applications of riordan matrix functions to bernoulli and euler polynomials abstract we define riordan matrix functions associated with riordan arrays and study their algebraic properties we also give their applications in the construction of new classes of bernoulli and euler polynomials and bernoulli and euler numbers referred to as the duals and conjugate bernoulli and euler polynomials and dual and conjugate bernoulli and euler numbers respectively

row polynomial matrices of riordan arrays abstract let r r n k n k 0 be a riordan array define the row polynomials r n q k 0 n r n k q k and the row polynomial matrix r q r n k q n k 0 by r n k q j k n r n j q j k then r q is also a riordan array with the r n q located on the leftmost column of r q in this paper we investigate combinatorial properties of the matrix r q and the sequence r n q n 0 including their characterizations the q total positivity of r q and the q log convexity of r n q n 0

row sums and alternating sums of riordan arrays abstract here we use row sum generating functions and alternating sum generating functions to characterize riordan arrays and subgroups of the riordan group numerous applications and examples are presented which include the construction of girard waring type identities we also show the extensions to weighted sum generating functions called the expected value generating functions of riordan arrays

geometric properties of heisenberg type groups on montre qu il y a une certaine metrique riemannienne contravariante qui induit une longueur d arc

diagonally striped matrices and approximate inverse preconditioners the inverse of a banded matrix is in general dense if the structure of the original banded matrix is striped that is the non zero diagonals are separated by one or more zero diagonals the inverse may exhibit a similar striped structure the motivation for studying inverses of striped matrices is to obtain efficient preconditioners for systems arising from radiation transport equations whose matrices include dominant values along diagonal stripes linear systems whose system matrix has a striped inverse lend themselves to the use of a sparse approximate inverse spai preconditioner whose structure is derived from that of the actual inverse

some counter examples on totally positive riordan arrays abstract we consider a recent result which claims that if the series d t and h t are polya frequency formal power series then the riordan array d t h t is totally positive namely we pose and answer the question if the various types of the converse of this claim are also true

the elements of finite order in the riordan group over the complex field abstract an element of finite order in the riordan group over the real field must have order 1 or 2 if we extend all the entries to be complex numbers then it may have any finite order in the present paper we investigate the elements of finite order of the riordan group over the complex field this notion leads us to define an element of pseudo order k 2 and k pseudo involution respectively it turns out that the inverse of a k pseudo involution only differs from it in signs we clarify some relationship between the elements of pseudo order k and k pseudo involutions in particular k pseudo involutions for k 0 mod 4 are characterized by a single sequence the subgroups of the riordan group formed by the elements having pseudo order of a prime power p r are also introduced

umbral presentations for polynomial sequences abstract using random variables as motivation this paper presents an exposition of formalisms developed in 1 2 for the classical umbral calculus a new construction and a variety of examples are presented culminating in several descriptions of sequences of binomial type in terms of umbral polynomials

matrices determined by a linear recurrence relation among entries abstract a matrix a a ij is called a 7 matrix if its entries satisfy the recurrence relation a i 1 j 1 a i 1 j a ij where are fixed numbers a 7 matrix is completely determined by its first row and first column in this paper we determine the structure of 7 matrices and investigate the sequences represented by columns of infinite 7 matrices

boundary element analysis of two dimensional elastoplastic contact problems in this paper a boundary element method for two dimensional elastoplastic stress analysis of frictional contact problems is presented the bodies in contact are treated as separate regions the contact conditions are used to join the different system of equations for different regions of contacted bodies and hence an overall system of equation is obtained an incremental and iterative procedure can be used to find the contact load or the contact extent and the proper contact conditions to include the plastic deformation in the analysis the initial strain algorithm is employed elastic perfectly plastic or work hardening material behaviour can be assumed for the numerical analysis an isoparametric three noded line elements are used to represent the boundary and eight noded quadrilateral or six noded triangular elements are used for the interior of the domain the displacement rates and traction rates are assumed to vary quadratically and the shape functions for the interior strain rates are also of quadratic type as an example the behaviour of an elastic and elastoplastic body with a smooth circular inclusion under the increasing load is presented

johann faulhaber and sums of powers early 1 7th century mathematical publications of johann faulhaber contain some remarkable theorems such as the fact that the r fold summation of lm 2m nm is a polynomial in n n r when m is a positive odd number the present paper explores a computation based approach by which faulhaber may well have discovered such results and solves a 360 year old riddle that faulhaber presented to his readers it also shows that similar results hold when we express the sums in terms of central factorial powers instead of ordinary powers faulhaber s coefficients can moreover be generalized to noninteger exponents obtaining asymptotic series for 1 2 na in powers of n 1 n 1 1

