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

uniform characterizations of non uniform complexity measures non uniform complexity measures originated in automata and formal languages theory are characterized in terms of well known uniform complexity classes the initial index of languages is introduced by means of several computational models it is shown to be closely related to context free cost boolean circuits straight line programs and turing machines with sparse oracles and time or space bounds

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

fourth order difference equation for the first associated of classical discrete orthogonal polynomials we derive the fourth order difference equation satisfied by the first associated of classical orthogonal polynomials of a discrete variable we give it explicitly for first associated of hahn polynomials from which can be derived by a limiting process the equation satisfied by first associated of all classical families continuous and discrete

backlund transformations for nonlinear parabolic equations the general results backlund transformations bts are considered for nonlinear parabolic equations of the form u t u xx h u x u x t 0 the most general form of both the bt and the class is adopted and we show that the only nonlinear equations possessing bts are the slight generalizations of burgers equation obtained by adding a forcing term u t u xx 2 uu x a x t 0

the askey scheme for hypergeometric orthogonal polynomials viewed from asymptotic analysis many limits are known for hypergeometric orthogonal polynomials that occur in the askey scheme we show how asymptotic representations can be derived by using the generating functions of the polynomials for example we discuss the asymptotic representation of the meixner pollaczek jacobi meixner and krawtchouk polynomials in terms of laguerre polynomials

on the hopcroft s minimization technique for dfa and dfca we show that the absolute worst case time complexity for hopcroft s minimization algorithm applied to unary languages is reached only for deterministic automata or cover automata following the structure of the de bruijn words a previous paper by berstel and carton gave the example of de bruijn words as a language that requires o nlogn steps in the case of deterministic automata by carefully choosing the splitting sets and processing these sets in a fifo mode for the list of the splitting sets in the algorithm we refine the previous result by showing that the berstel carton example is actually the absolute worst case time complexity in the case of unary languages for deterministic automata we show that the same result is valid also for the case of cover automata and an algorithm based on the hopcroft s method used for minimization of cover automata we also show that a lifo implementation for the splitting list will not achieve the same absolute worst time complexity for the case of unary languages both in the case of regular deterministic finite automata or in the case of the deterministic finite cover automata as defined by s yu

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 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

recurrence relations for the connection coefficients of orthogonal polynomials of a discrete variable we give explicitly recurrence relations satisfied by the connection coefficients between two families of the classical orthogonal polynomials of a discrete variable i e associated with the names of charlier meixner krawtchouk and hahn also a recurrence relation is given for the coefficients in the formula expressing the nth associated polynomial in terms of the original polynomials

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

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

expansion of multivariable polynomials in products of orthogonal polynomials in one variable we propose an approach to develop multivariable polynomials in multiple series of orthogonal polynomials in one variable the action of a partial differential operator on a series of products of classical orthogonal polynomials is first analyzed and permits the generation of recurrence relations for the expansion coefficients like in the one variable case polynomial solutions of linear partial differential equations pdes with polynomial coefficients are also examined giving new results on harmonic polynomials based on the appell property of hermite polynomials

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

results on the associated classical orthogonal polynomials abstract let p k x be any system of the classical orthogonal polynomials and let p k x c be the corresponding associated polynomials of order c c n second order recurrence relation in k is given for the connection coefficient a n 1 k c in p n 1 x c k 0 n 1 a n 1 k c p k x this result is obtained thanks to a new explicit form of the fourth order differential equation satisfied by p n 1 c

algorithms for the integration and derivation of chebyshev series general formulas for the mth integral and derivative of a chebyshev polynomial of the first or second kind are presented the result is expressed as a finite series of the same kind of chebyshev polynomials these formulas permit to accelerate the determination of such integrals or derivatives besides it is presented formulas for the mth integral and derivative of finite chebyshev series and a numerical algorithm for the direct evaluation of the mth derivative of such a series

