information entropy of orthogonal polynomials the boltzmann shannon information entropy of quantum mechanical systems in central potentials can be expressed in terms of the entropy e n of the classical orthogonal polynomials e email protected p 2 n x logp 2 n x w x dx here an explicit formula for the entropy of general orthogonal polynomials is obtained in terms of the eigenvalues of jacobi matrix associated with the orthogonal polynomials p n x and the weighted moments of p 2 n x m n email protected x kp 2 n x w x dx k 1 2

asymptotic formulae for the quantum renyi entropies of position application to the infinite well general asymptotic formulae are derived by means of the wkb approximation for the continuous and discrete renyi entropies of position of one dimensional quantum systems in energy eigenstates in terms of the corresponding entropies for a microcanonical ensemble of analogous classical systems these results are checked in the simplest particular case of the infinite potential well where the asymptotic formula for continuous entropies holds as an exact identity for the discrete entropies analytical expressions are obtained from which the asymptotic formulae given for the limiting cases of large and small measurement resolution can both be verified

constructible differentially finite algebraic series in several variables we extend the concept of cdf series to the context of several variables and show that the series solution of first order differential equations y x t y and functional equation y x t y with x cdf in two variables are cdf series we also give many effective closure properties for cdf series in several variables

positive solutions for three point boundary value problems with dependence on the first order derivative abstract a new fixed point theorem in a cone is applied to obtain the existence of at least one positive solution for the second order three point boundary value problem x f t x x 0 0 x 0 0 x 1 x where f is a nonnegative continuous function 0 0 1

a holonomic systems approach to special functions identities abstract we observe that many special functions are solutions of so called holonomic systems bernstein s deep theory of holonomic systems is then invoked to show that any identity involving sums and integrals of products of these special functions can be verified in a finite number of steps this is partially substantiated by an algorithm that proves terminating hypergeometric series identities and that is given both in english and in maple

entropic integrals of hyperspherical harmonics and spatial entropy of d dimensional central potentials the information entropy of a single particle in a quantum mechanical d dimensional central potential is separated in two parts one depends only on the specific form of the potential radial entropy and the other depends on the angular distribution spatial entropy the latter is given by an entropic like integral of the hyperspherical harmonics which is expressed in terms of the entropy of the gegenbauer polynomials this entropy is expressed in terms of the values of the quadratic logarithmic potential of gegenbauer polynomials cn t at the zeros of these polynomials then this potential for integer is given as a finite expansion of chebyshev polynomials of even order whose coefficients are shown to be wilson polynomials

rational function certification of multisum integral q identities the method of rational function certification for proving terminating hypergeometric identities is extended from single sums or integrals to multi integral sums and q integral sums

analytic and algorithmic aspects of generalized harmonic sums and polylogarithms in recent three loop calculations of massive feynman integrals within quantum chromodynamics qcd and e g in recent combinatorial problems the so called generalized harmonic sums in short s sums arise they are characterized by rational or real numerator weights also different from 1 in this article we explore the algorithmic and analytic properties of these sums systematically we work out the mellin and inverse mellin transform which connects the sums under consideration with the associated poincare iterated integrals also called generalized harmonic polylogarithms in this regard we obtain explicit analytic continuations by means of asymptotic expansions of the s sums which started to occur frequently in current qcd calculations in addition we derive algebraic and structural relations like differentiation with respect to the external summation index and different multi argument relations for the compactification of s sum expressions finally we calculate algebraic relations for infinite s sums or equivalently for generalized harmonic polylogarithms evaluated at special values the corresponding algorithms and relations are encoded in the computer algebra package harmonicsums

