerre polynomials a formula expressing the laguerre coefficients of a general order derivative of an infinitely differentiable function in terms of its original coefficients is proved and a formula expressing explicitly the derivatives of laguerre polynomials of any degree and for any order as a linear combination of suitable laguerre polynomials is deduced a formula for the laguerre coefficients of the moments of one single laguerre polynomial of certain degree is given formulae for the laguerre coefficients of the moments of a general order derivative of an infinitely differentiable function in terms of its laguerre coefficients are also obtained a simple approach in order to build and solve recursively for the connection coefficients between jacobi laguerre and hermite laguerre polynomials is described an explicit formula for these coefficients between jacobi and laguerre polynomials is given of which the ultra spherical polynomials of the first and second kinds and legendre polynomials are important special cases an analytical formula for the connection coefficients between hermite and laguerre polynomials is also obtained

modified clebsch gordan type expansions for products of discrete hypergeometric polynomials abstract starting from the second order difference hypergeometric equation satisfied by the set of discrete orthogonal polynomials p n we find the analytical expressions of the expansion coefficients of any polynomial rm x and of the product rm x qj x in series of the set p n these coefficients are given in terms of the polynomial coefficients of the second order difference equations satisfied by the involved discrete hypergeometric polynomials here qj x denotes an arbitrary discrete hypergeometric polynomial of degree j the particular cases in which r m corresponds to the non orthogonal families x m the rising factorials or pochhammer polynomials x m and the falling factorial or stirling polynomials x m are considered in detail the connection problem between discrete hypergeometric polynomials which here corresponds to the product case with m 0 is also studied and its complete solution for all the classical discrete orthogonal hypergeometric cdoh polynomials is given also the inversion problems of cdoh polynomials associated to the three aforementioned nonorthogonal families are solved

a note on the chebyshev coefficients of the moments of the general order derivative of an infinitely differentiable function an expression for the chebyshev coefficients of the moments of the general order derivative of an infinitely differentiable function in terms of its chebyshev coefficients is suggested

gauss periods orders and cryptographical applications experimental results on the multiplicative orders of gauss periods in finite fields are presented these results indicate that gauss periods have high order and are often primitive self dual normal elements in finite fields it is shown that gauss periods can be exponentiated in quadratic time an application is an efficient pseudorandom bit generator

an instability phenomenon in spectral methods the eigenvalues of chebyshev and legendre spectral differentiation matrices which determine the allowable time step in an explicit time integration are extraordinarily sensitive to rounding errors and other perturbations on a grid of n points per space dimension machine rounding leads to errors in the eigenvalues of size o n 2 this phenomenon may lead to inconsistency between predicted and observed time step restrictions one consequence of it is that spectral differentiation by interpolation in legendre points which has a favorable o n 1 time step restriction for the model problem u t u x in theory is subject to an o n 2 restriction in practice the same effect occurs with chebyshev points for the model problem u t xu x another consequence is that a spectral calculation with a fixed time step may be stable in double precision but unstable in single precision we know of no other examples in numerical computation of this kind of precision dependent stability

on moments of classical orthogonal polynomials abstract in this work using the inversion coefficients and some connection coefficients between some polynomial sets we give explicit representations of the moments of all the orthogonal polynomials belonging to the askey wilson scheme generating functions for some of these moments are also provided

recurrences and explicit formulae for the expansion and connection coefficients in series of ordinary bessel polynomials abstract a formula expressing explicitly the derivatives of ordinary bessel polynomials of any degree and for any order in terms of the ordinary bessel polynomials themselves is proved another explicit formula which expresses the ordinary bessel expansion coefficients of a general order derivative of an infinitely differentiable function in terms of its original ordinary bessel coefficients is also given a formula for the ordinary bessel coefficients of the moments of one single ordinary bessel polynomial of certain degree is proved formula for the ordinary bessel coefficients of the moments of a general order derivative of an infinitely differentiable function in terms of its ordinary bessel coefficients is also obtained application of these formulae for solving ordinary differential equations with varying coefficients by reducing them to recurrence relations in the expansion coefficients of the solution is explained an algebraic symbolic approach using mathematica in order to build and solve recursively for the connection coefficients between generalized bessel and ordinary bessel polynomials is described explicit formula for these coefficients between jacobi and ordinary bessel polynomials is given of which the ultraspherical polynomials and its consequences are important special cases two analytical formulae for the connection coefficients between laguerre ordinary bessel and hermite ordinary bessel are also developed

on the limit relations between classical continuous and discrete orthogonal polynomials abstract limit relations between classical continuous jacobi laguerre hermite and discrete charlier meixner kravchuk hahn orthogonal polynomials are well known and can be described by relations of type lim pn x qn x deeper information on these limiting processes can be obtained from the expansion p n x k 0 r k x n k in this paper a method for the recursive computation of coefficients rk x n is designed being the main tool the consideration of a closely related connection problem which can be solved also recurrently by using an algorithm recently developed by the authors