the spectral connection matrix for classical orthogonal polynomials of a single parameter abstract in this paper we study the so called connection problem of given a polynomial expressed in the basis of one set of orthogonal polynomials computing the coefficients with respect to a different set of orthogonal polynomials we restrict our current study to the classical real orthogonal polynomials of the hermite laguerre and gegenbauer including legendre families the computational tool for this work is the class of quasiseparable matrices while the relationships between orthogonal polynomials and rank structured matrices such as quasiseparable matrices are very well known in this paper we investigate a more recently considered relationship we prove that while the matrix that implements the desired connection is not itself quasiseparable it is an eigenvector matrix of one that is quasiseparable we suggest to refer to this structured matrix as the spectral connection matrix finally we present a simple algorithm exploiting the computationally favorable properties of quasiseparable matrices to implement the desired change of basis by exploiting the quasiseparable structure this algorithm enjoys an order of magnitude reduction of complexity as compared to the simple method of inverting the connection matrix directly while not the focus of the paper some very preliminary numerical experimentation shows some positive indications that even with this reduction in complexity the accuracy of the resulting change of basis algorithm is comparable to that of inverting the connection matrix directly

on the numerical evaluation of linear recurrences we present some remarks on the numerical evaluation of recurrence relations rounding error bounds are presented of the numerical scheme and some numerical examples are given in particular we analyse conversion recurrences from different families of orthogonal polynomials the limit case of jacobi sobolev polynomials random recurrences and perturbed gegenbauer polynomials in all these examples the theoretical bounds give sharp relative rounding error estimations the parallel evaluation of recurrences are also considered and numerical tests on a cray t3d are presented

a class of backlund transformations for equations of the type uxy f u ux the backlund transformation u f u u x uy is considered for equations of the form uxy f u ux the nonlinear forms of f u ux that admit such transformations are completely classified new backlund transformations for modified sine gordon modified sinh gordon and modified liouville equations are obtained

more on deterministic and nondeterministic finite cover automata finite languages are an important sub regular language family which were intensively studied during the last two decades in particular from a descriptional complexity perspective an important contribution to the theory of finite languages are the deterministic and the recently introduced nondeterministic finite cover automata dfcas and nfcas respectively as an alternative representation of finite languages by ordinary finite automata we compare these two types of cover automata from a descriptional complexity point of view showing that these devices have a lot in common with ordinary finite automata in particular we study how to adapt lower bound techniques for nondeterministic finite automata to nfcas such as e g the biclique edge cover technique solving an open problem from the literature moreover the trade off of conversions between dfcas and nfcas as well as between finite cover automata and ordinary finite automata are investigated finally we present some results on the average size of finite cover automata

δ sobolev orthogonal polynomials of meixner type asymptotics and limit relation let qn x n be the sequence of monic polynomials orthogonal with respect to the sobolev type inner product p x r x s u0 p x r x u1 p x r x where 0 f x f x 1 f x denotes the forward difference operator and u0 u1 is a coherent pair of positive definite linear functionals being u1 the meixner linear functional in this paper relative asymptotics for the qn x n sequence with respect to meixner polynomials on compact subsets of c 0 is obtained this relative asymptotics is also given for the scaled polynomials in both cases we deduce the same asymptotics as we have for the self coherent pair that is when u0 u1 is the meixner linear functional furthermore we establish a limit relation between these orthogonal polynomials and the laguerre sobolev orthogonal polynomials which is analogous to the one existing between meixner and laguerre polynomials in the askey scheme

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

on a structure formula for classical q orthogonal polynomials the classical orthogonal polynomials are given as the polynomial solutions pn x of the di1erential equation x y x x y x ny x 0 where x turns out to be a polynomial of at most second degree and x is a polynomial of 5rst degree in a similar way the classical discrete orthogonal polynomials are the polynomial solutions of the di1erence equation x 7 y x x 7 y x ny x 0 where 7 y x y x 1 y x and y x y x y x 1 denote the forward and backward di1erence operators respectively finally the classical q orthogonal polynomials of the hahn tableau are the polynomial solutions of the q di1erence equation x dqd1 qy x x dqy x q ny x 0 where dqf x f qx f x q 1 x q denotes the q di1erence operator we show by a purely algebraic deduction without using the orthogonality of the families considered that a structure formula of the type x d1 qpn x npn 1 x npn x npn 1 x n n 1 2 3 is valid moreover our approach does not only prove this assertion but generates the form of this structure formula a similar argument applies to the discrete and continuous cases and yields x pn x npn 1 x npn x npn 1 x n n and x p x npn 1 x npn x npn 1 x n n