automatic average case analysis of algorithms abstract many probabilistic properties of elementary discrete combinatorial structures of interest for the average case analysis of algorithms prove to be decidable this paper presents a general framework in which such decision procedures can be developed it is based on a combination of generating function techniques for counting and complex analysis techniques for asymptotic estimations we expose here the theory of exact analysis in terms of generating functions for four different domains the iterative recursive and unlabelled labelled data type domains we then present some major components of the associated asymptotic theory and exhibit a class of naturally arising functions that can be automatically analyzed a fair fragment of this theory is also incorporated into a system called lambda upsilon omega in this way using computer algebra one can produce automatically non trivial average case analyses of algorithms operating over a variety of decomposable combinatorial structures at a fundamental level this paper is part of a global attempt at understanding why so many elementary combinatorial problems tend to have elementary asymptotic solutions in several cases it proves possible to relate entire classes of elementary combinatorial problems whose structure is well defined with classes of elementary special functions and classes of asymptotic forms relative to counting probabilities or average case complexity

the holstein polaron problem revisited the holstein hamiltonian was proposed half a century ago since then decades of research have come up empty handed in the pursuit of a closed form solution an exact solution to the two site holstein model is presented in this paper the obtained results provide a clear image of the hamiltonian structure and allow for the investigation of the symmetry energy level crossings and polaronic characteristics of the system the main mathematical tool is a three term recurrence relation between the wave function amplitudes which was obtained using the properties of a family of orthogonal functions namely the poisson charlier polynomials it is shown that with the appropriate choice of basis the eigenfunctions of the problem naturally fall into two families parities associated with the discrete symmetry of the hamiltonian the asymptotic solution to the recurrence relation is found by using the birkhoff expansion the asymptotic sets the truncation criterion for the wave function which ensures the accurate calculation of the energy levels for any strength of electron phonon interaction the level crossing of states with different parities is discussed and the exact points of broken symmetry are found analytically the results are used as the building blocks for studying a four site system the inherited symmetries lead to the formation of a sparse matrix that is convenient for numerical calculations

asymptotics of quantum spin networks at a fixed root of unity a classical spin network consists of a ribbon graph i e an abstract graph with a cyclic ordering of the vertices around each edge and an admissible coloring of its edges by natural numbers the standard evaluation of a spin network is an integer number in a previous paper we proved an existence theorem for the asymptotics of the standard evaluation of an arbitrary classical spin network when the coloring of its edges are scaled by a large natural number in the present paper we extend the results to the case of an evaluation of quantum spin networks of arbitrary valency at a fixed root of unity as in the classical case our proofs use the theory of g functions of andre together with some new results concerning holonomic and q holonomic sequences of wilf zeilberger

liouville green approximations for a class of linear oscillatory difference equations of the second order abstract an asymptotic approximation theorem is proved for the solutions of linear oscillatory three term recurrence equations in a certain class this result represents a discrete analogue of the well known liouville green or wkbj theorem rigorously proved by olver for second order linear differential equations as in olver s theorem precise estimates are obtained for the error terms and the double asymptotic nature of the approximation is made clear applications to certain orthogonal polynomials are also given

g functions and multisum versus holonomic sequences abstract the purpose of the paper is three fold a we prove that every sequence which is a multidimensional sum of a balanced hypergeometric term has an asymptotic expansion of gevrey type 1 with rational exponents b we construct a class of g functions that come from enumerative combinatorics and c we give a counterexample to a question of zeilberger that asks whether holonomic sequences can be written as multisums of balanced hypergeometric terms the proofs utilize the notion of a g function introduced by siegel and its analytic arithmetic properties shown recently by andre

third order integrability conditions for homogeneous potentials of degree 1 we prove an integrability criterion of order 3 for a homogeneous potential of degree 1 in the plane still this criterion depends on some integer and it is impossible to apply it directly except for families of potentials whose eigenvalues are bounded to address this issue we use holonomic and asymptotic computations with error control of this criterion and apply it to the potential of the form v r r 1h exp i with h c z degh 3 we then find all meromorphically integrable potentials of this form

current trends in asymptotics some problems and some solutions abstract i discuss five topics of current interest in asymptotic analysis the use of probabilistic methods to estimate the growth of combinatorial sequences asymptotic methods in the theory of random walks the asymptotic estimation of solutions of difference equations asymptotic expansions in generalized scales and the computation by asymptotic methods of distributions whose moments are known i give a number of examples and mention some outstanding unsolved problems