fast accurate and small scale direct trajectory optimization using a gegenbauer transcription method this paper reports a novel direct gegenbauer ultraspherical transcription method gtm for solving continuous time optimal control oc problems ctocps with linear nonlinear dynamics and path constraints in elgindy et al 2012 1 we presented a gtm for solving nonlinear ctocps directly for the state and the control variables and the method was tailored to find the best path for an unmanned aerial vehicle mobilizing in a stationary risk environment this article extends the gtm to deal further with problems including higher order time derivatives of the states by solving the ctocp directly for the control u t and the highest order time derivative x n t n z the state vector and its derivatives up to the n 1 th order derivative can then be stably recovered by successive integration moreover we present our solution method for solving linear quadratic regulator lqr problems as we aim to cover a wider collection of ctocps with the concrete aim of comparing the efficiency of the current work with other classical discretization methods in the literature the proposed numerical scheme fully parameterizes the state and the control variables using gegenbauer expansion series for problems with various order time derivatives of the state variables arising in the cost function dynamical system or path terminal constraints the gtm seeks to fully parameterize the control variables and the highest order time derivatives of the state variables the time horizon is mapped onto the closed interval 0 1 the dynamical system characterized by differential equations is transformed into its integral formulation through direct integration the resulting problem on the finite interval is then transcribed into a nonlinear programming nlp problem through collocation at the gegenbauer gauss gg points the integral operations are approximated by optimal gegenbauer quadratures in a certain optimality sense the reduced nlp problem is solved in the gegenbauer spectral space and the state and the control variables are approximated on the entire finite horizon the proposed method achieves discrete solutions exhibiting exponential convergence using relatively small scale number of collocation points the advantages of the proposed direct gtm over other traditional discretization methods are shown through four well studied oc test examples the present work is a major breakthrough in the area of computational oc theory as it delivers significantly more accurate solutions using considerably smaller numbers of collocation points states and controls expansion terms moreover the gtm produces very small scale nlp problems which can be solved very quickly using the modern nlp software

stability of galerkin and inertial algorithms with variable time step size this paper provides galerkin and inertial algorithms for solving a class of nonlinear evolution equations spatial discretization can be performed by either spectral or finite element methods time discretization is done by euler explicit or euler semi implicit difference schemes with variable time step size moreover the boundedness and stability of these algorithms are studied by comparison we find that the boundedness and stability of inertial algorithm are superior to the ones of galerkin algorithm in the case of explicit scheme and the boundedness and stability of two algorithms are same in the case of semi implicit scheme

connection problems for polynomial solutions of nonhomogeneous differential and difference equations abstract we consider nonhomogeneous hypergeometric type differential difference and q difference equations whose nonhomogeneity is a polynomial q n x the polynomial solution of these problems is expanded in the q n x basis and also in a basis p n x related in a natural way with the homogeneous hypergeometric equation we give an algorithm building a recurrence relation for the expansion coefficients in both bases that we solve explicitly in many cases involving classical orthogonal polynomials finally some concrete applications and extensions are given

joint models for efficient estimation in proportional hazards regression models in survival studies information lost through censoring can be partially recaptured through repeated measures data which are predictive of survival in addition such data may be useful in removing bias in survival estimates due to censoring which depends upon the repeated measures here we investigate joint models for survival t and repeated measurements given a vector of covariates mixture models indexed as f t f t are well suited for assessing covariate effects on survival time our objective is efficiency gains using non parametric models for in order to avoid introducing bias by misspecification of the distribution for we model t as a piecewise exponential distribution with proportional hazards covariate effect the component t has a multinomial model the joint likelihood for survival and longitudinal data is maximized using the em algorithm the estimate of covariate effect is compared to the estimate based on the standard proportional hazards model and an alternative joint model based estimate we demonstrate modest gains in efficiency when using the joint piecewise exponential joint model in a simulation the estimated efficiency gain over the standard proportional hazards model is 6 4 per cent in clinical trial data the estimated efficiency gain over the standard proportional hazards model is 10 2 per cent copyright 2003 john wiley sons ltd

hazard regression with interval censored data in a recent paper kooperberg stone and truong 1995a introduced hazard regression hare in which linear splines and their tensor products are used to estimate the conditional log hazard function based on possibly censored positive response data and one or more covariates model selection is carried out in an adaptive fashion using maximum likelihood estimation of the unknown coefficients rao and wald statistics to carry out stepwise addition and deletion of basis functions and the bayesian information criterion bic to select the final model in the present paper the hare methodology is extended to accommodate interval censored data time dependent covariates and cubic splines the presence of interval censored data means that the log likelihood function may no longer be concave presenting additional numerical challenges the extended methodology is applied to a data set containing both interval censoring and time dependent covariates the new software will be available in a future release of s plus