a full tangent stiffness field boundary element formulation for geometric and material non linear problems of solid mechanics the field boundary element method naturally admits the solution algorithm in the incompressible regimes of fully developed plastic flow this is not the case with the generally popular finite element method without further modifications to the method such as reduced integration or a mixed method for treating the dilatational deformation the analyses by the field boundary element method for geometric and material non linear problems are generally carried out by an incremental algorithm where the velocities or displacement increments on the boundary are treated as the primary variables and an initial strain iteration method is commonly used to obtain the state of equilibrium for problems such as buckling and diffused tensile necking involving very large strains such a solution scheme may not be able to capture the bifurcation phenomena or the convergence will be unacceptably slow when the post bifurcation behaviour needs to be analysed to avoid this predicament a full tangent stiffness field boundary element formulation which takes the initial stress velocity gradient displacement gradient coupling terms accurately into account is presented in this paper here the velocity field both inside and on the boundary are treated as primary variables the large strain plasticity constitutive equation employed is based on an endochronic model of combined isotropic kinematic hardening plasticity using the concepts of material director triad and the associated plastic spin a generalized mid point radial return algorithm is presented for determining the objective increments of stress from the computed velocity gradients numerical results are presented for problems of diffuse necking involving very large strains and plastic instability in initially perfect elastic plastic plates under tension these results demonstrate the clear superiority of the full tangent stiffness algorithm over the initial strain algorithm in the context of the integral equation formulations for large strain plasticity

a priori sparsity patterns for parallel sparse approximate inverse preconditioners parallel algorithms for computing sparse approximations to the inverse of a sparse matrix either use a prescribed sparsity pattern for the approximate inverse or attempt to generate a good pattern as part of the algorithm this paper demonstrates that for pde problems the patterns of powers of sparsified matrices psms can be used a priori as effective approximate inverse patterns and that the additional effort of adaptive sparsity pattern calculations may not be required psm patterns are related to various other approximate inverse sparsity patterns through matrix graph theory and heuristics concerning the pde s green s function a parallel implementation shows that psm patterned approximate inverses are significantly faster to construct than approximate inverses constructed adaptively while often giving preconditioners of comparable quality

q double riordan matrices abstract in this paper the concept of q double riordan matrices is given by using binary operations q 2 and 1 q 2 we give a q analogue of the fundamental theorem of double riordan matrices and study the structure of the entries of the multiplications of q double riordan matrices as an application q analogue of the pascal fibonacci array can be obtained by a q double riordan representation

a formula to construct all involutions in riordan matrix groups abstract we get a formula for all finite riordan involutions using the natural approximation induced by the inverse limit construction we also describe all infinite riordan matrices which are involutions in the riordan group finally we get some consequences

half riordan array sequences abstract we construct sequences of vertical and horizontal halves of various riordan arrays by using the vertical and horizontal halve riordan array transform operators the sequence characterizations of halves of riordan arrays are presented and applied to the halve riordan array transformation the conditions of transforming a riordan array to a pseudo involution riordan array by using the halve riordan array transform are given the condition of preserving the elements of a certain subgroup of the riordan group under the halve riordan array transformation are shown other properties of the halves of riordan arrays and their entries such as related recurrence relations double variable generating functions combinatorial interpretations are also studied

double parameter recurrences for polynomials in bi infinite riordan matrices and some derived identities abstract we present some double parameter recurrence relations for families of polynomials in bi infinite riordan matrices from this point of view we treat some riordan families of generalized fibonacci and lucas polynomials we use these to obtain some identities related mainly with binomial coefficients and fibonacci numbers

matrix characterizations of riordan arrays abstract here we discuss two matrix characterizations of riordan arrays p matrix characterization and a matrix characterization p matrix is an extension of the stieltjes matrix defined in 28 and the production matrix defined in 8 by modifying the marked succession rule introduced in 21 a combinatorial interpretation of the p matrix is given the p matrix characterizations of some subgroups of riordan group are presented which are used to find some algebraic structures of the subgroups we also give the p matrix characterizations of the inverse of a riordan array and the product of two riordan arrays a matrix characterization is defined in 20 and it is proved to be a useful tool for a riordan array while on the other side the a sequence characterization is very complex sometimes by using the fundamental theorem of riordan arrays a method of construction of a matrix characterizations from riordan arrays is given the converse process is also discussed several examples and applications of two matrix characterizations are presented

riordan group involutions we study involutions in the riordan group especially those with combinatorial meaning we give a new determinantal criterion for a matrix to be a riordan involution and examine several classes of examples a complete characterization of involutions in the appell subgroup is developed in another direction we find several examples that generalize the rna matrix but are of independent interest