summation transformation and expansion formulas for bibasic series etude des extensions a deux bases de la formule de transformation d euler et de la formule de developpement de fields wimp etablissement d une formule de transformation a quatre bases independantes d une formule d inversion de lagrange et de quelques formules de sommation de degres 2 3 et 4

solvability of a three point nonlinear boundary value problem for a second order ordinary differential equation in case f x u u ps x p x u pjx u with pk 0 l 4 r locally integrable the boundary value problem l lt 1 2 was studied by kiguradze and lomtatidze i the purpose of this paper is to obtain existence and uniqueness theorems for the boundary value problem 1 1 t 1 2 under natural conditions on f using degree theoretic arguments we note that if u is a solution of 1 1 1 2 then there exists at least one 5 e q l such that u c 0 accordingly the boundary value problem

quantum information entropies and orthogonal polynomials this is a survey of the present knowledge on the analytical determination of the shannon information entropies for simple quantum systems single particle systems in central potentials emphasis is made on d dimensional harmonic oscillator and coulombian potentials in both position and momentum spaces first of all these quantities are explicitly shown to be controlled by the entropic integrals of some classical orthogonal polynomials hermite laguerre and gegenbauer then the connection of these integrals with more common mathematical objects such as the logarithmic potential energy and l p norms of orthogonal polynomials is brie y described third its asymptotic behaviour is discussed for both general and varying weights the explicit computation of these integrals is carried out for the chebyshev and gegenbauer polynomials which have a bounded orthogonality interval as well as for hermite polynomials to illustrate the di culties encountered when the interval is unbounded these results have allowed us to nd the position and momentum entropies of the ground and excited states of the physical systems mentioned above c 2001 elsevier science b v all rights reserved

lattice green s functions of the higher dimensional face centered cubic lattices we study the face centered cubic fcc lattice in up to six dimensions in particular we are concerned with lattice green functions lgfs and return probabilities computer algebra techniques such as the method of creative telescoping are used for deriving an ode for a given lgf for the four and five dimensional fcc lattices we give rigorous proofs of the odes that were conjectured by guttmann and broadhurst additionally we find the ode of the lgf of the six dimensional fcc lattice a result that was not believed to be achievable with current computer hardware

rational approximations for values of derivatives of the gamma function the arithmetic nature of euler s constant is still unknown and even getting good rational approximations to it is difficult recently aptekarev managed to find a third order linear recurrence with polynomial coefficients which admits two rational solutions a n and b n such that a n b n converges sub exponentially to viewed as 1 where r is the usual gamma function although this is not yet enough to prove that it is a major step in this direction in this paper we present a different but related approach based on simultaneous pade approximants to euler s functions from which we construct and study a new third order recurrence that produces a sequence in z whose height grows like the factorial and that converges sub exponentially to log z for any complex number z 0 where log is defined by its principal branch we also show how our approach yields in theory rational approximations of numbers related to s 1 for any integer s 1 in particular we construct a sixth order recurrence which provides simultaneous rational approximations of factorial height converging sub exponentially to the numbers and 1 2 1 2 2 2

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

lattice green functions the seven dimensional face centred cubic lattice we present a recursive method to generate the expansion of the lattice green function of the d dimensional face centred cubic fcc lattice we produce a long series for d 7 then we show and recall that in order to obtain the linear differential equation annihilating such a long power series the most economic way amounts to producing the non minimal order differential equations we use the method to obtain the minimal order linear differential equation of the lattice green function of the seven dimensional fcc lattice we give some properties of this irreducible order eleven differential equation we show that the differential galois group of the corresponding operator is included in this order eleven operator is non trivially homomorphic to its adjoint and we give a decomposition of this order eleven operator in terms of four order one self adjoint operators and one order seven self adjoint operator furthermore using the landau conditions on the integral we forward the regular singularities of the differential equation of the d dimensional lattice and show that they are all rational numbers we evaluate the return probability in random walks in the seven dimensional fcc lattice we show that the return probability in the d dimensional fcc lattice decreases as d 2 as the dimension d goes to infinity