on the coefficients of differentiated expansions and derivatives of jacobi polynomials a formula expressing explicitly the derivatives of jacobi polynomials of any degree and for any order in terms of the jacobi polynomials themselves is proved another explicit formula which expresses the jacobi expansion coefficients of a general order derivative of an infinitely differentiable function in terms of its original jacobi coefficients is also given the results for the special case of ultraspherical polynomials are considered the results for chebyshev polynomials of the first and second kinds and for legendre polynomials are also noted an application of how to use jacobi polynomials for solving ordinary and partial differential equations is described

recurrence relations for connection coefficients between two families of orthogonal polynomials we describe a simple approach in order to build recursively the connection coefficients between two families of orthogonal polynomial solutions of second and fourth order differential equations

factoring polynomials over finite fields using differential equations and normal bases the deterministic factorization algorithm for polynomials over finite fields that was recently introduced by the author is based on a new type of linearization of the factorization problem the main ingredients are differential equations in rational function fields and normal bases of field extensions for finite fields of characteristic 2 it is known that this algorithm has several advantages over the classical berlekamp algorithm we develop the algorithm in a general framework and we show that it is feasible for arbitrary finite fields in the sense that the linearization can be achieved in polynomial time

mappings and accuracy for chebyshev pseudo spectral approximations abstract the effect of mappings on the approximation by chebyshev collocation of functions which exhibit localized regions of rapid variation is studied a general strategy is introduced whereby mappings are adaptively constructed which map specified classes of rapidly varying functions into low order polynomials a particular family of mappings constructed in this way is tested on a variety of rapidly varying functions similar to those occurring in applications it is shown that the mapped function can be approximated much more accurately by chebyshev polynomial approximations than the original function the effect on the approximation of introducing subdomains is also studied the accuracy of the pseudo spectral approximation is very sensitive to the location of the interface although this sensitivity is reduced when mappings are employed within the subdomains

on the construction of recurrence relations for the expansion and connection coefficients in series of jacobi polynomials formulae expressing explicitly the jacobi coefficients of a general order derivative integral of an infinitely differentiable function in terms of its original expansion coefficients and formulae for the derivatives integrals of jacobi polynomials in terms of jacobi polynomials themselves are stated a formula for the jacobi coefficients of the moments of one single jacobi polynomial of certain degree is proved another formula for the jacobi coefficients of the moments of a general order derivative of an infinitely differentiable function in terms of its original expanded coefficients is also given a simple approach in order to construct and solve recursively for the connection coefficients between jacobi jacobi polynomials is described explicit formulae for these coefficients between ultraspherical and jacobi polynomials are deduced of which the chebyshev polynomials of the first and second kinds and legendre polynomials are important special cases two analytical formulae for the connection coefficients between laguerre jacobi and hermite jacobi are developed

variate generation for accelerated life and proportional hazards models we use accelerated life and proportional hazards lifetime models to account for the effects of covariates on a random lifetime we find that variate generation algorithms for monte carlo simulation in both the renewal and nonhomogeneous poisson process cases are a simple extension of the inverse cumulative distribution function cdf technique

analysis of time dependent covariates in a regressive relative survival model relative survival is a method for assessing prognostic factors for disease specific mortality however most relative survival models assume that the effect of covariate on disease specific mortality is fixed in time which may not hold in some studies and requires adapted modelling we propose an extension of the esteve et al regressive relative survival model that uses the counting process approach to accommodate time dependent effect of a predictor s on disease specific mortality this approach had shown its robustness and the properties of the counting process give a simple and attractive computational solution to model time dependent covariates our approach is illustrated with the data from the stanford heart transplant study and with data from a hospital based study on invasive breast cancer advantages of modelling time dependent covariates in relative survival analysis are discussed

piecewise exponential models for survival data with covariates a general class of models for analysis of censored survival data with covariates is considered if n individuals are observed over a time period divided into i n intervals it is assumed that lambda j t the hazard rate function of the time to failure of the individual j is constant and equal to lambda ij 0 on the i th interval and that the vector ell log lambda ij j 1 ldots n i 1 ldots i n lies in a linear subspace the maximum likelihood estimate hat ell of ell provides a simultaneous estimate of the underlying hazard rate function and of the effects of the covariates maximum likelihood equations and conditions for existence of hat ell are given the asymptotic properties of linear functionals of hat ell are studied in the general case where the true hazard rate function lambda 0 t is not a step function and i n increases without bound as the maximum interval length decreases in comparison with recent work on regression analysis of survival data the asymptotic results are obtained under more relaxed conditions on the regression variables

