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

polya number and first return of bursty random walk rigorous solutions abstract the recurrence properties of random walks can be characterized by polya number i e the probability that the walker has returned to the origin at least once in this paper we investigate polya number and first return for bursty random walk on a line in which the walk has different step size and moving probabilities using the concept of the catalan number we obtain exact results for first return probability the average first return time and polya number for the first time we show that polya number displays two different functional behavior when the walk deviates from the recurrent point by utilizing the lagrange inversion formula we interpret our findings by transferring polya number to the closed form solutions of an inverse function we also calculate polya number using another approach which corroborates our results and conclusions finally we consider the recurrence properties and polya number of two variations of the bursty random walk model

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

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

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

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

superexponential estimates and weighted lower bounds for the square function author s ivanisvili paata treil sergei abstract we prove the following superexponential distribution inequality for any integrable g on 0 1 d with zero average and any lambdag0 x in 0 1 d g geq lambda leq e lambda 2 2 d s g infty 2 where s g denotes the classical dyadic square function in 0 1 d the estimate is sharp when dimension d tends to infinity in the sense that the constant 2 d in the denominator cannot be replaced by c2 d with 01 they work with a special square function s infty and their result does not imply the estimates for the classical square function using good lambda inequalities technique we then obtain unweighted and weighted l p lower bounds for s to get the corresponding good lambda inequalities we need to modify the classical construction we also show how to obtain our superexponential distribution inequality although with worse constants from the weighted l 2 lower bounds for s obtained in 5

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

cocharacter sequences are holonomic abstract let c n n 0 be a codimension sequence of a characteristic zero p i algebra then the c n satisfy a linear homogenous recurrence relation whose coefficients are rational functions of n

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 extremal distributions and sharp l sub p bounds for sums of multilinear forms in this paper we present a study of the problem of approximating the expectations of functions of statistics in independent and dependent random variables in terms of the expectations of functions of the component random variables we present results providing sharp analogues of the burkholder rosenthal inequalities and related estimates for the expectations of functions of sums of dependent nonnegative r v s and conditionally symmetric martingale differences with bounded conditional moments as well as for sums of multilinear forms among others we obtain the following sharp inequalities e nk 1xk t 2max nk 1extk nk 1ak t for all nonnegative r v s x1 xn with e xk x1 xk 1 ak extk k 1 n 1 x003c t x003c 2undefined e nk 1xk t e t 1 max nk 1bk nk 1ask t s for all nonnegative r v s x1 xn with e xsk x1 xk 1 ask e xtk x1 xk 1 bk k 1 n 1 x003c t x003c 2undefined 0 x003c s le t 1undefined or t 2 0 x003c s le 1undefined where 1 is a poisson random variable with parameter 1 as applications new decoupling inequalities for sums of multilinear forms are presented and sharp khintchine marcinkiewicz zygmund inequalities for generalized moving averages are obtained the results can also be used in the study of a wide class of nonlinear statistics connected to problems of long range dependence and in an econometric setup in particular in stabilization policy problems and in the study of properties of moving average and autocorrelation processes the results are based on the iteration of a series of key lemmas that capture the essential extremal properties of the moments of the statistics involved

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

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

gevrey series in quantum topology our aim is to prove that two formal power series of importance to quantum topology are gevrey these series are the kashaev invariant of a knot reformulated by huynh and the second author and the gromov norm of the lmo of an integral homology 3 sphere it follows that the power series associated to a simple lie algebra and a homology sphere is gevrey contrary to the case of analysis our formal power series are not solutions to dieren tial equations with polynomial coecien ts the rst author has conjectured and in some cases proved in joint work with costin that our formal power series have resurgent borel transform with geometrically interesting set of singularities

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

on the small ball inequality in all dimensions abstract let h r denote an l normalized haar function adapted to a dyadic rectangle r 0 1 d we show that for choices of coefficients r we have the following lower bound on the l norms of the sums of such functions where the sum is over rectangles of a fixed volume n d 1 2 r 2 n r h r x l 0 1 d 2 n r 2 n r for some 0 1 2 the point of interest is the dependence upon the logarithm of the volume of the rectangles with n d 1 2 on the left above the inequality is trivial while it is conjectured that the inequality holds with n d 2 2 this is known in the case of d 2 michel talagrand the small ball problem for the brownian sheet ann probab 22 3 1994 1331 1354 mr 95k 60049 and a recent paper of two of the authors dmitriy bilyk michael t lacey on the small ball inequality in three dimensions duke math j 2006 in press arxiv math ca 0609815 proves a partial result towards the conjecture in three dimensions in this paper we show that the argument of dmitriy bilyk michael t lacey on the small ball inequality in three dimensions duke math j 2006 in press arxiv math ca 0609815 can be extended to arbitrary dimension we also prove related results in the subjects of the irregularity of distribution and approximation theory the authors are unaware of any prior results on these questions in any dimension d 4