harmonic oscillations in a quasi relativistic regime a square root anharmonic oscillator potential is analysed as a model which simulates a quasi relativistic squeezing of the harmonic oscillator spectrum several eligible namely perturbative variational hill determinant and riccati pade methods of construction of its bound states are compared

entropic integrals of orthogonal hypergeometric polynomials with general supports the boltzmann shannon information entropy of probability measures which involve the continuous hypergeometric type polynomials fpn x g orthogonal with respect to a general weight function x is determined by two integral quantities one with kernel p 2 x x lnp 2 x called as entropy of the polynomial pn x and another one with kernel p 2 x x ln x here an explicit expression for the latter quantity and for a broader family of related integrals is obtained in terms only of the second order dierential equation satised by the involved polynomials for illustration the general formula is applied to evaluate the integrals corresponding to the three classical families of continuous orthogonal polynomials on the real axis of hypergeometric type hermite laguerre and jacobi c 2000 elsevier science b v all rights reserved

directed compact percolation near a wall iii exact results for the mean length and number of contacts existing exact results for the percolation probability and mean cluster size for compact percolation near a dry wall are extended to the mean cluster length and the mean number of contacts with the wall the results are derived from our previous work on vesicles near an attractive wall and involve elliptic integrals as opposed to the simple rational forms found for the percolation probability and cluster size below the results for the cluster length satisfy previously conjectured differential equations a closed expression is conjectured for the mean size above in terms of a hypergeometric function

an algorithmic proof theory for hypergeometric ordinary and q multisum integral identities it is shown that every proper hypergeometric multisum integral identity orq identity with a fixed number of summations and or integration signs possesses a short computer constructible proof we give a fast algorithm for finding such proofs most of the identities that involve the classical special functions of mathematical physics are readily reducible to the kind of identities treated here we give many examples of the method including computer generated proofs of identities of mehta dyson selberg hille hardy q saalschutz and others the prospect of using the method for proving multivariate identities that involve an arbitrary number of summations integrations is discussed

proof of ira gessel s lattice path conjecture abstract we present a computer aided yet fully rigorous proof of ira gessel s tantalizingly simply stated conjecture that the number of ways of walking 2n steps in the region x y 0 y 0 of the square lattice with unit steps in the east west north and south directions that start and end at the origin equals

square lattice ising model susceptibility series expansion method and differential equation for χ 3 in a previous paper 2004 j phys a math gen 37 9651 68 we have given the fuchsian linear differential equation satisfied by 3 the three particle contribution to the susceptibility of the isotropic square lattice ising model this paper gives the details of the calculations with some useful tricks and tools allowing one to obtain long series in polynomial time the method is based on series expansion in the variables that appear in the n 1 dimensional integrals representing the n particle contribution to the isotropic square lattice ising model susceptibility the integration rules are straightforward due to remarkable formulae we derived for these variables we obtain without any numerical approximation 3 as a fully integrated series in the variable w s 2 1 s2 where s sh 2k with k j kt the conventional ising model coupling constant we also give some perspectives and comments on these results

information entropy of conditionally exactly solvable potentials we evaluate shannon entropy for the position and momentum eigenstates of some conditionally exactly solvable potentials which are isospectral to harmonic oscillator and whose solutions are given in terms of exceptional orthogonal polynomials the bialynicki birula mycielski inequality has also been tested for a number of states

fisher information of special functions and second order differential equations we investigate a basic question of analytic information theory namely the evaluation of the fisher information and the relative fisher information with respect to a non negative function for the probability distributions obtained by squaring the special functions of mathematical physics which are solutions of second order differential equations we obtain explicit expressions for these information theoretic properties via the expectation values of the coefficients of the differential equation we illustrate our approach for various nonrelativistic d dimensional wavefunctions and some special functions of physicomathematical interest emphasis is made in the nikiforov uvarov hypergeometric type functions which include and generalize the hermite functions and the gauss and kummer hypergeometric functions among others