primitive normal polynomials over finite fields in this note we significantly extend the range of published tables of primitive normal polynomials over finite fields for each pn 1050 with p 97 we provide a primitive normal polynomial of degree n over fp moreover each polynomial has the minimal number of nonzero coefficients among all primitive normal polynomials of degree n over fp the roots of such a polynomial generate a primitive normal basis of fpn over fp and so are of importance in many computational problems we also raise several conjectures concerning the distribution of such primitive normal polynomials including a refinement of the primitive normal basis theorem

jacobi spectral galerkin method for the integrated forms of second order differential equations abstract efficiently direct solvers based on the jacobi galerkin method for the integrated forms of second order elliptic equations in one and two space variables are presented they are based on appropriate base functions for the galerkin formulation which lead to discrete systems with specially structured matrices that can be efficiently inverted the homogeneous dirichlet boundary conditions are satisfied exactly by expanding the unknown variable into a polynomial basis of functions which are built upon the jacobi polynomials the direct solution algorithm here developed for the homogeneous dirichlet problem in two dimensions relies upon a diagonalization process numerical results indicating the high accuracy and effectiveness of these algorithms are presented

an accurate solution of the poisson equation by the chebyshev tau method abstract a new tau method is presented for the two dimensional poisson equation comparison of the results for the test problem u x y sin 4 x sin 4 y with those computed by haidvogel and zang using the matrix diagonalization method and dang vu and delcarte using the chebyshev collocation method indicates that our method would be more accurate

high order gegenbauer integral spectral element method integrated with an adaptive chebyshev optimization strategy for solving linear singularly perturbed differential equations abstract we propose a high order adaptive spectral element method based on gegenbauer polynomial basis for solving singularly perturbed differential equations with dirichlet boundary conditions and a boundary layer at one end the method reduces the singularly perturbed boundary value problem into a single system of linear algebraic equations when the solution is smooth on the solution domain or two systems of linear algebraic equations when the solution has a thin boundary layer the core of the new adaptive strategy lies in constructing a mathematical model of the last few gegenbauer coefficients in the gegenbauer truncated series approximating the second order derivative of the solution which can be readily converted into a minimization problem in a least squares sense and solved using fast optimization techniques an error and convergence analysis of the method for sufficiently smooth solutions is presented a numerical test example with various parameter settings is presented to verify the accuracy effectiveness and applicability of the proposed method numerical comparisons with other rival methods in the literature are included to demonstrate further the power of the proposed method

minimal recurrence relations for connection coefficients between classical orthogonal polynomials discrete case abstract we present a simple approach in order to compute recursively the connection coefficients between two families of classical discrete orthogonal polynomials charlier meixner kravchuk hahn i e the coefficients cm n in the expression p n x n m 0 c m n q m x where pn x and qm x belong to the aforementioned class of polynomials this is scv2 done by adapting a general and systematic algorithm recently developed by the authors to the discrete classical situation moreover extensions of this method allow to give new addition formulae and to estimate cm n asymptotics in limit relations between some families

expansions in series of orthogonal hypergeometric polynomials abstract let us consider an arbitrary hypergeometric polynomial qj x and a set of orthogonal hypergeometric polynomials p n x in the domain of orthogonality here the expansion coefficients of xm and xmqj x m n0 in series of the set p n x are found in terms of the polynomials x and x characterizing the second order differential equations satisfied by the involved hypergeometric polynomials the resulting general expressions which are given in an explicit and compact form are used to produce known for checking and unknown expansions for various concrete classical orthogonal polynomials

bivariate krawtchouk polynomials in this paper we present a recurrent procedure to solve an inversion problem for monic bivariate krawtchouk polynomials written in vector column form giving its solution explicitly as a by product a general connection problem between two vector column of monic bivariate krawtchouk families is also explicitly solved moreover in the non monic case and also for krawtchouk families several expansion formulas are given but for polynomials written in scalar form

transverse limits in the askey tableau limit relations between classical discrete charlier meixner kravchuk hahn to classical continuous jacobi laguerre hcrmite orthogonal polynomials to be called transverse limits are nicely presented in the askey tableau and can be described by relations of type lim 0 p n x s q n s where p n s and q n s stand for the discrete and continuous family respectively deeper information on these limiting processes can be obtained by expanding the discrete polynomial family as p n x s y k r k s n in this paper a method for computing the coefficients r k s n is designed the main tool being the consideration of some closely related connection and inversion problems

a note on the chebyshev coefficients of the general order derivative of an infinitely differentiable function abstract a formula expressing the chebyshev coefficients of the general order derivative of an infinitely differentiable function in terms of its chebyshev coefficients is suggested