riordan arrays and d orthogonality abstract using the theory of the riordan group and d orthogonal polynomials we shall show that sequences of d orthogonal polynomials can be also generated by the riordan group we interpret some families of d orthogonal polynomials within the framework of the riordan group

horizontal and vertical formulas for exponential riordan matrices and their applications abstract in this paper we show that an infinite lower triangular matrix a a i j i j n 0 is an exponential riordan matrix a e g f given by i j a i j z i i g f j j if and only if there exist both a horizontal pair h n h n n 0 and a vertical pair v n v n n 0 of sequences that represent all the elements in the matrix as a consequence we obtain that if the horizontal and vertical pairs of an exponential riordan matrix are identical then the matrix is an involution in addition this concept can be applied to obtain the determinants of the production matrix and some conditions for the d orthogonality of the sheffer polynomial sequences

algebraic aspects of some riordan arrays related to binary words avoiding a pattern we consider some riordan arrays related to binary words avoiding a pattern p which can be easily studied by means of an a matrix rather than their a sequence both concepts allow us to define every element as a linear combination of other elements in the array the a sequence is unique and corresponds to a linear dependence from the previous row the a matrix is not unique and corresponds to a linear dependence from several previous rows however for the problems considered in the present paper we show that the a matrix approach is more convenient we provide explicit algebraic generating functions for these riordan arrays and obtain many statistics on the corresponding languages we thus obtain a deeper insight of the languages l p of binary words avoiding p having a number of 0s less or equal to the number of 1s

riordan graphs i structural properties abstract in this paper we use the theory of riordan matrices to introduce the notion of a riordan graph the riordan graphs are a far reaching generalization of the well known and well studied pascal graphs and toeplitz graphs and also some other families of graphs the riordan graphs are proved to have a number of interesting fractal properties which can be useful in creating computer networks with certain desirable features or in obtaining useful information when designing algorithms to compute values of graph invariants the main focus in this paper is the study of structural properties of families of riordan graphs obtained from infinite riordan graphs which includes a fundamental decomposition theorem and certain conditions on riordan graphs to have an eulerian trail cycle or a hamiltonian cycle we will study spectral properties of the riordan graphs in a follow up paper

simple proofs of open problems about the structure of involutions in the riordan group abstract we prove that if d g x f x is an element of order 2 in the riordan group then g x exp x xf x for some antisymmetric function x z also we prove that every element of order 2 in the riordan group can be written as bmb 1 for some element b and m 1 1 in the riordan group these proofs provide solutions to two open problems presented by l shapiro l w shapiro some open questions about random walks involutions limiting distributions and generating functions adv appl math 27 2001 585 596

riordan graphs ii spectral properties abstract the authors of this paper have used the theory of riordan matrices to introduce the notion of a riordan graph in 3 riordan graphs are proved to have a number of interesting fractal properties and they are a far reaching generalization of the well known and well studied pascal graphs and toeplitz graphs and also some other families of graphs the main focus in 3 is the study of structural properties of families of riordan graphs obtained from certain infinite riordan graphs in this paper we use a number of results in 3 to study spectral properties of riordan graphs our studies include but are not limited to the spectral graph invariants for riordan graphs such as the adjacency eigenvalues signless laplacian eigenvalues nullity positive and negative inertia indices and rank we also study determinants of riordan graphs in particular giving results about determinants of catalan graphs

an application of riordan arrays to the transient analysis of m m 1 queues abstract riordan arrays are used extensively in many contexts as a combinatorial tool for solving enumeration problems such as lattice path counting problems in this paper we first show how riordan array method can be applied to the transient analysis of m m 1 queue with zero customer at the initial point second we extend this method to the generalized riordan array with multiple support functions in order to deal with the transient analysis of m m 1 queue with non zero customers at the initial point numerical examples are also given to show how easy and quick the transient probability obtained from the riordan method can be computed

a symbolic treatment of riordan arrays abstract we approach riordan arrays and their generalizations via umbral symbolic methods this new approach allows us to derive fundamental aspects of the theory of riordan arrays as immediate consequences of the umbral version of the classical abel s identity for polynomials in particular we obtain a novel non recursive formula for riordan arrays and derive from this new formula some known recurrences and a new recurrence relation for riordan arrays

schroder matrix as inverse of delannoy matrix abstract using riordan arrays we introduce a generalized delannoy matrix by weighted delannoy numbers it turns out that delannoy matrix pascal matrix and fibonacci matrix are all special cases of the generalized delannoy matrices meanwhile schroder matrix and catalan matrix also arise in involving inverses of the generalized delannoy matrices these connections are the focus of our paper the half of generalized delannoy matrix is also considered in addition we obtain a combinatorial interpretation for the generalized fibonacci numbers