logarithmic potential of hermite polynomials and information entropies of the harmonic oscillator eigenstates the problem of calculating the information entropy in both position and momentum spaces for the nth stationary state of the one dimensional quantum harmonic oscillator reduces to the evaluation of the logarithmic potential vn t hn x 2 ln x t e x2 dx at the zeros of the hermite polynomial hn x here a closed analytical expression for vn t is obtained which in turn yields an exact analytical expression for the entropies when the exact location of the zeros of hn x is known an inequality for the values of vn t at the zeros of hn x is conjectured which leads to a new nonvariational upper bound for the entropies finally the exact formula for vn t is written in an alternative way which allows the entropies to be expressed in terms of the even order spectral moments of the hermite polynomials the asymptotic n 1 limit of this alternative expression for the entropies is discussed and the conjectured upper bound for the entropies is proved to be asymptotically valid

entropic uncertainty relations for the infinite well we calculate the entropic and standard uncertainty relations for the position and momentum of the infinite potential well as functions of its quantum states we show that the entropic uncertainty relations express more adequately the uncertainty principle than the standard ones which use the dispersions as measures of the uncertainties for two non commuting observables

positive solutions of fourth order singular three point eigenvalue problems abstract in this paper by establishing a new comparison theorem and constructing upper and lower solutions some sufficient conditions of existence of positive solutions for the following singular fourth order three point eigenvalue problem u 4 t f t u t 0 1 u 0 u u 1 0 u 0 u u 1 0 are established due to schauder s fixed point theorem for large enough where 0 1 are constants f can be singular at t 0 and or 1 u 0 moreover some peculiar cases are discussed and some further results are obtained

asymptotics of information entropies of some toda like potentials the spreading of the quantum probability density for the highly excited states of a single particle system with an exponential type potential on the positive semiaxis is quantitatively determined in both position and momentum spaces by means of the boltzmann shannon information entropy this problem boils down to the calculation of the asymptotics of the entropy like integrals of the modified bessel function of the second kind also called the mcdonald function or basset function the dependence of the two physical entropies on the large quantum number n is given in detail it is shown that the semiclassical wkb position space entropy grows slower than the corresponding quantity of not only the harmonic oscillator but also the single particle systems with any power type potential of the form v x x2k x r and k n the momentum space entropy calculated with a method based on the properties of the mcdonald function is rigorously found to have a behavior of the form ln ln n in strong contrast with the

positive solutions for fourth order differential equations with deviating arguments and integral boundary conditions abstract in this paper we investigate integral boundary value problems for fourth order differential equations with deviating arguments we discuss our problem both for advanced or delayed arguments we establish sufficient conditions under which such problems have positive solutions to obtain the existence of multiple at least three positive solutions we use a fixed point theorem due to avery and peterson an example is also included to illustrate that corresponding assumptions are satisfied the results are new

entropy of orthogonal polynomials with freud weights and information entropies of the harmonic oscillator potential the information entropy of the harmonic oscillator potential v x 1 2 x2 in both position and momentum spaces can be expressed in terms of the so called entropy of hermite polynomials i e the quantity sn h h2n x log h2n x e x2dx these polynomials are instances of the polynomials orthogonal with respect to the freud weights w x exp x m m 0 here a very precise and general result of the entropy of freud polynomials recently established by aptekarev et al j math phys 35 4423 4428 1994 specialized to the hermite kernel case m 2 leads to an important refined asymptotic expression for the information entropies of very excited states i e for large n in both position and momentum spaces to be denoted by s and s respectively briefly it is shown that for large values of n s 1 2log log 2n e o 1 and s 1 2log log 2n e o 1 so that s s log 2 2n e2 o 1 in agreement with the generalized indetermination relation of byalinicki birula and mycielski commun m