data generation for the cox proportional hazards model with time dependent covariates a method for medical researchers the proliferation of longitudinal studies has increased the importance of statistical methods for time to event data that can incorporate time dependent covariates the cox proportional hazards model is one such method that is widely used as more extensions of the cox model with time dependent covariates are developed simulations studies will grow in importance as well an essential starting point for simulation studies of time to event models is the ability to produce simulated survival times from a known data generating process this paper develops a method for the generation of survival times that follow a cox proportional hazards model with time dependent covariates the method presented relies on a simple transformation of random variables generated according to a truncated piecewise exponential distribution and allows practitioners great flexibility and control over both the number of time dependent covariates and the number of time periods in the duration of follow up measurement within this framework an additional argument is suggested that allows researchers to generate time to event data in which covariates change at integer valued steps of the time scale the purpose of this approach is to produce data for simulation experiments that mimic the types of data structures applied that researchers encounter when using longitudinal biomedical data validity is assessed in a set of simulation experiments and results indicate that the proposed procedure performs well in producing data that conform to the assumptions of the cox proportional hazards model

generating survival times to simulate cox proportional hazards models with time varying covariates simulations and monte carlo methods serve an important role in modern statistical research they allow for an examination of the performance of statistical procedures in settings in which analytic and mathematical derivations may not be feasible a key element in any statistical simulation is the existence of an appropriate data generating process one must be able to simulate data from a specified statistical model we describe data generating processes for the cox proportional hazards model with time varying covariates when event times follow an exponential weibull or gompertz distribution we consider three types of time varying covariates first a dichotomous time varying covariate that can change at most once from untreated to treated e g organ transplant second a continuous time varying covariate such as cumulative exposure at a constant dose to radiation or to a pharmaceutical agent used for a chronic condition third a dichotomous time varying covariate with a subject being able to move repeatedly between treatment states e g current compliance or use of a medication in each setting we derive closed form expressions that allow one to simulate survival times so that survival times are related to a vector of fixed or time invariant covariates and to a single time varying covariate we illustrate the utility of our closed form expressions for simulating event times by using monte carlo simulations to estimate the statistical power to detect as statistically significant the effect of different types of binary time varying covariates this is compared with the statistical power to detect as statistically significant a binary time invariant covariate copyright 2012 john wiley sons ltd

results for some inversion problems for classical continuous and discrete orthogonal polynomials explicit expressions for the coefficients in the expansion of classical discrete orthogonal polynomials charlier meixner krawtchouck hahn hahn eberlein into the falling factorial basis are given the corresponding inversion problems are solved explicitly this is done by using a general algorithm recently developed by the authors which is also applied to this kind of inversion problem but relating the basis and the classical continuous orthogonal polynomials of jacobi laguerre hermite and bessel

chebyshev expansion method for solving second and fourth order elliptic equations in this paper we introduce a new spectral method based on chebyshev polynomials for solving second and fourth order elliptic equations moreover the suggested method is applicable for a wide area of differential equations an explicit formula for the chebyshev polynomials in terms of arbitrary order of their derivatives is presented also a formula for the successive integration of chebyshev polynomials in terms of chebyshev polynomials is proved numerical results indicate that the suggested method is significantly more accurate than that based on the chebyshev tau method the present results are in satisfactory agreement with the exact solutions

representations of orthogonal polynomials abstract zeilberger s algorithm provides a method to compute recurrence and differential equations from given hypergeometric series representations and an adaption of almquist and zeilberger computers recurrence and differential equations for hyperexponential integrals further versions of this algorithm allow the computation of recurrence and differential equations from rodrigues type formulas and from generating functions in particular these algorithms can be used to compute the differential difference and recurrence equations for the classical continuous and discrete orthogonal polynomials from their hypergeometric representations and from their rodrigues representations and generating functions in recent work we used an explicit formula for the recurrence equation of families of classical continuous and discrete orthogonal polynomials in terms of the coefficients of their differential difference equations to give an algorithm to identify the polynomial system from a given recurrence equation in this article we extend these results by presenting a collection of algorithms with which any of the conversions between the differential difference equation the hypergeometric representation and the recurrence equation is possible the main technique is again to use explicit formulas for structural identities of the given polynomial systems

the ultraspherical coefficients of the moments of a general order derivative of an infinitely differentiable function abstract a formula for the ultraspherical coefficients of the moments of one single ultraspherical polynomial of certain degree is given formulae for the ultraspherical coefficients of the moments of a general order derivative of an infinitely differentiable function in terms of its ultraspherical coefficients are also obtained the corresponding formulae for the important special cases of chebyshev polynomials of the first and second kinds and of legendre polynomials are deduced two interesting numerical applications of how to use these formulae for solving ordinary differential equations with varying coefficients by reducing them to recurrence relations of lowest order in the ultraspherical expansion coefficients in the sense of lewanowicz are discussed comparisons with the results obtained by optimal algorithm of lewanowicz 1976 are also made

primitive polynomials over finite fields in this note we extend the range of previously published tables of primitive polynomials over finite fields for each p 1050 with p 97 we provide a primitive polynomial of degree n over fp moreover each polynomial has the minimal number of nonzero coefficients among all primitives of degree n over fp

