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

linearization and krein like functionals of hypergeometric orthogonal polynomials the krein like r functionals of the hypergeometric orthogonal polynomials pn x with the kernel of the form xs x pm1 x pmr x being x the weight function on the interval r are determined by means of the srivastava linearization method the particular 2 functionals which are particularly relevant in quantum physics are explicitly given in terms of the degrees and the characteristic parameters of the polynomials they include the well known power moments and the novel krein like moments moreover various related types of exponential and logarithmic functionals are also investigated

generic formulas for the values at the singular points of some special monic classical hq ω orthogonal polynomials it is well known that the classical orthogonal polynomials of jacobi bessel laguerre and hermite are solutions of a sturm liouville problem of the type s x y n email protected x y n email protected ny n 0 where s and t are polynomials such that email protected 2 and email protected 1 and l n is a constant independent of x recently based on the hypergeometric character of the solutions of this differential equation w koepf and m masjed jamei a generic formula for the values at the boundary points of monic classical orthogonal polynomials j comput appl math 191 2006 98 105 found a generic formula only in terms of the coefficients of s and t for the values of the classical orthogonal polynomials at the singular points of the above differential hypergeometric equation in this paper we generalize the mentioned result giving the analogous formulas for both the classical q orthogonal polynomials of the q hahn tableau and the classical d w orthogonal polynomials both are special cases of the classical h q w orthogonal polynomials which are solutions of the hypergeometric type difference equation s x h q wh 1 q wy email protected x h q wy email protected ny n 0 where h q w is the difference operator introduced by hahn and s t and l n being as above our approach is algebraic and it does not require hypergeometric functions

axioms for bases of closed regular fuzzy matroids this paper represents an attempt to define a set of bases which can determine a closed regular fuzzy matroid and to study some properties of section fuzzy matroids elementary section fuzzy matroids and elementary fuzzy matroids axioms for bases of both elementary fuzzy matroids and closed regular fuzzy matroids are presented respectively it is proved that the set of bases of all closed regular fuzzy matroids on a set equipped with a suitable order is order isomorphic to the set of systems of independent sets of all closed regular fuzzy matroids on this set equipped with the inclusion order

factorization of fourth order differential equations for perturbed classical orthogonal polynomials we factorize the fourth order differential equations satisfied by the laguerre hahn orthogonal polynomials obtained from some perturbations of classical orthogonal polynomials such as the rth associated for genetic r the general co recursive the general co recursive associated the general co dilated and the general co modified classical orthogonal polynomials moreover we find four linearly independent solutions of the fourth order differential equations and show that the factorization obtained for modifications of classical orthogonal polynomials is still valid with some minor changes when the polynomial family modified is semi classical finally we extend the validity of the results obtained for the associated classical orthogonal polynomials with integer order of association from integers to reals

fuzzy shortest path problems incorporating interactivity among paths abstract this paper deals with a shortest path problem on a network in which a fuzzy number instead of a real number is assigned to each arc length such a problem is ill posed because each arc cannot be identified as being either on the shortest path or not therefore based on the possibility theory we introduce the concept of degree of possibility that an arc is on the shortest path every pair of distinct paths from the source node to any other node is implicitly assumed to be noninteractive in the conventional approaches this assumption is unrealistic and involve inconsistencies to overcome this drawback we define a new comparison index between the sum of fuzzy numbers by considering interactivity among fuzzy numbers an algorithm is presented to determine the degree of possibility for each arc on a network finally this algorithm is evaluated by means of large scale numerical examples consequently we can find this approach is efficient even for real world practical networks

strong asymptotics of orthogonal polynomials with respect to exponential weights we consider asymptotics of orthogonal polynomials with respect to weights w x dx e q x dx on the real line where q x 2m k 0 qkx k q2m 0 denotes a polynomial of even order with positive leading coefficient the orthogonal polynomial problem is formulated as a riemann hilbert problem following 22 23 we employ the steepest descent type method introduced in 18 and further developed in 17 19 in order to obtain uniform plancherel rotach type asymptotics in the entire complex plane as well as asymptotic formulae for the zeros the leading coefficients and the recurrence coefficients of the orthogonal polynomials c 1999 john wiley sons inc

