An ordered partition of [n] = {1, 2, . . . , n} is a partition whose blocks are endowed with a linear order. Let OPn,k be the set of ordered partitions of [n] with k blocks and OPn,k(σ) be the set of ordered partitions in OPn,k that avoid a pattern σ. For any permutation pattern σ of length three, Godbole, Goyt, Herdan and Pudwell obtained formulas for the number of ordered partitions of [n] wi...

Remark. This identity is easily verified using the WZ method, in a generalized form [Z] that applies when the summand is a hypergeometric term times a WZ potential function. It holds for all positive n, since it holds for n=1,2,3 (check!), and since the sequence defined by the sum satisfies a certain (homog.) third order linear recurrence equation. To find the recurrence, and its proof, downloa...

We present an algorithm which decides the shift equivalence problem for Pfinite sequences. A sequence is called P-finite if it satisfies a homogeneous linear recurrence equation with polynomial coefficients. Two sequences are called shift equivalent if shifting one of the sequences s times makes it identical to the other, for some integer s. Our algorithm computes, for any two P-finite sequence...

We show that multiple orthogonal polynomials for r measures (μ1, . . . , μr) satisfy a system of linear recurrence relations only involving nearest neighbor multi-indices ~n ± ~ej, where ~ej are the standard unit vectors. The recurrence coefficients are not arbitrary but satisfy a system of partial difference equations with boundary values given by the recurrence coefficients of the orthogonal ...

The author continues to study series transformations for the Euler-Mascheroni constant γ . Here, we discuss in detail recently published results of A. I. Aptekarev and T. Rivoal who found rational approximations to γ and γ log q q ∈ Q>0 defined by linear recurrence formulae. The main purpose of this paper is to adapt the concept of linear series transformations with integral coefficients such t...

This paper looks at the approach of using generating functions to solve linear inhomogeneous recurrence equations with constant coefficients. It will be shown that the generating functions for these recurrence equations are rational functions. By decomposing a generating function into partial fractions, one can derive explicit formula as well as asymptotic estimates for its coefficients. Key–Wo...

lie in finitely many proper subspaces of Q. As an application of the subspace theorem Schmidt [16] described all norm form equations that have finitely many solutions. The subspace theorem has been further developed by Schlickewei [11, 12] and is proved in it’s most general form by Evertse and Schlickewei [3] (see also [13]). These investigations led to many applications, e.g. to the finiteness...

Dedicated to Prof. Jaroslav Kurzweil on the occasion of his 80th birthday Abstract. Asymptotic properties of the half-linear difference equation (∗) ∆(an|∆xn| α sgn∆xn) = bn|xn+1| α sgnxn+1 are investigated by means of some summation criteria. Recessive solutions and the Riccati difference equation associated to (∗) are considered too. Our approach is based on a classification of solutions of (...

In this paper, we derive new recurrence relations and generating matrices for the sums of usual Tribonacci numbers and 4n subscripted Tribonacci sequences, fT4ng ; and their sums. We obtain explicit formulas and combinatorial representations for the sums of terms of these sequences. Finally we represent relationships between these sequences and permanents of certain matrices. 1. Introduction Th...

Given a suitable weight on IR, there exist many (recursive) three term recurrence relations for the corresponding multivariate orthogonal polynomials. In principle, these can be obtained by calculating pseudoinverses of a sequence of matrices. Here we give an explicit recursive three term recurrence for the multivariate Jacobi polynomials on a simplex. This formula was obtained by seeking the b...