joint estimation of time dependent and non linear effects of continuous covariates on survival in order to yield more flexible models the cox regression model lambda t x lambda 0 t exp betax has been generalized using different non parametric model estimation techniques one generalization is the relaxation of log linearity in x lambda t x lambda 0 t exp r x another is the relaxation of the proportional hazards assumption lambda t x lambda 0 t exp beta t x these generalizations are typically considered independently of each other we propose the product model lambda t x lambda 0 t exp beta t r x which allows for joint estimation of both effects and investigate its properties the functions describing the time dependent beta t and non linear r x effects are modelled simultaneously using regression splines and estimated by maximum partial likelihood likelihood ratio tests are proposed to compare alternative models simulations indicate that both the recovery of the shapes of the two functions and the size of the tests are reasonably accurate provided they are based on the correct model by contrast type i error rates may be highly inflated and the estimates considerably biased if the model is misspecified applications in cancer epidemiology illustrate how the product model may yield new insights about the role of prognostic factors

proportional hazards tests and diagnostics based on weighted residuals summary nonproportional hazards can often be expressed by extending the cox model to include time varying coefficients e g for a single covariate the hazard function for subject i is modelled as exp fl t zi t a common example is a treatment effect that decreases with time we show that the function 3 t can be directly visualized by smoothing an appropriate residual plot also many tests of proportional hazards including those of cox 1972 gill schumacher 1987 harrell 1986 lin 1991 moreau o quigley mesbah 1985 nagelkerke oosting hart 1984 o quigley pessione 1989 schoenfeld 1980 and wei 1984 are related to time weighted score tests of the proportional hazards hypothesis and can be visualized as a weighted least squares line fitted to the residual plot

recurrences and explicit formulae for the expansion and connection coefficients in series of bessel polynomials a formula expressing explicitly the derivatives of bessel polynomials of any degree and for any order in terms of the bessel polynomials themselves is proved another explicit formula which expresses the bessel expansion coefficients of a general order derivative of an infinitely differentiable function in terms of its original bessel coefficients is also given a formula for the bessel coefficients of the moments of one single bessel polynomial of certain degree is proved a formula for the bessel coefficients of the moments of a general order derivative of an infinitely differentiable function in terms of its bessel coefficients is also obtained application of these formulae for solving ordinary differential equations with varying coefficients by reducing them to recurrence relations in the expansion coefficients of the solution is explained an algebraic symbolic approach using mathematica in order to build and solve recursively for the connection coefficients between bessel bessel polynomials is described an explicit formula for these coefficients between jacobi and bessel polynomials is given of which the ultraspherical polynomial and its consequences are important special cases two analytical formulae for the connection coefficients between laguerre bessel and hermite bessel are also developed

recurrence relations for connection coefficients between q orthogonal polynomials of discrete variables in the non uniform lattice x s q2s we obtain the structure relations for q orthogonal polynomials in the exponential lattice and from these we construct the recurrence relation for the connection coefficients between two families of polynomials belonging to the classical class of discrete q orthogonal polynomials an explicit example is also given

martingale based residuals for survival models summary graphical methods based on the analysis of residuals are considered for the setting of the highly used cox 1972 regression model and for the andersen gill 1982 generalization of that model we start with a class of martingale based residuals as proposed by barlow prentice 1988 these residuals and or their transforms are useful for investigating the functional form of a covariate the proportional hazards assumption the leverage of each subject upon the estimates of 13 and the lack of model fit to a given subject 1 1 model consider a set of n independent subjects such that the counting process ni ni t t o for the ith subject in the set indicates the number of observed events experienced over time t the sample paths of the ni are step functions with jumps of size 1 and with ni 0 0 we assume that the intensity function for ni t is given by yi t da t zi t yi t ep z t dao t 1 where yi t is a 0 1 process indicating whether the ith subject is a risk at time t 13 is a vector of regression coefficients zi t is a p dimensional vector of covariate processes and ao is the baseline cumulative hazard function several familar survival models fit into this framework the andersen gill 1982 generalization of the cox 1972 model arises when ao t is completely unspecified the further restriction that yi t 1 until the first event or censoring and is 0 thereafter yields the cox model the parametric form ao t t yields a poisson model or an exponential if restricted to a single event per subject and ao t tp a weibull model our attention will focus primarily on the andersen gill and cox models however the

efficient spectral petrov galerkin methods for the integrated forms of third and fifth order elliptic differential equations using general parameters generalized jacobi polynomials abstract this article analyzes some algorithms for solving numerically the integrated forms of third and fifth order differential equations in one variable subject to homogenous and nonhomogeneous boundary conditions using a dual petrov galerkin method two new families of general parameters generalized jacobi polynomials are introduced and used for this purpose numerical results indicating the high accuracy and effectiveness of the proposed algorithms are presented