h type groups and iwasawa decompositions since their introduction by a kaplan kpl some ten years ago generalised heisenberg groups also known as groups of heisenberg type or h type groups have provided a framework in which to construct interesting examples in geometry and analysis see for instance c2 kp2 kp3 kpr k2 rl r2 tl tv the iwasawa n groups associated to all the real rank one simple lie groups are h type so one has a convenient vehicle for studying these in a unified way many problems on these simple lie groups can be reduced to a problem on h type groups via the so called noncompact picture and often problems on h type groups can be solved on all the groups of the family in one fell swoop as in for example ch ck dr out of this approach to studying simple lie groups several problems arise such as why only some h type groups correspond to simple lie groups of real rank one in this paper we discuss various features of iwasawa n groups which distinguish them in the class of all h type groups we shall show that all h type groups which possess certain geometric properties clearly possessed by iwasawa n groups satisfy a lie algebraic condition implicit in the work of b kostant kt2 that we shall call the j condition we shall also use elementary clifford algebra to classify the

a determinantal approach to appell polynomials a new definition by means of a determinantal form for appell 1880 1 polynomials is given general properties some of them new are proved by using elementary linear algebra tools finally classic and non classic examples are considered and the coefficients calculated by an ad hoc mathematica code for particular sequences of appell polynomials are given

representing polynomials as characteristic polynomials via the stieltjes transform abstract the stieltjes transform s a of an infinite lower triangular matrix a with nonzero diagonal entries is defined by s a a 1 a where a is the matrix obtained from a by deleting its initial row in this paper we express a sequence of polynomials as the characteristic polynomials of the stieltjes transforms using a highly structured infinite lower triangular matrix called a riordan matrix as a result computation of the zeros of such polynomials becomes amenable to iterative methods for computing eigenvalues or to eigenvalue location theorems such as the gersgorin theorem we also describe a finite analog of the polynomial correspondence and its relationship to eigenvalue regions as an application the recurrence relations for several polynomial sequences are obtained using the stieltjes transform

enumeration of directed site animals on two dimensional lattices studies the problem of directed site animals on the square triangular and hexagonal lattices closed form expressions are proposed for a s the number of animals of size s on the square and triangular lattices these expressions have been checked for s or 33 and s or 10 for the square and triangular lattices respectively by explicit enumeration they imply that a s varies as lambda 2 s 0 for large s where lambda 3 for the square lattice and lambda 4 for the triangular and theta 1 2 for both for the hexagonal lattice a s is found for s or 48 the results are consistent with lambda 2 0252 or 0 0005 and theta 1 2

functions and jordan canonical forms of riordan matrices abstract this paper collects results about riordan arrays in the framework of matrix functions actually the following methodology applies to any square matrix m m with exactly one eigenvalue of algebraic multiplicity m n generalized lagrange bases are used to construct hermite polynomials that interpolate a family of functions moreover we show a parallel application of such functions via jordan canonical forms and case studies are given

some inverse limit approaches to the riordan group abstract using the inverse limit tool we obtain the riordan group in its infinite and bi infinite representations from groups of finite riordan matrices we define two different reflections for bi infinite riordan matrices employing these definitions we give answers to the problems d d and d d in the riordan group so we describe the self complementary and self dual riordan matrices

elements of finite order in the riordan group and their eigenvectors abstract we consider elements of finite order in the riordan group r over a field f of characteristic 0 we solve for all n 2 two foundational questions posed by l shapiro 2001 for the case n 2 involutions given f x of finite compositional order n theorem 1 gives a formula for g x showing that those g x which make g x f x r a riordan element of order n are precisely those for which the matrix g x f x has an eigenvector h 0 h 1 t with h 0 0 theorem 2 classifies finite order riordan group elements up to conjugation in r and gives a normal form for finite order riordan arrays under similarity this leads to theorem 3 a formula for all eigenvectors of finite order riordan arrays and results in interesting combinatorial identities we relate our work to 2008 and 2013 papers of cheon and kim which motivated this paper and we solve their open question this circle of ideas closes with a new proof of c marshall s theorem which finds the unique f x given bi invertible g x such that g x f x is an involution

recurrence relations for polynomial sequences via riordan matrices we give recurrence relations for any family of generalized appell polynomials unifying so some known recurrences for many classical sequences of polynomials our main tool to get our goal is the riordan group we use the product of riordan matrices to interpret some relationships between different polynomial families moreover using the hadamard product of series we get a general recurrence relation for the polynomial sequences associated to the so called generalized umbral calculus

