<Emphasis Type="Italic">n</Emphasis>-Colour even self-inverse compositions

An n-colour even self-inverse composition is defined as an n-colour selfinverse composition with even parts. In this paper, we get generating functions, explicit formulas and recurrence formulas for n-colour even self-inverse compositions. One new binomial identity is also obtained.     

与自反的n-color有序分拆相关的一些恒等式

首先,给出了偶数2v的自反的n-color有序分拆与v+1,v-1的n-color有序分拆之间的一个组合双射,并利用相应的计数公式得到了一个组合恒等式.其次,给出了正整数自反的n-color有序分拆数与Fibonacci数、Lucas数之间的一个关系式,并利用此关系式给出了偶数与奇数的自反的n-color有序分拆之间的一个组合双射.最后,给出了一些涉及正整数v的自反的n-color有序分拆数与其它有约束条件的有序分拆数之间的分拆恒等式.     

Self-inverse Sheffer sequences and Riordan involutions

In this short note, we focus on self-inverse Sheffer sequences and involutions in the Riordan group. We translate the results of Brown and Kuczma on self-inverse sequences of Sheffer polynomials to describe all involutions in the Riordan group.     

二项式型自反多项式

In this paper, we define the self-inverse sequences related to sequences of polynomials of binomial type, and give some interesting results of these sequences. Moreover, we study the self-inverse sequences related to the Laguerre polynomials.     

由几何反演导出的分形几何

本文介绍了由几何反演而导出的自反分形概念.构造了一个对称的,分叉自反分形曲线和一个自反分形尘集.推广了分形密切的概念而且提出了一个新的概念——分形包络.最后给出了“肥皂”和“鸡蛋”的两个自反分形的实例.     

Euler-Seidel method for certain combinatorial numbers and a new characterization of Fibonacci sequence

In this paper we use the Euler-Seidel method for deriving new identities for hyperharmonic and r-Stirling numbers. The exponential generating function is determined for hyperharmonic numbers, which result is a generalization of Gosper’s identity. A classification of second order recurrence sequences is also given with the help of this method.      

New special function recurrences giving new indefinite integrals

Sequences of new recurrence relations are presented for Bessel functions, parabolic cylinder functions and associated Legendre functions. The sequences correspond to values of an integer variable r and are generalizations of each conventional recurrence relation, which correspond to r=1. The sequences can be extended indefinitely, though the relations become progressively more intricate as r increases. These relations all have the form of a first-order linear inhomogeneous differential equation, which can be solved by an integrating factor. This gives a very general indefinite integral for each recurrence. The method can be applied to other special functions which have conventional recurrence relations. All results have been checked numerically using Mathematica.     

Recurrence relation with two indices and plane compositions

The aim of this paper is to study analytical and combinatorial methods to solve a special type of recurrence relation with two indices. It is shown that the recurrence relation enumerates a natural combinatorial object called a plane composition. In addition, further interesting recurrence relations arise in the study of statistics for these plane compositions.     

关于正整数的有序分拆的Inplace恒等式的一些注记

本文给出了一类特殊的称之为Inplace有序分拆的两个递推关系式的组合证明. 同时, 我们也得到了关于Inplace 1-2 有序分拆,回文的有序分拆的一些新的恒等式.     

A Mathematica package for q-holonomic sequences and power series

We describe a Mathematica package for dealing with q-holonomic sequences and power series. The package is intended as a q-analogue of the Maple package gfun and the Mathematica package GeneratingFunctions. It provides commands for addition, multiplication, and substitution of these objects, for converting between various representations (q-differential equations, q-recurrence equations, q-shift equations), for computing sequence terms and power series coefficients, and for guessing recurrence equations given initial terms of a sequence. C. Koutschan partially supported by the Austrian Science Foundation (FWF) grants SFB F1305.     

Lp-Approximable Sequences of Vectors and Limit Distribution of Quadratic Forms of Random Variables

The properties of L₂-approximable sequences established here form a complete toolkit for statistical results concerning weighted sums of random variables, where the weights are nonstochastic sequences approximated in some sense by square-integrable functions and the random variables are “two-wing” averages of martingale differences. The results constitute the first significant advancement in the theory of L₂-approximable sequences since 1976 when Moussatat introduced a narrower notion of L₂-generated sequences. The method relies on a study of certain linear operators in the spaces L_p and l_p. A criterion of L_p-approximability is given. The results are new even when the weight generating function is identically 1. A central limit theorem for quadratic forms of random variables illustrates the method.     