minimal cover automata for finite languages abstract a cover automaton a of a finite language l is a finite deterministic automaton dfa that accepts all words in l and possibly other words that are longer than any word in l a minimal deterministic finite cover automaton dfca of a finite language l usually has a smaller size than a minimal dfa that accept l thus cover automata can be used to reduce the size of the representations of finite languages in practice in this paper we describe an efficient algorithm that for a given dfa accepting a finite language constructs a minimal deterministic finite cover automaton of the language we also give algorithms for the boolean operations on deterministic cover automata i e on the finite languages they represent

applications of q taylor theorems we establish two new q analogues of a taylor series expansion for polynomials using special askey wilson polynomial bases combining these expansions with an earlier expansion theorem we derive inverse relations and evaluate certain linearization coefficients byproducts include new summation theorems new results on a q exponential function and quadratic transformations for q series

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

quick construction of recurrence relations for the jacobi coefficients abstract recurrence relations are given for the jacobi series coefficients of functions which satisfy linear differential equations of first to third order with polynomial coefficients

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

incremental construction of minimal deterministic finite cover automata we present a fast incremental algorithm for constructing minimal deterministic finite cover automata dfca for a given language since it was shown that the minimal dfca for a language l has less states than the minimal deterministic finite automata dfa for the same language l this technique seems to be the best choice for incrementally building the automaton for a large language especially when the number of states in the dfca is significantly less than the number of states in the corresponding minimal dfa we have implemented the proposed algorithm and have tested it against the best known dfca minimization technique

expansion of multivariate polynomials in products of univariate orthogonal polynomials discrete case we present a method to develop multivariate polynomials in multiple series of products of univariate classical orthogonal polynomials of a discrete variable the method is based on the ability to apply difference operators on such series in order to generate recurrence relations for the expansion coefficients like in the one variable case

dual birth and death processes and orthogonal polynomials we discuss connections between birth and death processes and orthogonal polynomials we survey certain spectral properties of orthogonal polynomials and spectra of birth and death processes birth and death processes with linear birth and death rates are studied in some detail we also mention some results concerning birth and death processes whose transition rates are rational functions

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

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

a sufficient condition to polynomially compute a minimum separating dfa the computation of a minimal separating automaton msa for regular languages has been studied from many different points of view from synthesis of automata or grammatical inference to the minimization of incompletely specified machines or compositional verification in the general case the problem is np complete but this drawback does not prevent the problem from having a real application in the above mentioned fields in this paper we propose a sufficient condition that guarantees that the computation of the msa can be carried out with polynomial time complexity

on limit relations between some families of bivariate hypergeometric orthogonal polynomials in this paper we deal with limit relations between bivariate hypergeometric polynomials we analyze the limit relation from trinomial distribution to bivariate gaussian distribution obtaining the limit transition from the second order partial difference equation satisfied by bivariate hypergeometric kravchuk polynomials to the second order partial differential equation verified by bivariate hypergeometric hermite polynomials as a consequence the limit relation between both families of orthogonal polynomials is established a similar analysis between bivariate hahn and bivariate appell orthogonal polynomials is also presented

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

on the connection and linearization problem for discrete hypergeometric q polynomials abstract in the present paper starting from the second order difference hypergeometric equation on the non uniform lattice x s satisfied by the set of discrete hypergeometric orthogonal q polynomials p n we find analytical expressions of the expansion coefficients of any q polynomial r m x s on x s and of the product r m x s q j x s 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 q polynomials

quasi orthogonality and differential equations abstract the following theorem is proved quasi orthogonal polynomials corresponding to the semi classical or laguerre hahn families and the orthogonal polynomials themselves verify differential equations of same order reducible to two or four by the hahn theorem

q taylor and interpolation series for jackson q difference operators abstract we prove q taylor series for jackson q difference operators absolute and uniform convergence to the original function are proved for analytic functions we derive interpolation results for entire functions of q exponential growth which is less than ln q 1 0 q 1 from its values at the nodes a q n n n a is a non zero complex number with absolute and uniform convergence criteria

