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

on a system of two nonlinear difference equations abstract in this paper we study the oscillatory behavior the boundedness of the solutions and the global asymptotic stability of the positive equilibrium of the system of two nonlinear difference equationsxn 1 a yn xn p yn 1 a xn yn q n 0 1 p qare positive integers

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

on some solvable systems of difference equations abstract we show that the following systems of difference equations x n 1 u n 1 v n y n 1 w n 1 s n n n 0 where u n v n w n s n are some of the sequences x n or y n with real initial values x 0 and y 0 are solvable in fourteen out of sixteen possible cases two cases are left unsolved probably the most interesting is the result in the case u n x n v n x n w n x n s n y n where a fascinating formula is obtained in an elegant way by using some ad hoc ideas

domains of undefinable solutions of some equations and systems of difference equations for some classes of difference equations and systems of difference equations we find the sets of initial values for which solutions of these equations and systems are not well defined so called domains of undefinable solutions

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

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

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

more on a rational second order difference equation abstract here we give natural explanations for some recent relations for the solutions of the following difference equation z n 2 z n 1 3 z n 2 2 z n 1 2 n n 0 and extend them for the case of complex initial values we also present some other properties related to solutions of the equation as well as some properties of the solutions of an associate difference equation of first order

on fourteen solvable systems of difference equations abstract in this paper we mainly consider the systems of difference equations x n 1 1 p n q n y n 1 1 r n s n n n 0 where each of the sequences p n q n r n and s n represents either the sequence x n or the sequence y n with nonzero real initial values x 0 and y 0 then we solve fourteen out of sixteen possible systems it is noteworthy to depict that the solutions are presented in terms of fibonacci numbers for twelve systems of these fourteen systems

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

on some systems of difference equations abstract in this note we show that the following systems of difference equations u n 1 v n 1 v n v n 1 u n 1 u n u n 1 v n 1 u n v n 1 u n 1 v n u n 1 u n 1 v n v n 1 v n 1 u n with n n 0 and complex initial values u 0 and v 0 are solvable explicitly by means of riccati equations and based on the formulae for the general solutions we present the behaviour of their solutions as n while the result is natural for the first system it is a bit surprising for the other two systems

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

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

on the difference equation xn xn 2 bn cnxn 1xn 2 abstract we prove that the difference equation x n x n 2 b n c n x n 1 x n 2 n n 0 where b n n n 0 and c n n n 0 are sequences periodic with period two and the initial values x 2 x 1 are real numbers can be solved explicitly some applications of obtained formulae are given

on the difference equation xn xn kb cxn 1 xn k abstract we describe the behavior of well defined solutions of the difference equation x n x n k b cx n 1 x n k n n 0 where k n parameters b c and the initial values x k x 1 are real numbers these results among others improve and extend our results in s stevic more on a rational recurrence relation appl math e notes 4 2004 80 85

on a solvable system of difference equations of fourth order it is shown that the next system of k difference equationsx n j a n j x n k j b n j i 1 kx n i s j i 1 c n j n n 0 j 1 k where a n j b n j c n j n n 0 j 1 k and initial values x i j i j 1 k are real numbers and where s n 1 k is defined by s km j j 1 j 1 k 1 s km k 1 m n 0 can be solved in closed form in an elegant way some applications of obtained formulas for the case when the sequences a n j b n j and c n j are k periodic are also given

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

boundedness character of a class of difference equations abstract a complete picture regarding the boundedness character of positive solutions to the following difference equation x n max a x n 1 p x n k p n n 0 where k 2 and the parameters a and p are positive real numbers is given in particular for the case p k 1 0 k k k 1 k 1 we prove that all solutions to the equation are bounded we also present corresponding results concerning the following closely related difference equation x n a x n 1 p x n k p n n 0

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 hall higman shult type theorem for arbitrary finite groups we shall extend a fixed point theorem of shult to arbitrary finite groups this will have applications to the study of group automorphisms

solutions of a max type system of difference equations we show in an elegant way that the general solution to the following max type system of difference equations xn 1 maxaxn ynxn yn 1 maxayn xnyn n n0 for the case y0 x0 a 0 y0 x0 max a 1 a is given by xn afk n 1 n x0fk n y0fk n 1 n n n y1 x0 y0 and yn y0fk n 1 1afk n 1 n 1x0fk n 1 1 1 n n 2 where k n n for n even k n n 1 for n odd n n0 n n mod2 for n n0 and where the sequence fn n n is the fibonacci sequence correcting and giving a proof of a wrong result in the paper d simsek b demir c cinar on the solutions of the system of difference equations xn 1 max a xn yn xn yn 1 max a yn xn yn discrete dyn nat soc 2009 2009 11 pp article id 325296