asymptotic and interlacing properties of zeros of exceptional jacobi and laguerre polynomials abstract in this paper we state and prove some properties of the zeros of exceptional jacobi and laguerre polynomials generically the zeros of exceptional polynomials fall into two classes the regular zeros which lie in the interval of orthogonality and the exceptional zeros which lie outside that interval we show that the regular zeros have two interlacing properties one is the natural interlacing between zeros of consecutive polynomials as a consequence of their sturm liouville character while the other one shows interlacing between the zeros of exceptional and classical polynomials a heine mehler type formula is provided for the exceptional polynomials which allows to derive the asymptotic behaviour of their regular zeros for large degree n and fixed codimension m we also describe the location and the asymptotic behaviour of the m exceptional zeros which converge for large n to fixed values

on combinatorial optimization problems on matroids with uncertain weights in this paper the combinatorial optimization problem on weighted matroid is considered it is assumed that the weights in the problem are ill known and they are modeled as fuzzy intervals the optimality of solutions and the optimality of elements are characterized this characterization is performed in the setting of possibility theory a method of choosing a solution under uncertainty is also proposed

differential equations in the spectral parameter we determine all the potentialsv x for the schrodinger equation 2 v x k2 such that some family of eigenfunctions satisfies a differential equation in the spectral parameterk of the formb k k o x o for each suchv x we determine the algebra of all possible operatorsb and the corresponding functions x

the computational complexity of the criticality problems in a network with interval activity times abstract the paper analyzes the criticality in a network with interval activities duration times a natural generalization of the criticality notion for a path an activity and an event for the case of network with interval activity duration times is given the computation complexity of five problems linked to the introduced criticality notion is presented

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

critical path analysis in the network with fuzzy activity times a natural generalization of the criticality notion in a network with fuzzy activity times is given it consists in direct application of the extension principle of zadeh to the notion of criticality of a path an activity an event treated as a function of the activities duration times in the network there are shown some relations between the notion of fuzzy criticality introduced in the paper and the notion of interval criticality criticality in the network with interval activity times proposed by the authors in another paper two methods of calculation of the path degree of criticality according to the proposed concept of fuzzy criticality are presented

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

combined state adding and state deleting approaches to type iii multi step rationally extended potentials applications to ladder operators and superintegrability type iii multi step rationally extended harmonic oscillator and radial harmonic oscillator potentials characterized by a set of k integers m1 m2 mk such that m1 m2 mk with mi even resp odd for i odd resp even are considered the state adding and state deleting approaches to these potentials in a supersymmetric quantum mechanical framework are combined to construct new ladder operators the eigenstates of the hamiltonians are shown to separate into mk 1 infinite dimensional unitary irreducible representations of the corresponding polynomial heisenberg algebras these ladder operators are then used to build a higher order integral of motion for seven new infinite families of superintegrable two dimensional systems separable in cartesian coordinates the finite dimensional unitary irreducible representations of the polynomial algebras of such systems are directly determined from the ladder operator action on the constituent one dimensional hamiltonian eigenstates and provide an algebr

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

second structure relation for q semiclassical polynomials of the hahn tableau the q classical orthogonal polynomials of the q hahn tableau are characterized from their orthogonality condition and by a first and a second structure relation unfortunately for the q semiclassical orthogonal polynomials a generalization of the classical ones we find only in the literature the first structure relation in this paper a second structure relation is deduced in particular by means of a general finite type relation between a q semiclassical polynomial sequence and the sequence of its q differences such a structure relation is obtained