bivariate delta evolution equations and convolution polynomials computing polynomial expansions of solutions abstract this paper describes an application of rota and collaborator s ideas about the foundation on combinatorial theory to the computing of solutions of some linear functional partial differential equations we give a dynamical interpretation of the convolution families of polynomials concretely we interpret them as entries in the matrix representation of the exponentials of certain contractive linear operators in the ring of formal power series this is the starting point to get symbolic solutions for some functional partial differential equations we introduce the bivariate convolution product of convolution families to obtain symbolic solutions for natural extensions of functional evolution equations related to delta operators we put some examples to show how these symbolic methods allow us to get closed formulas for solutions of genuine partial differential equations we create an adequate framework to base theoretically some of the performed constructions and to get some existence and uniqueness results

generalized humbert polynomials via generalized fibonacci polynomials in this paper we define the generalized p q fibonacci polynomials un m x and generalized p q lucas polynomials vn m x and further introduce the generalized humbert polynomials un m r x as the convolutions of un m x we give many expressions expansions recurrence relations and differential recurrence relations of un m r x and study the matrices and determinants related to the polynomials un m x vn m x and un m r x finally we present an algebraic interpretation for the generalized humbert polynomials un m r x it can be found that various well known polynomials are special cases of un m x vn m x or un m r x therefore by introducing the general polynomials un m x vn m x and un m r x we have a unified approach to dealing with many special polynomials in the literature

palindromes and pseudo involution multiplication abstract a riordan array g f is called a pseudo involution if g f m or equivalently m g f where m 1 z is an involution this paper presents a palindromic property of pseudo involutions which seems both novel and useful if a and b are both pseudo involutions then so is the triple product aba with this it follows that if a b c are pseudo involutions so is any palindromic word using these symbols

sequence characterizations of double riordan arrays and their compressions abstract inspired by the fibonacci tree shown in the recent book catalan numbers by richard stanley we present a combinatorial way to construct double riordan arrays by using eco technique the sequence characterizations of double riordan arrays and subgroups of the double riordan group are found as an extension of pascal fibonacci triangle the compression forms of double riordan arrays called double quasi riordan arrays are defined the connections of lower triangular matrices and double riordan arrays as well as their compressions are given those results are also extended to the case of high order riordan arrays which are defined in the paper the pairs of sheffer polynomials and pairs of summation formulas associated with double riordan arrays are defined and discussed

shift operators defined in the riordan group and their applications abstract in this paper we discuss a linear operator t defined in riordan group r by using the upper shift matrix u and lower shift matrix u t namely for each r r t r u r u t some isomorphic properties of the operator t and the structures of its range sets for different domains are studied by using the operator t and the properties of bell subgroup of r the riordan type chu vandermonde identities and the riordan equivalent identities of format last theorem and beal conjecture are shown the applications of the shift operators to the complementary riordan arrays and to the riordan involutions and riordan pseudo involutions are also presented

a unified approach for the catalan matrices by using riordan arrays abstract we study generalized catalan matrices based on the riordan array and fuss catalan numbers a unified combinatorial interpretation for the entries of the generalized catalan matrices is presented by means of m dyck paths some properties and examples of the generalized catalan matrices are given including a new convolution formula for the generalized catalan numbers finally we present applications of generalized catalan matrices to the problems in counting the hill free and lower peak free m dyck paths

fuss catalan matrices their weighted sums and stabilizer subgroups of the riordan group abstract in this paper we present the riordan arrays called fuss catalan matrices which are constructed by the convolutions of the generating functions of the fuss catalan numbers we also discuss weighted sums of the fuss catalan matrices using such matrices as transformations of recursive sequences and their connection with stabilizer subgroups of the riordan group

a combinatorial equivalence relation for formal power series abstract in this paper we examine an equivalence relation on the set of formal power series with nonzero constant term this is done both in terms of functional equations and also by interlacing two concepts from riordan group theory the a sequence and the bell subgroup the best known example gives an equivalence class 1 z 1 1 z c z t z q z where c z t z and q z are generating functions of the catalan numbers ternary numbers and quaternary numbers respectively a power series for one member of an equivalence class can be transformed into power series for the rest of members in the equivalence class and interpretations in terms of weighted lattice paths can also be given

on the chromatic number of integral circulant graphs integral circulant graphs are a generalization of unitary cayley graphs recently studied by klotz and sander the integral circulant graph x n d has vertices 0 1 n 1 and two vertices a and b are adjacent iff gcd x y n d where d d d n 1 d 3 we construct a family of counterexamples using exhaustive search algorithm for graph coloring and disprove the conjecture in this case