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

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

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

minimal consistent dfa revisited we address the classic problem of polynomial computation of a minimal finite state representation compatible with two finite input sets that contain those strings that have to be accepted and rejected we extend the previously defined uniformly complete sample and prove that it is a special case of the sufficient condition we propose

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

on the compressibility of finite languages and formal proofs abstract we consider the minimal number of productions needed for a grammar to cover a finite language l as the grammatical complexity of l we study this measure for several types of word and tree grammars and show that it is closely connected to well known measures for the complexity of formal proofs in first order predicate logic we construct an incompressible sequence of finite word languages and transfer this and several other results about the complexity of word and tree languages to formal proofs

a basic class of symmetric orthogonal polynomials of a discrete variable abstract by using a generalization of sturm liouville problems in discrete spaces a basic class of symmetric orthogonal polynomials of a discrete variable with four free parameters which generalizes all classical discrete symmetric orthogonal polynomials is introduced the standard properties of these polynomials such as a second order difference equation an explicit form for the polynomials a three term recurrence relation and an orthogonality relation are presented it is shown that two hypergeometric orthogonal sequences with 20 different weight functions can be extracted from this class moreover moments corresponding to these weight functions can be explicitly computed finally a particular example containing all classical discrete symmetric orthogonal polynomials is studied in detail

recurrence equations and their classical orthogonal polynomial solutions the classical orthogonal polynomials are given as the polynomial solutions p n x of the differential email protected x y x email protected x y x email protected ny x 0 where s x is a polynomial of at most second degree and t x is a polynomial of first degree in this paper a general method to express the coefficients a n b n and c n of the recurrence equationp n 1 x a nx b n p n x c np n 1 x in terms of the given polynomials s x and t x is used to present an algorithm to determine the classical orthogonal polynomial solutions of any given holonomic three term recurrence equation i e a homogeneous linear three term recurrence equation with polynomial coefficients in a similar way classical discrete orthogonal polynomial solutions of holonomic three term recurrence equations can be determined by considering their corresponding difference email protected x email protected y x email protected x dy x email protected ny x 0 where dy x y x 1 y x and y x y x y x 1 denote the forward and backward difference operators respectively and a similar approach applies to classical q orthogonal polynomials being solutions of the q difference email protected x d qd 1 qy x email protected x d qy x email protected q ny x 0 whered qf x f qx f x q 1 x q 1 denotes the q difference operator

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

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

on linearization and connection coefficients for generalized hermite polynomials abstract we consider the problem of finding explicit formulas recurrence relations and sign properties for both connection and linearization coefficients for generalized hermite polynomials most of the computations are carried out by the computer algebra system maple using appropriate algorithms

testing based on identifiable p systems using cover automata and x machines we represent a significant advance on the issue of testing for implementations specified by p systems with transformation and communicating rules using the x machine framework and the concept of cover automaton we devise a testing approach for such systems that under well defined conditions our approach ensures that the implementation conforms to the specification we investigate the issue of identifiability for p systems that is an essential prerequisite for testing these systems and establishes a fundamental set of properties for identifiable p systems this paper represents a significant advance on the issue of testing for implementations specified by p systems with transformation and communicating rules using the x machine framework and the concept of cover automaton it devises a testing approach for such systems that under well defined conditions it ensures that the implementation conforms to the specification it also investigates the issue of identifiability for p systems that is an essential prerequisite for testing implementations based on such specifications and establishes a fundamental set of properties for identifiable p systems