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

fuzzy scheduling modelling flexible constraints vs coping with incomplete knowledge abstract an overview of some fuzzy set based approaches to scheduling is proposed emphasizing two distinct uses of fuzzy sets representing preference profiles and modelling uncertainty distributions the first setting leads to a valued non compensatory generalization of constraint directed scheduling the other setting yields a possibility theoretic counterpart of pert where probability distributions of activity durations are changed into possibility distributions for the purpose of modelling incomplete information it is pointed out that a special case of the latter interval valued pert is a difficult ill known problem regarding the determination of critical activities latest starting times and floats lastly when flexible constraints and uncertain processing times are to be jointly considered the use of possibilistic decision theory leads to the computation of robust schedules

on stieltjes relations painleve iv hierarchy and complex monodromy a generalization of the stieltjes relations for the fourth painleve piv transcendents and their higher analogues determined by dressing chains is proposed it is proven that if a rational function from a certain class satisfies these relations it must be a solution of some higher piv equation the approach is based on the interpretation of the stieltjes relations as local trivial monodromy conditions for certain schrodinger equations in the complex domain as a corollary a new class of schrodinger operators with trivial monodromy is constructed in terms of the piv transcendents

hypergeometric series solutions of linear operator equations let k be a field and l k x k x be a linear operator acting on the ring of polynomials in x over the field k we provide a method to find a suitable basis b k x of k x and a hypergeometric term c k such that y x k 0 c kb k x is a formal series solution to the equation l y x 0 this method is applied to construct hypergeometric representations of orthogonal polynomials from the differential difference equations or recurrence relations they satisfied both the ordinary cases and the q cases are considered

characteristic polynomials of random matrices number theorists have studied extensively the connections between the distribution of zeros of the riemann function and of some generalizations with the statistics of the eigenvalues of large random matrices it is interesting to compare the average moments of these functions in an interval to their counterpart in random matrices which are the expectation values of the characteristic polynomials of the matrix it turns out that these expectation values are quite interesting for instance the moments of order 2k scale for unitary invariant ensembles as the density of eigenvalues raised to the power k 2 the prefactor turns out to be a universal number i e it is independent of the specific probability distribution an equivalent behaviour and prefactor had been found as a conjecture within number theory the moments of the characteristic determinants of random matrices are computed here as limits at coinciding points of multi point correlators of determinants these correlators are in fact universal in dyson s scaling limit in which the difference between the points goes to zero the size of the matrix goes to infinity and their product remains finite

multi indexed jacobi polynomials and maya diagrams multi indexed jacobi polynomials are defined by the wronskian of four types of eigenfunctions of the poschl teller hamiltonian we give a correspondence between multi indexed jacobi polynomials and pairs of maya diagrams and we show that any multi indexed jacobi polynomial is essentially equal to some multi indexed jacobi polynomial of two types of eigenfunction as an application we show a wronskian type formula of some special eigenstates of the deformed poschl teller hamiltonian

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

rational extensions of the quantum harmonic oscillator and exceptional hermite polynomials we prove that every rational extension of the quantum harmonic oscillator that is exactly solvable by polynomials is monodromy free and therefore can be obtained by applying a finite number of state deleting darboux transformations on the harmonic oscillator equivalently every exceptional orthogonal polynomial system of hermite type can be obtained by applying a darboux crum transformation to the classical hermite polynomials exceptional hermite polynomial systems only exist for even codimension 2m and they are indexed by the partitions of m we provide explicit expressions for their corresponding orthogonality weights and differential operators and a separate proof of their completeness exceptional hermite polynomials satisfy a 2l 3 recurrence relation where l is the length of the partition explicit expressions for such recurrence relations are given

minmax regret approach and optimality evaluation in combinatorial optimization problems with interval and fuzzy weights this paper deals with a general combinatorial optimization problem in which closed intervals and fuzzy intervals model uncertain element weights the notion of a deviation interval is introduced which allows us to characterize the optimality and the robustness of solutions and elements the problem of computing deviation intervals is addressed and some new complexity results in this field are provided possibility theory is then applied to generalize a deviation interval and a solution concept to fuzzy ones

some methods for evaluating the optimality of elements in matroids with ill known weights in this paper a class of matroidal combinatorial optimization problems with imprecise weights of elements is considered the imprecise weights are modeled by intervals and fuzzy intervals the concepts of possible and necessary optimality under imprecision are recalled some efficient methods for evaluating the possible and necessary optimality of elements in the interval valued problems are proposed some efficient algorithms for computing the exact degrees of possible and necessary optimality of elements in the fuzzy valued problems are designed