some algebraic structure of the riordan group abstract in this paper we highlight some of the algebraic properties of the riordan group we present some new properties and provide proofs for some known properties for instance proofs of properties of the power bell and derivative subgroups are presented we also give a new property for generating stochastic riordan matrices which are elements of the stochastic riordan subgroup centralizers stabilizers and certain riordan group isomorphisms are also given and proved we conclude by proving some properties connecting similar riordan matrices and pseudo involutions in the riordan group

identities induced by riordan arrays historically there exist two versions of the riordan array concept the older one better known as recursive matrix consists of bi infinite matrices d n k n k is an element of z k n implies d n k 0 deals with formal laurent series and has been mainly used to study algebraic properties of such matrices the more recent version consists of infinite lower triangular arrays d n k n k is an element of n k n implies d n k 0 deals with formal power series and has been used to study combinatorial problems here we show that every riordan array induces two characteristic combinatorial sums in three parameters n k m is an element of z these parameters can he specialized and generate an indefinite number of other combinatorial identities which are valid in the hi infinite realm of recursive matrices

parametric catalan numbers and catalan triangles abstract here presented a generalization of catalan numbers and catalan triangles associated with two parameters based on the sequence characterization of bell type riordan arrays among the generalized catalan numbers a class of large generalized catalan numbers and a class of small generalized catalan numbers are defined which can be considered as an extension of large schroder numbers and small schroder numbers respectively using the characterization sequences of bell type riordan arrays some properties and expressions including the taylor expansions of generalized catalan numbers are given a few combinatorial interpretations of the generalized catalan numbers are also provided finally a generalized motzkin numbers and motzkin triangles are defined similarly an interrelationship among parametrical catalan triangle pascal triangle and motzkin triangle is presented based on the sequence characterization of bell type riordan arrays

advanced boundary element analysis of two and three dimensional problems of elasto plasticity an advanced formulation of the boundary element method has been developed for inelastic analysis based on an initial stress approach the iterative solution algorithm makes use of an accelerated initial stress approach in which the past history of initial stresses are used to obtain an initial estimate for the current increment in the present analysis the geometry and functions are represented by higher order quadratic shape functions to model complex geometries and rapid functional variations accurately the methods of numerical integration of the kernels particularly the singular type are substantially improved by devising suitable automatic sub segmentation routines that incorporate the recent developments in mapping procedures the formulations have been implemented for two dimensional plane stress plane strain and three dimensional elasto plasticity problems

riordan arrays and related polynomial sequences abstract in this paper we generalize the results due to luzon and moron and present a characterization of the generalized riordan arrays using this characterization we establish a recurrence for the generalized sheffer sequences and study some special types of polynomial sequences including the generalized lucas u sequence and v sequence and the generalized humbert polynomial sequence in particular by appealing to the factorizations of riordan arrays the umbral composition of polynomial sequences and the hadamard product of series we establish various relations among these polynomial sequences thus we give a unified approach to some well known polynomial sequences appeared in combinatorics and the theory of special functions

recursive matrices and umbral calculus two major objectives can be seen to guide much recent work in enumeration 1 to single out a limited variety of recurrences for numerical sequences which will encompass counting problems of wide enough type 2 to recover from empirical data an underlying set theoretic structure which would reveal the source of the given recursion we are here concerned with the first of these objectives though the eventual understanding of the second is tacitly present if only as a goal we noticed the coincidence of several computations which similar as they are in retrospect had failed to realize their kinship on leafing through the unique assembly of recursively solvable combinatorial problems in comtet s and sloane s invaluable collections one is struck by the repeated occurrence of one and the same kind of double recursion more strikingly the same recursion is seen to occur in the polynomial sequences of the umbra1 calculus of roman and rota see 25 everywhere the lagrange inversion formula for power series plays a pivotal role much work is nowadays going into the unraveling of the everdeeper layers of combinatorial significance of this formula both in the ordinary case and in its as yet partially worked out noncommutative and qanalogs andrews foata garsia gessel joni raney reiner schtitzenberger to name but a few whatever their origins the identities abutting lagrange inversion are expressed by integers alone this suggests not only a hidden set theoretic layer but a characteristic free generalization as well this generalization is the central theme of our work we define a monoid of infinite matrices recursive matrices for short the entries of these matrices give the sought out recursion for example that for coefficients of binomial and sheffer polynomials and factor sequences as well as that of the special sequences recently introduced by roman in 26 546 002 l 8693 82 040546 28 02 00 0