On the properties of sequences of fuzzy-valued Choquet integrable functions

In this paper, we focus on investigating the properties of sequences of fuzzy-valued Choquet (for short, (C)-) integrable functions. Firstly, the concept of uniform (C)-integrabiliy and other new concepts like uniform absolute continuity and uniform boundedness for sequences of fuzzy-valued (C)-integrable functions are introduced and then the relations among them are discussed. As the applications of these concepts, we also present several convergence theorems for sequences of fuzzy-valued (C)-integrable functions by using uniform (C)-integrability.     

Connection coefficients between orthogonal polynomials and the canonical sequence: an approach based on symbolic computation

We deal with the problem of obtaining closed formulas for the connection coefficients between orthogonal polynomials and the canonical sequence. We use a recurrence relation fulfilled by these coefficients and symbolic computation with the Mathematica language. We treat the cases of Gegenbauer, Jacobi and a new semi-classical sequence.     

An analogue of Montel's theorem for some classes of rational functions

For sequences of rational functions, analytic in some domain, a theorem of Montel's type is proved. As an application, sequences of rational functions of the best L _p -approximation with an unbounded number of finite poles are considered.     

Semiconjugate factorizations of higher order linear difference equations in rings

We use a new nonlinear method to study linear difference equations with variable coefficients in a non-trivial ring R. If the homogeneous part of the linear equation has a solution in the unit group of a ring with identity (a unitary solution), then we show that the equation decomposes into two linear equations of lower orders. This decomposition, known as a semiconjugate factorization in the nonlinear theory, is based on sequences of ratios of consecutive terms of a unitary solution. Such sequences, which may be called eigensequences, are well suited to variable coefficients; for instance, they provide a natural context for the expression of the Poincaré–Perron theorem. As applications, we obtain new results for linear difference equations with periodic coefficients and for linear recurrences in rings of functions (e.g. the recurrence for the modified Bessel functions).     

New Cyclic Difference Sets with Singer Parameters Constructed from d-Homogeneous Functions

In this paper, for a prime power q, new cyclic difference sets with Singer para- meters ((q ⁿ –1/q–1), (q ^n–1–1/q–1), (q ^n–2–1/q–1)) are constructed by using q-ary sequences (d-homogeneous functions) of period q ⁿ –1 and the generalization of GMW difference sets is proposed by combining the generation methods of d-form sequences and extended sequences. When q is a power of 3, new cyclic difference sets with Singer parameters ((q ⁿ –1/q–1), (q ^n–1–1/q–1), (q ^n–2–1/q–1)) are constructed from the ternary sequences of period q ⁿ –1 with ideal autocorrelation introduced by Helleseth, Kumar, and Martinsen.     

Cauchy, Ferrers-Jackson and Chebyshev polynomials and identities for the powers of elements of some conjugate recurrence sequences

In this paper some decompositions of Cauchy polynomials, Ferrers-Jackson polynomials and polynomials of the form x ²ⁿ + y ²ⁿ , n ∈ ℕ, are studied. These decompositions are used to generate the identities for powers of Fibonacci and Lucas numbers as well as for powers of the so called conjugate recurrence sequences. Also, some new identities for Chebyshev polynomials of the first kind are presented here.     

KK-groups for generalized operator algebras. II

The extension of Kasparovs bivariant K-theory to inverse limits of C ^* -algebras admits exact Puppe sequences in both variables. Two exact sequences generalizing Milnor's lim-lim¹ sequences are established. For CW complexes the extended K-theory is representable K-theory.     

One interesting family of diophantine triplets

In this note properties of two sequences generated by the recurrence relation G _n+2 = 4G _n+1 ? G _n , are studied. It is shown that one of the sequences leads to a family of diophantine triplets. Some interesting properties of these sequences are also established.     

Partitions and Compositions Defined by Inequalities

We consider sequences of integers (₁,..., _k) defined by a system of linear inequalities _{i j>i}a_ijj with integer coefficients. We show that when the constraints are strong enough to guarantee that all _i are nonnegative, the generating function for the integer solutions of weight n has a finite product form , where the b_i are positive integers that can be computed from the coefficients of the inequalities. The results are proved bijectively and are used to give several examples of interesting identities for integer partitions and compositions. The method can be adapted to accommodate equalities along with inequalities and can be used to obtain multivariate forms of the generating function. We show how to extend the technique to obtain the generating function when the coefficients a_i,i+1 are allowed to be rational, generalizing the case of lecture hall partitions. Our initial results were conjectured thanks to the Omega package (G.E. Andrews, P. Paule, and A. Riese, European J. Comb. 22(7) (2001), 887–904).Research supported by NSA grants MDA 904-00-1-0059 and MDA 904-01-0-0083.