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

a bochner type characterization theorem for exceptional orthogonal polynomials abstract it was recently conjectured that every system of exceptional orthogonal polynomials is related to a classical orthogonal polynomial system by a sequence of darboux transformations in this paper we prove this conjecture which paves the road to a complete classification of all exceptional orthogonal polynomials in some sense this paper can be regarded as the extension of bochner s result for classical orthogonal polynomials to the exceptional class as a supplementary result we derive a canonical form for exceptional operators based on a bilinear formalism and prove that every exceptional operator has trivial monodromy at all primary poles

connectedness of refined goetschel voxman fuzzy matroids this paper studies connectedness of goetschel voxman fuzzy matroids briefly g v fuzzy matroids an analog of connectedness of crisp finite matroids based on the results of fuzzy circuits given by goetschel and voxman the transitivity theorem concerning fuzzy circuits of g v fuzzy matroids is established and thus the useful notion of refined g v fuzzy matroid is introduced the connectedness of refined g v fuzzy matroids is then defined by using an equivalence relation on the set of all fuzzy points on the ground set and some expected properties of these g v fuzzy matroids are presented additionally five kinds of fuzzy matroids are compared

an improved algorithm for selecting p items with uncertain returns according to the minmax regret criterion we consider the problem of selecting a subset of p investments of maximum total return out of a set of n available investments with uncertain returns where uncertainty is represented by interval estimates for the returns and the minmax regret objective is used we develop an algorithm that solves this problem in o min p n p n time this improves the previously known complexity o min p n p 2n

l m fuzzy matroids in this paper an extension of matroids is introduced in lattice valued fuzzy set theory such a generalized matroid is a pair made of a finite set and a fuzzy subfamily of its lattice valued fuzzy subsets satisfying three axioms it uses two lattices one for membership grades of elements to fuzzy sets and another for membership degrees of fuzzy sets in a fuzzy family of fuzzy sets it is a logical generalization of the notions of matroids and fuzzifying matroids in our definition of generalized fuzzy matroids each l fuzzy subset can be regarded to be independent of other ones to some degree the relevance of generalized fuzzy matroids is shown in the setting of fuzzy vector spaces and fuzzy graphs

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

on the rational monodromy free potentials with sextic growth we study the rational potentials v x with sextic growth at infinity such that the corresponding one dimensional schrodinger equation has no monodromy in the complex domain for all values of the spectral parameter we investigate in detail the subclass of such potentials which can be constructed by the darboux transformations from the well known class of quasiexactly solvable potentials v x6 x2 l l 1 x2 we show that in contrast with the case of quadratic growth there are monodromy free potentials which have quasirational eigenfunctions but which cannot be given by this construction we discuss the relations between the corresponding algebraic varieties and present some elementary solutions of the calogero moser problem in the external field with sextic potential

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

exceptional orthogonal polynomials and generalized schur polynomials we show that the exceptional orthogonal polynomials can be viewed as confluent limits of the generalized schur polynomials introduced by sergeev and veselov

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

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

the spectral class of the quantum mechanical harmonic oscillator the purpose of this paper is to study the so calledspectral classq of anharmonic oscillatorsq d2 q having the same spectrum n 2n n 0 as the harmonic oscillatorq0 d2 x2 1 thenorming constants t n mathop lim limits x uparrow infty ell g 1 n e n x mathord left vphantom e n x e n right kern nulldelimiterspace e n x of the eigenfunctions ofq form a complete set of coordinates inq in terms of which the potential may be expressed asq x2 1 2d2 lg with theta det left delta ij e ti 1 int limits x infty e i 0 e j 0 0 leqq i j infty right e n 0 being thenth eigenfunctionq0 the spectrum and norming constants are canonically conjugate relative to the bracket f g fd gdx to wit i j 0 ti 2 j 1 or 0 according to whetheri j or not and ti tj 0 this prompts an investigation of the symplectic geometry ofq the function is related to the theta function of a singular algebraic curve numerical results are also presented