evaluating non analytic functions of matrices abstract the paper revisits the classical problem of evaluating f a for a real function f and a matrix a with real spectrum the evaluation is based on expanding f in chebyshev polynomials and the focus of the paper is to study the convergence rates of these expansions in particular we derive bounds on the convergence rates which reveal the relation between the smoothness of f and the diagonalizability of the matrix a we present several numerical examples to illustrate our analysis

an empirical comparison of statistical tests for assessing the proportional hazards assumption of cox s model in the analysis of survival data using the cox proportional hazard ph model it is important to verify that the explanatory variables analysed satisfy the proportional hazard assumption of the model this paper presents results of a simulation study that compares five test statistics to check the proportional hazard assumption of cox s model the test statistics were evaluated under proportional hazards and the following types of departures from the proportional hazard assumption increasing relative hazards decreasing relative hazards crossing hazards diverging hazards and non monotonic hazards the test statistics compared include those based on partitioning of failure time and those that do not require partitioning of failure time the simulation results demonstrate that the time dependent covariate test the weighted residuals score test and the linear correlation test have equally good power for detection of non proportionality in the varieties of non proportional hazards studied using illustrative data from the literature these test statistics performed similarly

accurate spectral solutions for the parabolic and elliptic partial differential equations by the ultraspherical tau method we present a double ultraspherical spectral methods that allow the efficient approximate solution for the parabolic partial differential equations in a square subject to the most general inhomogeneous mixed boundary conditions the differential equations with their boundary and initial conditions are reduced to systems of ordinary differential equations for the time dependent expansion coefficients these systems are greatly simplified by using tensor matrix algebra and are solved by using the step by step method numerical applications of how to use these methods are described numerical results obtained compare favorably with those of the analytical solutions accurate double ultraspherical spectral approximations for poisson s and helmholtz s equations are also noted numerical experiments show that spectral approximation based on chebyshev polynomials of the first kind is not always better than others based on ultraspherical polynomials

efficient algorithms for construction of recurrence relations for the expansion and connection coefficients in series of al salam carlitz i polynomials two formulae expressing explicitly the derivatives and moments of al salam carlitz i polynomials of any degree and for any order in terms of al salam carlitz i themselves are proved two other formulae for the expansion coefficients of general order derivatives d p f x and for the moments xd p f x of an arbitrary function f x in terms of its original expansion coefficients are also obtained application of these formulae for solving q difference equations with varying coefficients by reducing them to recurrence relations in the expansion coefficients of the solution is explained an algebraic symbolic approach using mathematica in order to build and solve recursively for the connection coefficients between al salam carlitz i polynomials and any system of basic hypergeometric orthogonal polynomials belonging to the q hahn class is described

primitive normal bases for finite fields it is proved that any finite extension of a finite field has a normal basis consisting of primitive roots introduction let q be a prime power q 1 we denote by f9 a finite field of q elements it is well known that for every positive integer m there exists a normal basis of f m over f9 i e a basis of the form q q 2 qp aaq aq2 a9 1 with a e f m it is also well known that the multiplicative group f of fqm is cyclic i e that for some a e pq we have f an n e z such an element a is called a primitive root of fqm following davenport 4 we call qq2 rn i a normal basis a a a9 a a a9 of fq over fq a primitive normal basis if a is a primitive root of fq carlitz 2 3 proved in 1952 that for all sufficiently large qm there exists a primitive normal basis of f m over fq davenport 4 proved in 1968 that a primitive normal basis exists for all m if q is prime in the present paper this result is extended to the general case theorem for every prime power q 1 and every positive integer m there exists a primitive normal basis of fqm over fq section 1 contains an exposition of certain results due to ore 7 concerning the galois module structure of finite fields these lead to an alternative formulation of the theorem in section 2 we describe an improved version of the method of carlitz and davenport which handles all but finitely many pairs q m in section 3 we determine which are the remaining pairs and they are dealt with in section 4 we denote the cardinality of a set s by s and the group of units of a ring r with 1 by r if f g are polynomials in one variable we mean by g i f that g divides f and is monic i e has leading coefficient one the same notation for divisibility is used for positive integers received march 5 1986 1980 mathematics subject classification 1985 revision primary 11t30 12e20

efficient spectral galerkin algorithms for direct solution of second order equations using ultraspherical polynomials it is well known that spectral methods tau galerkin collocation have a condition number of o n4 n is the number of retained modes of polynomial approximations this paper presents some efficient spectral algorithms which have a condition number of o n2 based on the ultraspherical galerkin methods for second order elliptic equations in one and two space variables the key to the efficiency of these algorithms is to construct appropriate base functions which lead to systems with specially structured matrices that can be efficiently inverted the complexities of the algorithms are a small multiple of nd 1 operations for a d dimensional domain with n 1 d unknowns while the convergence rates of the algorithms are exponential for problems with smooth solutions