pi algebras and their cocharacters abstract all pi algebras r satisfy identities of the form f x y x 1 y 1 x 2 y 2 y n 1 x n theorem a the existence of these identities imply also that cocharacters x n r lie in a hook shaped strip of width depending on the degree of the minimal identity of r theorem c this extends a characterization of rings satisfying a capelli identity theorem b

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

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

elimination of variables in linear solvable polynomial algebras and holonomicity abstract let k be a field of characteristic 0 based on the gelfand kirillov dimension computation of modules over solvable polynomial k algebras where solvable polynomial algebras are in the sense of a kandri rody and v weispfenning 1990 j symbolic comput 9 1 26 we prove that the elimination lemma obtained from d zeilberger 1990 j comput appl math 32 321 368 by using holonomic modules over the weyl algebra a n k and used in the automatic proving of special function identities holds for a class of solvable polynomial k algebras without any holonomicity restriction this opens a way to the solution of the extension contraction problem stemming from the automatic proving of multivariate identities with respect to the finiteness in the sense of f chyzak and b salvy 1998 j symbolic comput 26 187 227 it also yields a holonomicity so that automatic proving of multivariate identities may be dealt with by manipulating polynomial function coefficients instead of rational functions

on generating series of finitely presented operads abstract given an operad p with a finite grobner basis of relations we study the generating functions for the dimensions of its graded components p n under moderate assumptions on the relations we prove that the exponential generating function for the sequence dim p n is differential algebraic and in fact algebraic if p is a symmetrization of a non symmetric operad if in addition the growth of the dimensions of p n is bounded by an exponent of n or a polynomial of n in the non symmetric case then moreover the ordinary generating function for the above sequence dim p n is rational we give a number of examples of calculations and discuss conjectures about the above generating functions for more general classes of operads

elementary theory of free non abelian groups we prove that any two non abelian free groups have the same elementary theory and that this theory is decidable these results solve two questions that were raised by tarski in 1945

polycyclic by finite groups and first order sentences abstract a finitely generated group g is said to be quasi finitely axiomatizable qfa in the sense of a nies 2003 see 5 if there exists a first order sentence satisfied by g such that each finitely generated group satisfying is isomorphic to g we give a complete algebraic characterization of polycyclic by finite groups which are qfa a polycyclic by finite group g is qfa if and only if for each subgroup h of finite index in g the centre z h is included in the isolator of the derived subgroup h moreover we show that in this context there is an equivalence between being qfa and being prime these results generalize the ones that f oger 2006 obtains in 8 in the context of nilpotent by finite groups further we show that each polycyclic by finite group satisfies a sentence such that each finitely generated group which satisfies is polycyclic by finite and has the same hirsch number this result is both stronger and weaker than the ones of g sabbagh and j s wilson 1991 in 15 on the one hand of d raphael 1996 in 11 on the other hand they proved the same by using infinitely many sentences but with a restriction on the complexity of the sentences these results are corollaries of our main theorem which is an adaptation to the class of polycyclic by finite groups of a theorem by f oger and g sabbagh 2006 in 9 in the context of finite by nilpotent groups

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

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

the colored homflypt function is q holonomic we prove that the homflypt polynomial of a link colored by partitions with a fixed number of rows is a q holonomic function by specializing to the case of knots colored by a partition with a single row it proves the existence of an a q superpolynomial of knots in 3 space as was conjectured by string theorists our proof uses skew howe duality that reduces the evaluation of web diagrams and their ladders to a poincare birkhoff witt computation of an auxiliary quantum group of rank the number of strings of the ladder diagram the result is a concrete and algorithmic web evaluation algorithm that is manifestly q holonomic

nonarithmetic superrigid groups counterexamples to platonov s conjecture margulis showed that most arithmetic groups are superrigid platonov conjectured conversely that finitely generated linear groups which are superrigid must be of arithmetic type we construct counterexamples to platonov s conjecture