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

parameter based fisher s information of orthogonal polynomials the fisher information of the classical orthogonal polynomials with respect to a parameter is introduced its interest justified and its explicit expression for the jacobi laguerre gegenbauer and grosjean polynomials found

invariance properties of wronskian type determinants of classical and classical discrete orthogonal polynomials abstract given a finite set f f 1 f k of nonnegative integers written in increasing order of magnitude and a classical discrete family p n n of orthogonal polynomials charlier meixner krawtchouk or hahn we consider the casorati determinant det p f i x j 1 i j 1 k in this paper we prove an invariance property of this kind of casorati determinants when the set f is substituted by the set i f 0 1 2 max f max f f f f our approach uses orthogonal polynomials that are eigenfunctions of higher order difference operators krall discrete polynomials these polynomials are orthogonal with respect to certain christoffel transforms of the classical discrete measures by passing to the limit this invariance property is extended to wronskian type determinants whose entries are hermite laguerre and jacobi polynomials

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

zeros of wronskians of hermite polynomials and young diagrams abstract for a certain class of partitions a simple qualitative relation is observed between the shape of the young diagram and the pattern of zeros of the wronskian of the corresponding hermite polynomials in the case of the two term wronskian w h n h n k we give an explicit formula for the asymptotic shape of the zero set as n some empirical asymptotic formulas are given for the zero sets of three term and four term wronskians

the fourth painleve equation and associated special polynomials in this article rational solutions and associated polynomials for the fourth painleve equation are studied these rational solutions of the fourth painleve equation are expressible as the logarithmic derivative of special polynomials the okamoto polynomials the structure of the roots of these okamoto polynomials is studied and it is shown that these have a highly regular structure the properties of the okamoto polynomials are compared and contrasted with those of classical orthogonal polynomials further representations are given of the associated rational solutions in the form of determinants through schur functions

a new approach to the fuzzification of matroids in this paper the concept of closed fuzzy pre matroids is generalized to l fuzzy set theory when l is a complete lattice it is also called an l fuzzifying matroid in the definition of l fuzzifying matroids each subset can be regarded as an independent set to some degree when l is completely distributive an l fuzzifying matroid can be characterized by means of its l fuzzifying rank function an l fuzzifying matroid and its l fuzzifying rank function are one to one corresponding

on parameter derivatives of a family of polynomials in two variables the purpose of the present paper is to give the parameter derivative representations of the form p n k x y m 0 n 1 j 0 m d n j m p m j x y j 0 k e n j k p n j x y for a family of orthogonal polynomials of variables x and y with being a parameter and 0 k n n k 0 1 2 first we shall present the representations of the parameter derivatives of the generalized gegenbauer polynomials c n x with the help of the parameter derivatives of the classical jacobi polynomials p n x i e p n x and p n x then by using these derivatives we investigate the parameter derivatives for two variable analogues of the generalized gegenbauer polynomials furthermore we discuss orthogonality properties of the parametric derivatives of these polynomials

on the complexity of a class of combinatorial optimization problems with uncertainty we consider a robust minmax regret version of the problem of selecting p elements of minimum total weight out of a set of m elements with uncertainty in weights of the elements we present a polynomial algorithm with the order of complexity o min p m p 2m for the case where uncertainty is represented by means of interval estimates for the weights we show that the problem is np hard in the case of an arbitrary finite set of possible scenarios even if there are only two possible scenarios this is the first known example of a robust combinatorial optimization problem that is np hard in the case of scenario represented uncertainty but is polynomially solvable in the case of the interval representation of uncertainty