combinatorics of a generalized narayana identity abstract the narayana identity is a well known formula that expresses the classical catalan numbers as sums of the ordinary narayana numbers in this paper we generalize the narayana identity to a family of riordan arrays including the array of ballot numbers the classical catalan triangle and several generalized catalan triangles recently studied a combinatorial description based on non crossing partitions is given for this identity for the column recursive rule and for the sheffer sequence associated with any array of the family

an extension of the generalized pascal matrix and its algebraic properties abstract the extended generalized pascal matrix can be represented in two different ways as a lower triangular matrix n x y or as a symmetric n x y these matrices generalize p n x q n x and r n x which are defined by zhang and by call and velleman a product formula for n x y has been found which generalizes the result of call and velleman it is shown that not only can n x y be factorized by special summation but also n x y as q n xy s t y 1 x or n x y p n t y x finally the inverse of n x y and the values of det n x y det n 1 x y det n x y and det n 1 x y are given

a new bem formulation for tow and three dimensional elastoplasticity using particular integras new two and three dimensional bem formulations are developed for elastoplastic stress analysis these new procedures differ from previous work in that volume integration is not required to incorporate the nonlinear effects in the analysis instead initial stress rates are introduced in the bem system via particular integrals the present formulation is implemented in a general purpose multi region system and examples are presented to demonstrate the accuracy and versatility of the method

an algebraic structure for faber polynomials abstract let be the set of functions convergent for all z 1 with a laurent series of the form f z z n 0 a n z n in this paper we prove that the set of faber polynomial sequences over and the set of their normalized kth derivative sequences form groups which are isomorphic to the hitting time subgroup and the bell k subgroup of the riordan group respectively further a relationship between such faber polynomial sequences and lucas and sheffer polynomial sequences is derived

convolutions of the generalized morgan voyce polynomials we consider the following classes of polynomials b n m s x c n m s x b n m s x and c n m s x which are convolutions of the generalized morgan voyce polynomials where s is a nonnegative integer these convolutions are related to the classical morgan voyce chebyshev and jacobsthal polynomials

notes on the total positivity of riordan arrays abstract let r d t h t be a riordan array we show that if both d t and h t are polya frequency formal power series then r is totally positive

riordan arrays and harmonic number identities let the numbers p r n k be defined by p r n k p r h n 1 h k 1 h n r h k r where p r x 1 x r 1 ry r 0 x 1 1 x 2 r 1 x r and y r are the exponential complete bell polynomials by observing that the numbers p r n k generate two riordan arrays we establish several general summation formulas from which series of harmonic number identities are obtained in particular some of these harmonic number identities also involve other special combinatorial sequences such as the stirling numbers of both kinds the lah numbers the bernoulli numbers and polynomials and the cauchy numbers of both kinds

finite and infinite dimensional lie group structures on riordan groups abstract we introduce a frechet lie group structure on the riordan group we give a description of the corresponding lie algebra as a vector space of infinite lower triangular matrices we describe a natural linear action induced on the frechet space k n by any element in the lie algebra we relate this to a certain family of bivariate linear partial differential equations we obtain the solutions of such equations using one parameter groups in the riordan group we show how a particular semidirect product decomposition in the riordan group is reflected in the lie algebra we study the stabilizer of a formal power series under the action induced by riordan matrices we get stabilizers in the fraction field k x using bi infinite representations we provide some examples the main tool to get our results is the paper 18 where the riordan group was described using inverse sequences of groups of finite matrices

a general umbral calculus in infinitely many variables on etudie des suites de fonctions factorielles satisfaisant des analogues des recurrences binomiales et de sheffer et on montre comment un calcul operatoriel ressemblant au calcul umbral peut etre construit dessus

riordan arrays associated with laurent series and generalized sheffer type groups a relationship between a pair of laurent series and riordan arrays is formulated in addition a type of generalized sheffer groups is defined by using riordan arrays with respect to power series with non zero coefficients the isomorphism between a generalized sheffer group and the group of the riordan arrays associated with laurent series is established furthermore appell associated bell and hitting time subgroups of the groups are defined and discussed a relationship between the generalized sheffer groups with respect to different type of power series is presented the equivalence of the defined riordan array pairs and generalized stirling number pairs is given a type of inverse relations of various series is constructed by using pairs of riordan arrays finally several applications involving various arrays polynomial sequences special formulas and identities are also presented as illustrative examples

a determinantal approach to sheffer sequences abstract in this paper by using the theory of riordan arrays and the relations between sheffer sequences and riordan arrays we give a determinantal definition for sheffer sequences based on this new definition some general properties of sheffer sequences are reproved and the determinantal representations of some well known sheffer sequences are presented