asymptotic relations between the hahn type polynomials and meixner pollaczek jacobi meixner and krawtchouk polynomials it has been shown in ferreira et al asymptotic relations in the askey scheme for hypergeometric orthogonal polynomials adv in appl math 31 1 2003 61 85 lopez and temme approximations of orthogonal polynomials in terms of hermite polynomials methods appl anal 6 1999 131 146 the askey scheme for hypergeometric orthogonal polynomials viewed from asymptotic analysis j comput appl math 133 2001 623 633 that the three lower levels of the askey table of hypergeometric orthogonal polynomials are connected by means of asymptotic relations in ferreira et al limit relations between the hahn polynomials and the hermite laguerre and charlier polynomials submitted for publication we have established new asymptotic connections between the fourth level and the two lower levels in this paper we continue with that program and obtain asymptotic expansions between the fourth level and the third level we derive 16 asymptotic expansions of the hahn dual hahn continuous hahn and continuous dual hahn polynomials in terms of meixner pollaczek jacobi meixner and krawtchouk polynomials from these expansions we also derive three new limits between those polynomials some numerical experiments show the accuracy of the approximations and in particular the accuracy in the approximation of the zeros of those polynomials

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

bivariate second order linear partial differential equations and orthogonal polynomial solutions in this paper we construct the main algebraic and differential properties and the weight functions of orthogonal polynomial solutions of bivariate second order linear partial differential equations which are admissible potentially self adjoint and of hypergeometric type general formulae for all these properties are obtained explicitly in terms of the polynomial coefficients of the partial differential equation using vector matrix notation moreover rodrigues representations for the polynomial eigensolutions and for their partial derivatives of any order are given as illustration these results are applied to a two parameter monic appell polynomials finally the non monic case is briefly discussed 2011 elsevier inc

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

an nlogn algorithm for hyper minimizing a minimized deterministic automaton we improve a recent result a badr hyper minimization in o n 2 internat j found comput sci 20 4 2009 735 746 for hyper minimized finite automata namely we present an o nlogn algorithm that computes for a given deterministic finite automaton dfa an almost equivalent dfa that is as small as possible such an automaton is called hyper minimal here two finite automata are almost equivalent if and only if the symmetric difference of their languages is finite in other words two almost equivalent automata disagree on acceptance on finitely many inputs in this way we solve an open problem stated in a badr v geffert i shipman hyper minimizing minimized deterministic finite state automata rairo theor inf appl 43 1 2009 69 94 and by badr moreover we show that minimization linearly reduces to hyper minimization which shows that the time bound o nlogn is optimal for hyper minimization independently similar results were obtained in p gawrychowski a jez hyper minimisation made efficient in proc 34th int symp mathematical foundations of computer science in lncs vol 5734 springer 2009 pp 356 368

a distributional study of discrete classical orthogonal polynomials abstract for the sequences of discrete classical orthogonal polynomials charlier meixner hahn we can find a functional u which satisfies the difference distributional equation u u where and are polynomials of degrees 2 and 1 respectively from this it follows that these polynomials are solutions of a second order difference equation also they can be represented by a rodrigues type formula the sequence of difference polynomials derived from them constitutes an orthogonal polynomial sequence their weight functions satisfy a pearson type difference equation a structure relation pn 1 anpn 2 bnpn 1 cnpn also holds

general linearization formulae for products of continuous hypergeometric type polynomials the linearization of products of wavefunctions of exactly solvable potentials often reduces to the generalized linearization problem for hypergeometric polynomials hps of a continuous variable which consists of the expansion of the product of two arbitrary hps in series of an orthogonal hp set here this problem is algebraically solved directly in terms of the coefficients of the second order differential equations satisfied by the involved polynomials general expressions for the expansion coefficients are given in integral form and they are applied to derive the connection formulae relating the three classical families of hypergeometric polynomials orthogonal on the real axis hermite laguerre and jacobi as well as several generalized linearization formulae involving these families the connection and linearization coefficients are generally expressed as finite sums of terminating hypergeometric functions which often reduce to a single function of the same type when possible these functions are evaluated in closed form in some cases sign properties of the coefficients such as positivity or non negativity conditions are derived as a by product from their resulting explicit representations