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

on some periodic systems of max type difference equations we show that all positive solutions to the system of max type difference equations xn 1 max1 i m1f1ixn ki 1 1 1 xn ki 2 1 2 xn ki l 1 l n xn s 1 xn 2 max1 i m2f2ixn ki 1 2 1 xn ki 2 2 2 xn ki l 2 l n xn s 2 xn l max1 i mlflixn ki 1 l 1 xn ki 2 l 2 xn ki l l l n xn s l n n0 where s l mj ki t j n j t 1 l and for a fixed j i 1 mj and where the functions fji 0 l n0 0 j 1 l i 1 mj satisfy some conditions are eventually periodic with not necessarily prime period s a related result for the corresponding system of min type difference equations is also proved

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

on the system xn 1 ynx n k yn k 1 an bnynxn k yn 1 xnyn k xn k 1 cn dnxnyn k it is shown that the system of difference equationsx n 1 y nx n ky n k 1 a n b ny nx n k y n 1 x ny n kx n k 1 c n d nx ny n k email protected n 0 where email protected n the sequences a n b n c n d n email protected n 0 and the initial values x i y i i 0 email protected are real numbers can be solved in closed form asymptotic behavior of well defined solutions of the system for the case when all the sequences a n b n c n and d n are constant is studied in detail

on the dynamics of the difference equation xn 1 1bnxn xn 1 in this note we study the following non autonomous equation x n 1 1b nx n x n 1 n 0 1 where b n 0 is a period 2 sequence and the initial values x 1 x 0 0 we show that the unique prime period 2 solution of this equation is globally asymptotically stable which answer open problem 5 proposed by camouzis and ladas 6

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

on the system of difference equations xn cnyn 3 an bnyn 1xn 2y n 3 yn γnxn 3 αn βnxn 1yn 2 xn 3 we show that the system of difference equationsx n c ny n 3a n b ny n 1x n 2y n 3 y email protected nx n email protected email protected nx n 1y n 2x n 3 email protected n 0 where the sequences a n b n c n a n b n c n email protected n 0 and initial values x i y i email protected 1 2 3 are real numbers such that c n 0 c n 0 email protected n 0 can be solved in closed form and for the case when all the sequences a n b n c n a n b n and c n are constant we describe the asymptotic behavior of well defined solutions of the system

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

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

on a nonlinear generalized max type difference equation abstract the boundedness character of positive solutions of the following max type difference equation x n max a x n 1 p x n k r n n 0 where k n 1 the parameters a and r are positive and p is a nonnegative real number is studied in this paper our main results considerably improve results appearing in the literature

on a system of difference equations abstract we show that the system of difference equations x n 1 ax n 1 by n x n 1 c y n 1 y n 1 x n y n 1 n n 0 where the parameters a b c and initial values x 1 x 0 y 1 y 0 are real numbers can be solved considerably improving the results in the literature

on a third order system of difference equations abstract we show that the system of difference equations x n 1 a 1 x n 2 b 1 y n z n 1 x n 2 c 1 y n 1 a 2 y n 2 b 2 z n x n 1 y n 2 c 2 z n 1 a 3 z n 2 b 3 x n y n 1 z n 2 c 3 n n 0 where the parameters a i b i c i i 1 2 3 and initial values x j y j z j j 0 1 2 are real numbers can be solved developing further the results in the literature

global stability of a max type difference equation we show that every positive solution to the difference equationx n maxa 1x n p 1 a 1 a 2x n p 2 a 2 a kx n p k a k n n 0 where p i i 1 k are natural numbers such that 1 0 a i 1 1 i 1 k converges to max 1 i ka i 1 a i 1 this result improves and complements the main result in our recent note s stevic global stability of a difference equation with maximum appl math comput 210 2009 525 529 since it also considers the case when a i 1 0 i 1 k

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

periodicity of a class of nonautonomous max type difference equations abstract in this note we prove several general and easily applicable results regarding the periodicity of positive solutions of the following difference equations x n max f 1 x n k 1 n f m x n k m n x n s n n 0 and x n min f 1 x n k 1 n f m x n k m n x n s n n 0 where 1 k 1 k m m s n and where the functions f i 0 n 0 0 1 i m satisfy some additional conditions these results considerably extend and unify numerous results on max type difference equations existing in the literature

on difference equations with powers as solutions and their connection with invariant curves abstract we look for nonlinear systems of difference equations which have power functions as solutions and we find a connection to the theory of invariant curves and to self reversed polynomials in particular we construct algebraic solutions of a special equation of invariant curves

on the nonautonomous difference equation xn 1 an xn 1pxnq abstract in this paper we study the asymptotic behavior and the periodicity of the positive solutions of the nonautonomous difference equation x n 1 a n x n 1 p x n q n 0 1 where an is a positive bounded sequence p q 0 and x 1 x0 are positive numbers

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