bijections and the riordan group one of the cornerstone ideas in mathematics is to take a problem and to look at it in a bigger space in this paper we examine combinatorial sequences in the context of the riordan group various subgroups of the riordan group each give us a different view of the original sequence in many cases this leads to both a combinatorial interpretation and to eco rewriting rules in this paper we will concentrate on just four of the subgroups of the riordan group to demonstrate some of the possibilities of this approach

structural properties of riordan matrices and extending the matrices abstract we consider an infinite lower triangular matrix l l n k n k n 0 and a sequence n n n 0 called the a b sequence such that every element l n 1 k 1 except lying in column 0 can be expressed as l n 1 k 1 i 0 n k m i l n ai k bi 0 0 where a and b are integers with a b m 0 and b 0 this concept generalizes the a sequence of a riordan matrix as a result we explore several structural properties of riordan matrices by means of a b sequences in particular if b

log convexity of aigner catalan riordan numbers abstract let t t n k n k 0 be an infinite lower triangular matrix defined by t 0 0 1 t n 1 0 j 0 n z j t n j t n 1 k 1 j k n a j k t n j for n k 0 where all z j a j k are nonnegative and a j k 0 unless j k 0 we show that the sequence t n 0 n 0 is log convex if the coefficient matrix a is tp 2 where z 0 z 1 z 2 and a a i j i j 0 this gives a unified proof of the log convexity of many well known combinatorial sequences including the catalan numbers the motzkin numbers the central binomial coefficients the schroder numbers the bell numbers and so on

cayley continuants in 1858 cayley considered a particular kind of tridiagonal determinants or continuants by a direct inspection of the first cases he conjectured an identity expressing these determinants in terms of certain other determinants considered by sylvester in 1854 then cayley proved the conjectured identity by induction but as he wrote he felt unsatisfied with his proof the main aim of this paper is to give a straightforward proof of cayley s identity using the method of formal series moreover we use this method and umbral calculus techniques to obtain several other identities cayley continuants appear in several contexts and in particular in enumerative combinatorics mittag leffler polynomials meixner polynomials of the first kind the falling and the raising factorials are just few instances of these continuants they can be interpreted in terms of weighted permutations moreover as we prove in this paper they also appear in the context of hankel determinants generated by certain catalan like numbers

riordan matrices in the reciprocation of quadratic polynomials we iterate contractive one degree polynomials with coefficients in the ring k x of formal power series to calculate the reciprocal in k x of a quadratic polynomial doing this we meet the structure of riordan array we interpret certain changes of variable as a riordan array we finish the paper by using our techniques to find new ways to get known formulas for the sum of powers of natural numbers involving stirling and eulerian numbers

recurrence relations for the sheffer sequences abstract in this paper using the production matrix of an exponential riordan array g t f t we give a recurrence relation for the sheffer sequence for the ordered pair g t f t we also develop a new determinant representation for the general term of the sheffer sequence as applications determinant expressions for some classical sheffer polynomial sequences are derived

the relevant prefixes of coloured motzkin walks an average case analysis in this paper we study some relevant prefixes of coloured motzkin walks otherwise called coloured motzkin words in these walks the three kinds of step can have a b and c colours respectively in particular when a b c 1 we have the classical motzkin walks while for a c 1 and b 0 we find the well known dyck walks by using the concept of riordan arrays and probability generating functions we find the average length of the relevant prefix in a walk of length n and the corresponding variance in terms of a b and c this result is interesting from a combinatorial point of view and also provides an average case analysis of algorithms related to the problem of ranking and generating uniformly at random the coloured motzkin words

addition formulas for polynomials built on classical combinatorial sequences abstract we prove addition formulas for some polynomials built on combinatorial sequences catalan numbers bell numbers or related to classical polynomials hermite these relations give also an explicit computation of several hankel determinants

an algebraic characterization of the set of succession rules qui dedit beneficium taceat narret qui accepit seneca merci maurice in this paper we will give a formal description of succession rules in terms of linear operators satisfying certain conditions this representation allows us to introduce a system of well defined operations into the set of succession rules and then to tackle problems of combinatorial enumeration simply by using operators instead of generating functions finally we will suggest several open problems whose solution should lead to an algebraic characterization of the set of succession rules

some discrete d orthogonal polynomial sets in this paper we characterize all polynomial sets which are at the same time d orthogonal and appell the resulting polynomials reduce to charlier polynomials for d 1 1 various properties of the obtained polynomials are singled out generating function recurrence relation of order d 1 and a difference equation of order d 1 we also explicitly express the d dimensional functional for which the d orthogonality holds