upper and lower solutions and topological degree this paper deals with second order nonlinear boundary value problems we suppose the existence of upper and lower solutions of the problems which are well ordered i e the lower solution is less than the upper one and we also consider the case of upper and lower solutions having the opposite ordering we prove the relation between the topological degree and strict upper and lower solutions in both cases and using this we get the existence and multiplicity results for the boundary value problems under consideration

multiplicity of positive solutions for second order three point boundary value problems we study the multiplicity of positive solutions for the second order three point boundary value problem u h t f u 0 t 0 1 u 0 0 u u 1 where 0 1 0 1 the methods employed are fixed point index theorems and leray schauder degree and upper and lower solutions

sister celine s technique and its generalizations abstract sister celine fasenmyer s technique for obtaining pure recurrence relations for hypergeometric polynomials is formalized and generalized in various directions applications include algorithms for verifying any given binomial coefficients identity and any identities involving sums and integrals of products of special functions this is shown to lead to a new approach to the theory of special functions which allows a natural definition of special functions of several variables

solutions of linear difference equations with variable coefficients the explicit solution of a linear difference equation of unbounded order with variable coefficients is presented as special cases the solutions of nonhomogeneous and homogeneous linear difference equations of order n with variable coefficients are obtained from these solutions we also get expressions for the product of companion matrices and the power of a companion matrix

a diagrammatic approach to the aj conjecture the aj conjecture relates a quantum invariant a minimal order recursion for the colored jones polynomial of a knot known as the hat a polynomial with a classical invariant namely the defining polynomial a of the mathrm psl 2 mathbb c character variety of a knot more precisely the aj conjecture asserts that the set of irreducible factors of the hat a polynomial after we set q 1 and excluding those of l degree zero coincides with those of the a polynomial in this paper we introduce a version of the hat a polynomial that depends on a planar diagram of a knot that conjecturally agrees with the hat a polynomial and we prove that it satisfies one direction of the aj conjecture our proof uses the octahedral decomposition of a knot complement obtained from a planar projection of a knot the r matrix state sum formula for the colored jones polynomial and its certificate

square lattice ising model susceptibility connection matrices and singular behaviour of χ 3 and χ 4 we present a simple but efficient way to calculate connection matrices between sets of independent local solutions defined at two neighbouring singular points of fuchsian differential equations of quite large orders such as those found for the third and fourth contribution 3 and 4 to the magnetic susceptibility of the square lattice ising model we deduce all the critical behaviours of the solutions 3 and 4 as well as the asymptotic behaviour of the coefficients in the corresponding series expansions we confirm that the newly found quadratic singularities of the fuchsian ode associated with 3 are not singularities of the particular solution 3 itself we use the previous connection matrices to get the exact expressions of all the monodromy matrices of the fuchsian differential equation for 3 and 4 expressed in the same basis of solutions these monodromy matrices are the generators of the differential galois group of the fuchsian differential equations for 3 and 4 whose analysis is just sketched here as far as the physics implications of the solutions are concerned we find challenging qualitative differences when comparing the corrections to scaling for the full susceptibility at high temperature respectively low temperature and the first two terms 1 and 3 respectively 2 and 4

resurrecting the asymptotics of linear recurrences abstract once on the forefront of mathematical research in america the asymptotics of the solutions of linear recurrence equations is now almost forgotten especially by the people who need it most namely combinatorists and computer scientists here we present this theory in a concise form and give a number of examples that should enable the practicing combinatorist and computer scientist to include this important technique in her or his asymptotics tool kit

the method of lower and upper solutions for fourth order singular m point boundary value problems abstract this paper investigates the existence of positive solutions for fourth order singular m point boundary value problems firstly we establish a comparison theorem then we define a partial ordering in c 2 0 1 c 4 0 1 and construct lower and upper solutions to give a necessary and sufficient condition for the existence of c 2 0 1 as well as c 3 0 1 positive solutions our nonlinearity f t x y may be singular at x y t 0 and or t 1