an adaptive pseudo spectral method for reaction diffusion problems the spectral interpolation error was considered for both the chebyshev pseudo spectral and galerkin approximations a family of functionals i sub r u with the property that the maximum norm of the error is bounded by i sub r u j sub r where r is an integer and j is the degree of the polynomial approximation was developed these functionals are used in the adaptive procedure whereby the problem is dynamically transformed to minimize i sub r u the number of collocation points is then chosen to maintain a prescribed error bound the method is illustrated by various examples from combustion problems in one and two dimensions

transformations of matrices into banded form abstract a class of full or triangular matrices r is described for which there exist banded matrices b such that the product br is also banded the banded matrices yield recursion relations for solving systems of linear equations examples of such matrices arising from second derivative operators acting on orthogonal function expansions are used to illustrate the main theorem and its application practical considerations in efficient implementation are discussed

gegenbauer approximation and its applications to differential equations on the whole line abstract a gegenbauer approximation is discussed several imbedding inequalities and inverse inequalities are obtained some approximation results are given by variable transformation differential equations on the whole line are changed to certain equations on a finite interval gegenbauer polynomials are used for their numerical solutions the stabilities and convergences of proposed schemes are proved the main idea and techniques used in this paper are also applicable to other multiple dimensional problems in unbounded domains

solving connection and linearization problems within the askey scheme and its q analogue via inversion formulas for the polynomial families pn x n belonging to the askey scheme or to its q analogue the hypergeometric represen tation provides a natural expansion of the form pn x n 0 dm n m x where the expanding basism x is in general a product of pochhammer symbols or q shifted factorials in this paper we solve the corresponding inversion problem i e we compute the coe6cients im n in the expansionn x n m 0 im n pm x which are then used as a tool for solving any connection and linearization problem within the askey scheme and its q analogue extensions of this approach for polynomials outside these two schemes are also given c

optimal gegenbauer quadrature over arbitrary integration nodes this paper treats definite integrations numerically using gegenbauer quadratures the novel numerical scheme introduces the idea of exploiting the strengths of the chebyshev legendre and gegenbauer polynomials through a unified approach and using a unique numerical quadrature in particular the developed numerical scheme employs the gegenbauer polynomials to achieve rapid rates of convergence of the quadrature for the small range of the spectral expansion terms for a large scale number of expansion terms the numerical quadrature has the advantage of converging to the optimal chebyshev and legendre quadratures in the l norm and l 2 norm respectively the key idea is to construct the gegenbauer quadrature through discretizations at some optimal sets of points of the gegenbauer gauss gg type in a certain optimality sense we show that the gegenbauer polynomial expansions can produce higher order approximations to the definite integrals 1 x if x dx of a smooth function f x c 1 1 for the small range by minimizing the quadrature error at each integration point x i through a pointwise approach the developed gegenbauer quadrature can be applied for approximating integrals with any arbitrary sets of integration nodes exact integrations are obtained for polynomials of any arbitrary degree n if the number of columns in the developed gegenbauer integration matrix gim is greater than or equal to n the error formula for the gegenbauer quadrature is derived moreover a study on the error bounds and the convergence rate shows that the optimal gegenbauer quadrature exhibits very rapid convergence rates faster than any finite power of the number of gegenbauer expansion terms two efficient computational algorithms are presented for optimally constructing the gegenbauer quadrature we illustrate the high order approximations of the optimal gegenbauer quadrature through extensive numerical experiments including comparisons with conventional chebyshev legendre and gegenbauer polynomial expansion methods the present method is broadly applicable and represents a strong addition to the arsenal of numerical quadrature methods

on the coefficients of integrated expansions of ultraspherical polynomials the problem of expressing the coefficients of an expansion of orthogonal polynomials that has been integrated an arbitrary number of times in terms of the coefficients of the original expansion is considered a formula is proved for the ultraspherical polynomials of which the chebyshev and legendre polynomials are important special cases

cox regression analysis of multivariate failure time data the marginal approach multivariate failure time data are commonly encountered in scientific investigations because each study subject may experience multiple events or because there exists clustering of subjects such that failure times within the same cluster are correlated in this paper i present a general methodology for analysing such data which is analogous to that of liang and zeger for longitudinal data analysis this approach formulates the marginal distributions of multivariate failure times with the familiar cox proportional hazards models while leaving the nature of dependence among related failure times completely unspecified the baseline hazard functions for the marginal models may be identical or different simple estimating equations for the regression parameters are developed which yield consistent and asymptotically normal estimators and robust variance covarinace estimators are constructed to account for the intra class correlation simulation results demonstrate that the large sample approximations are adequate for practical use and that ignoring the intra class correlation could yield rather misleading variance estimators the proposed methodology has been fully implemented in a simple computer program which also incorporates several alternative approaches detailed illustrations with data from four clinical or epidemiologic studies are provided