sharp martingale inequalities and applications to riesz transforms on manifolds lie groups and gauss space abstract we prove new sharp l p logarithmic and weak type inequalities for martingales under the assumption of differential subordination the l p estimates are feynman kac type versions of burkholder s celebrated martingale transform inequalities from the martingale l p inequalities we obtain that riesz transforms on manifolds of nonnegative bakry emery ricci curvature have exactly the same l p bounds as those known for riesz transforms in the flat case of r d from the martingale logarithmic and weak type inequalities we obtain similar inequalities for riesz transforms on compact lie groups and spheres combining the estimates for spheres with poincare s limiting argument we deduce the corresponding results for riesz transforms associated with the ornstein uhlenbeck semigroup thus providing some extensions of p a meyer s l p inequalities

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

on zeilberger s algorithm and its q analogue abstract gosper s and zeilberger s algorithms for summation of terminating hypergeometric series as well as the q versions of these algorithms are described in a very rigorous way the paper is a companion to maple v procedures implementing these algorithms it concludes with the help information for these procedures

the fuchsian differential equation of the square lattice ising model χ 3 susceptibility using an expansion method in the variables xi that appear in n 1 dimensional integrals representing the n particle contribution to the ising square lattice model susceptibility we generate a long series of coefficients for the three particle contribution 3 using an n4 polynomial time algorithm we give the fuchsian differential equation of order 7 for 3 that reproduces all the terms of our long series an analysis of the properties of this fuchsian differential equation is performed

optimum bounds for the distributions of martingales in banach spaces a general device is proposed which provides for extension of exponential inequalities for sums of independent real valued random variables to those for martingales in the 2 smooth banach spaces this is used to obtain optimum bounds of the rosenthal burkholder and chung types on moments of the martingales in 2 smooth banach spaces in turn it leads to best order bounds on moments of sums of independent random vectors in any separable banach spaces although the emphasis is put on infinite dimensional martingales most of the results seem to be new even for one dimensional martingales moreover the bounds on moments of the rosenthal burkholder type seem to be to a certain extent new even for sums of independent real valued random variables analogous inequalities for one dimensional supermartingales are given

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

random k noncrossing rna structures in this paper we introduce a combinatorial framework that provides an interpretation of rna pseudoknot structures as sampling paths of a markov process our results facilitate a variety of applications ranging from the energy based sampling of pseudoknot structures as well as the ab initio folding via hidden markov models our main result is an algorithm that generates rna pseudoknot structures with uniform probability this algorithm serves as a steppingstone to sequence specific as well as energy based transition probabilities the approach employs a correspondence between pseudoknot structures parametrized in terms of the maximal number of mutually crossing arcs and certain tableau sequences the latter can be viewed as lattice paths the main idea of this paper is to view each such lattice path as a sampling path of a stochastic process and to make use of d finiteness for the efficient computation of the corresponding transition probabilities

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

on the failure of lower square function estimates in the non homogeneous weighted setting we show that the classical a infty condition is not sufficient for a lower square function estimate in the non homogeneous weighted l 2 space we also show that under the martingale a 2 condition an estimate holds true but the optimal power of the characteristic jumps from 1 2 to 1 even when considering the classical a 2 characteristic this is in a sharp contrast to known estimates in the dyadic homogeneous setting as well as the recent positive results in this direction on the discrete time non homogeneous martingale transforms last we give a sharp a infty estimate for the n adic homogeneous case growing with n

densities of short uniform random walks in higher dimensions we study arithmetic properties of short uniform random walks in arbitrary dimensions with a focus on explicit hypergeometric evaluations of the moment functions and probability densities in the case of up to five steps somewhat to our surprise we are able to provide complete extensions to arbitrary dimensions for most of the central results known in the two dimensional case

first order rigidity of non uniform higher rank arithmetic groups if gamma is an irreducible non uniform higher rank characteristic zero arithmetic lattice for example mathrm sl n mathbb z n ge 3 and lambda is a finitely generated group that is elementarily equivalent to gamma then lambda is isomorphic to gamma

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

sharp square function inequalities for conditionally symmetric martingales let t be a conditionally symmetric martingale taking values in a hilbert space 1ei and let s i be its square function if vp is the smallest positive zero of the confluent hypergeometric function and ap is the largest positive zero of the parabolic cylinder function of parameter p then the following inequalities are sharp 3 vp iis llp 2 moreover the constants vp and ap for the cases mentioned above are also best possible for the marcinkiewicz paley inequalities for haar functions