asymptotic expansions for second order linear difference equations abstract formal series solutions are obtained for the difference equation y n 2 a n y n 1 b n y n 0 where a n and b n have asymptotic expansions of the form a n s 0 a s n s and b n s 0 b s n s for large values of n and b0 0 these solutions are characterized by the roots of the characteristic equation 2 a0 b0 0 our discussion is divided into three cases according to whether the roots are distinct or equal and do not satisfy the auxiliary equation a1 b1 0 or equal and do satisfy the auxiliary equation the last case is further divided into three subcases according to whether the roots of the indicial equation 1 2 a1 a2 b2 0 do not differ by a nonnegative integer or differ by a positive integer or are equal in all cases the formal series solutions will be shown to be asymptotic our approach is based on the method of successive approximations

spatial entropy of central potentials and strong asymptotics of orthogonal polynomials the boltzmann shannon information entropy of quantum mechanical systems in central potentials can be expressed in terms of the entropy sn of the classical orthogonal polynomials here an asymptotic formula for the entropy of general orthogonal polynomials on finite intervals is obtained it is shown that this entropy is intimately related to the relative entropy i 0 of the equilibrium measure 0 x and the weight function x of the polynomials to do so the theory of strong asymptotics of orthogonal polynomials on compact sets is used

three positive fixed points of nonlinear operators on ordered banach spaces we generalize the fixed point theorem of leggett williams which is a theorem giving conditions that imply the existence of three fixed points of an operator defined on a cone in a banach space we then show how to apply our theorem to prove the existence of three positive solutions to a second order discrete boundary value problem 2001 elsevier science ltd all rights reserved

on computing the coefficients of bivariate holonomic formal series in this work we study the problem of computing the coefficients of holonomic formal series in two commuting variables given a formal series x y n k 0 cnkxnyk specified by a holonomic system j 0d1 pj x y xj 0 and j 0d2qj x y yj 0 with a suitable finite set of intial conditions xayb x y we show that the coefficient xiyj x y can be computed in time o i j under the uniform cost criterion

strong asymptotics of laguerre polynomials and information entropies of two dimensional harmonic oscillator and one dimensional coulomb potentials the information entropies of the two dimensional harmonic oscillator v x y 1 2 x2 y2 and the one dimensional hydrogen atom v x 1 x can be expressed by means of some entropy integrals of laguerre polynomials whose values have not yet been analytically determined here we first study the asymptotical behavior of these integrals in detail by extensive use of strong asymptotics of laguerre polynomials then this result which is also important by itself in a context of both approximation theory and potential theory is employed to analyze the information entropies of the aforementioned quantum mechanical potentials for the very excited states in both position and momentum spaces it is observed in particular that the sum of position and momentum entropies has a logarithmic growth with respect to the main quantum number which characterizes the corresponding physical state finally the rate of convergence of the entropies is numerically examined

an exact solution of two friendly interacting directed walks near a sticky wall we find the exact solution of two interacting friendly directed walks modelling polymers on the square lattice these walks are confined to the quarter plane by a horizontal attractive surface to capture the effects of dna denaturation and adsorption we find the solution to the model s corresponding generating function by means of the obstinate kernel method specifically we apply this technique in two different instances to establish partial solutions for two simplified generating functions of the same underlying model that ignore either surface or shared contacts we then subsequently combine our two partial solutions to find the solution for the full generating function in terms of the two simpler variants this expression guides our analysis of the model where we find the system exhibits four phases and proceed to delineate the full phase diagram showing that all observed phase transitions are second order

rationality of the mobius function of a composition poset we consider the zeta and mobius functions of a partial order on integer compositions first studied by bergeron bousquet melou and dulucq the mobius function of this poset was determined by sagan and vatter we prove rationality of various formal power series in noncommuting variables whose coefficients are evaluations of the zeta function and the mobius function the proofs are either directly from the definitions or by constructing finite state automata we also obtain explicit expressions for generating functions obtained by specializing the variables to commutative ones we reprove sagan and vatter s formula for using this machinery these results are closely related to those of bjorner and reutenauer about subword order and we discuss a common generalization