generalized linearization problems we prove that if pi x and pj x are two families of semi classical orthogonal polynomials all the linearization coefficients li j k occurring in the product of these two families satisfy a linear recurrence relation involving only the k index this property also extends to the linearization coefficients arising from an arbitrary number of products of semi classical orthogonal polynomials

the three term recurrence relation and the differentiation formulas for hypergeometric type functions abstract the functions of hypergeometric type are the solutions y y x of the differential equation z y z y y 0 where are polynomials of degrees not higher than 2 and 1 respectively and is a constant here we consider a class of functions of hypergeometric type with the additional condition that 1 2 1 0 being a complex number in general moreover we assume that the coefficients of the polynomials and have no dependence on to this class of functions belong gauss kummer and hermite functions the classical orthogonal polynomials and many other functions encountered in linear and non linear physics we obtain two important structural properties of these functions i the so called three term recurrence relation which correlates three functions of successive orders and ii the differentiation formulas also called ladder or structure relations or even differential recurrence relations which relate the first derivative y z with the functions y z and y 1 z or y 1 z finally these three relationships are applied to the polynomials of hypergeometric type which form a broad subclass of functions y where is a positive integer number and the associated contour is closed for completeness the explicit expressions corresponding to all classical orthogonal polynomials jacobi laguerre hermite and bessel are tabulated

a time complexity gap for two way probabilistic finite state automata it is shown that if a two way probabilistic finite state automaton 2pfa m recognizes a nonregular language l with error probability bounded below frac 1 2 then there is a positive constant b depending on m such that for infinitely many inputs x the expected running time of m on input x must exceed 2 n b where n is the length of x this complements a result of freivalds showing that 2pfa s can recognize certain nonregular languages in exponential expected time it also establishes a time complexity gap for 2pfa s since any regular language can be recognized by some 2pfa in linear time other results give roughly exponential upper and lower bounds on the worst case increase in the number of states when converting a polynomial time 2pfa to an equivalent two way nondeterministic finite state automaton or to an equivalent one way deterministic finite state automaton

zeros of gegenbauer and hermite polynomials and connection coefficients in this paper sharp upper limit for the zeros of the ultraspherical polynomials are obtained via a result of obrechkoff and certain explicit connection coefficients for these polynomials as a consequence sharp bounds for the zeros of the hermite polynomials are obtained

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

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

bounded sequence testing from deterministic finite state machines the w andwp methods are the basis for conformance testing from a deterministic finite state machine dfsm when the conformance relation considered is equivalence however many dfsm applications use only input sequences of limited length in such cases the test data only need to establish that the implementation under test produces the specified responses for sequences of length less than or equal to the upper bound l this paper extends the w and wp methods to the case in which only bounded sequences are allowed the methods for bounded sequences are stronger than the originals since test suites for the unbounded case can be obtained as a particular case in which the upper bound l is sufficiently large from the new formulae furthermore the generalization is not straightforward as it is not sufficient to extract the sequences of length at most l from the test suites produced in the unbounded case or even all prefixes of length at most l of the original test sequences the practicality of the methods is also improved in comparison to the unbounded case the size of the test suites may be considerably reduced while the complexity of the test generation algorithms remains basically unchanged

backlund transformations for generalized nonlinear schrodinger equations a general class of backlund transformations are considered for equations of the form izy zxx f z z 0 where f z z is a function of z x iy and z x iy the nonlinear forms of this equation that admit such transformations are completely classified and shown to exist only when f z z z2z the nonlinear schrodinger equation z ln z z ln z z z 2 or suitable combinations of these functions the form f z z z z 2 leads to auto backlund transformations for the boussinesq equation

extensions of some results of p humbert on bezout s identity for classical orthogonal polynomials in this paper the bezout s identity is analyzed in the context of classical orthogonal polynomials solution of a second order differential equation of hypergeometric type differential equations relation with the starting family as well as recurrence relations and explicit representations are given for the bezout s pair extensions to classical orthogonal polynomials of a discrete variable and their q analogues are also presented applications of these results for the representation of the second kind functions are given