a sharper condition for the solvability of a three point second order boundary value problem letf 0 1 r2 rbe a function satisfying caratheodory s conditions ande t l1 0 1 let 0 1 r 1 1 be given this paper is concerned with the problem of existence of a solution for the three point boundary value problem formula omitted this problem was studied earlier by gupta ntouyas and tsamatos when 1 and when 1 with 1 in the general case this problem was studied by gupta as a multi point boundary value problem in this paper sharper existence conditions are obtained for the solvability of the above boundary value problem in the general case

asymptotics for entropy integrals associated with exponential weights we establish a first order asymptotic for the entropy integrals 1pn2 log pn2 w2 and 1pn2 log pnw 2 w2 where pn n 0 are the orthonormal polynomials associated with the exponential weight w2

computer based manipulation of systems of equations in elasticity problems with grobner bases abstract the classical method of buchberger s grobner bases for multivariate polynomial equations with a variety of applications in computer algebra is applied to three problems in elasticity concerning i a simple elastic truss ii an elementary plane elasticity problem in rectangular coordinates and iii the derivation of a compatibility partial differential equation for the stress components and the proof of biharmonicity of the same components also in plane elasticity the present results aim at the practical illustration of the possibility of deriving computer generated formulae in elasticity problems by using grobner bases as has been done in the case of geometric problems and the popular and modern computer algebra system maple v equipped with a related package has been used further generalizations of the present results are quite possible and some of these are discussed in brief

asymptotic values for degrees associated with strips of young diagrams abstract the sums s l n occur in the representations of the symmetric and the general linear groups in combinatorics and in pi algebras we give asymptotic values for these sums

triple solutions for second order three point boundary value problems abstract we establish the existence of at least three positive solutions to the second order three point boundary value problem u f t u 0 u 0 0 u u 1 where 0 lt f 0 1 0 0 is continuous we accomplish this by making growth assumptions on f which can apply to many more cases than the sublinear and superlinear ones discussed in recent works

a three point boundary value problem for nonlinear fourth order differential equations abstract in this paper the authors consider the nonlinear fourth order ordinary differential equation e u t g t f u 0 with the boundary conditions b u 0 u 1 u 0 u p u 1 0 some results on the existence and nonexistence of positive solutions to problem e b are obtained results on the existence of infinitely many positive solutions are also presented examples are included to demonstrate that the results are sharp

proof of george andrews s and david robbins s q tspp conjecture the conjecture that the orbit counting generating function for totally symmetric plane partitions can be written as an explicit product formula has been stated independently by george andrews and david robbins around 1983 we present a proof of this long standing conjecture

on a class of languages with holonomic generating functions we define a class of languages rcm obtained by considering regular languages linear constraints on the number of occurrences of symbols and morphisms the class rcm presents some interesting closure properties and contains languages with holonomic generating functions as a matter of fact rcm is related to one way 1 reversal bounded k counter machines and also to parikh automata on letters indeed rcm is contained in l nfcm but not in l dfcm and strictly includes l lpa we conjecture that l dfcm rcm

mellin transforms and asymptotics finite differences and rice s integrals high order differences of simple number sequences may be analysed asymptotically by means of integral representations residue calculus and contour integration this technique akin to mellin transform asymptotics is put in perspective and illustrated by means of several examples related to combinatorics and the analysis of algorithms like digital tries digital search trees quadtrees and distributed leader election

triple positive solutions of three point boundary value problems for fourth order differential equations using the avery and peterson fixed point theorem we investigate the existence of three positive solutions for the fourth order three point boundary value problem u 4 t h t f t u t u t 00 0 x 1 and h t is sign changing on 0 1 the interesting point is that the nonlinear term is allowed to depend on u