scholarly journals Some Basic đť‘‘-Orthogonal Polynomial Sets

2005 â—˝  
Vol 12 (4) â—˝  
pp. 583-593
Author(s):  
Ali Zaghouani
Keyword(s):  
Orthogonal Polynomial â—˝  
Recurrence Relation â—˝  
Generating Function â—˝  
Linear Change â—˝  
Change Of Variable

Abstract The purpose of this paper is to study the class of polynomial sets which are at the same time 𝑑-orthogonal and 𝑞-Appell. By a linear change of variable, the resulting set reduces to 𝑞-Al-Salam–Carlitz polynomials, for 𝑑 = 1. Various properties of the obtained polynomials are singled out: a generating function, a recurrence relation of order 𝑑 + 1. We also explicitly express a 𝑑-dimensional functional for which the d-orthogonality holds.

scholarly journals New Eulerian Numbers of Type $D$

10.37236/5514 â—˝  
2016 â—˝  
Vol 23 (1) â—˝  
Author(s):  
Anna Borowiec â—˝  
Wojciech Młotkowski
Keyword(s):  
Recurrence Relation â—˝  
Generating Function â—˝  
Probability Distributions â—˝  
Type A â—˝  
Eulerian Numbers â—˝  
Eulerian Polynomials â—˝  
Type D

We introduce a new array of type $D$ Eulerian numbers, different from that studied by Brenti, Chow and Hyatt. We find in particular the recurrence relation, Worpitzky formula and the generating function. We also find the probability distributions whose moments are Eulerian polynomials of type $A$, $B$ and $D$.

scholarly journals A Newton Interpolation Approach to Generalized Stirling Numbers

10.1155/2012/351935 â—˝  
2012 â—˝  
Vol 2012 â—˝  
pp. 1-17 â—˝  
Author(s):  
Aimin Xu
Keyword(s):  
Explicit Expression â—˝  
Recurrence Relation â—˝  
Generating Function â—˝  
Stirling Numbers â—˝  
Divided Differences â—˝  
Newton Interpolation â—˝  
Basic Properties â—˝  
Generalized Stirling Numbers

We employ the generalized factorials to define a Stirling-type pair{s(n,k;α,β,r),S(n,k;α,β,r)}which unifies various Stirling-type numbers investigated by previous authors. We make use of the Newton interpolation and divided differences to obtain some basic properties of the generalized Stirling numbers including the recurrence relation, explicit expression, and generating function. The generalizations of the well-known Dobinski's formula are further investigated.

scholarly journals The Maximum Independent Sets of de Bruijn Graphs of Diameter 3

10.37236/681 â—˝  
2011 â—˝  
Vol 18 (1) â—˝  
Author(s):  
Dustin A. Cartwright â—˝  
María Angélica Cueto â—˝  
Enrique A. Tobis
Keyword(s):  
Recurrence Relation â—˝  
Generating Function â—˝  
Independent Sets â—˝  
De Bruijn Graph â—˝  
Alphabet Size â—˝  
De Bruijn Graphs â—˝  
Exponential Generating Function â—˝  
De Bruijn

The nodes of the de Bruijn graph $B(d,3)$ consist of all strings of length $3$, taken from an alphabet of size $d$, with edges between words which are distinct substrings of a word of length $4$. We give an inductive characterization of the maximum independent sets of the de Bruijn graphs $B(d,3)$ and for the de Bruijn graph of diameter three with loops removed, for arbitrary alphabet size. We derive a recurrence relation and an exponential generating function for their number. This recurrence allows us to construct exponentially many comma-free codes of length 3 with maximal cardinality.

scholarly journals Closed Forms for Derangement Numbers in Terms of the Hessenberg Determinants

2016 â—˝  
Author(s):  
Feng Qi â—˝  
Jiao-Lian Zhao â—˝  
Bai-Ni Guo
Keyword(s):  
Recurrence Relation â—˝  
Generating Function â—˝  
Higher Order â—˝  
Order Derivative â—˝  
Exponential Generating Function â—˝  
Higher Order Derivative

In the paper, the authors find closed forms for derangement numbers in terms of the Hessenberg determinants, discover a recurrence relation of derangement numbers, present a formula for any higher order derivative of the exponential generating function of derangement numbers, and compute some related Hessenberg and tridiagonal determinants.

scholarly journals The proof of the Perepechko’s conjecture concerning near-perfect matchings on Cm x Pn cylinders of odd order

2019 â—˝  
Vol 13 (2) â—˝  
pp. 361-377
Author(s):  
Rade Doroslovacki â—˝  
Jelena Djokic â—˝  
Bojana Pantic â—˝  
Olga Bodroza-Pantic
Keyword(s):  
Recurrence Relation â—˝  
Generating Function â—˝  
Perfect Matchings â—˝  
Odd Order

For all odd values of m, we prove that the sequence of the numbers of near-perfect matchings on Cm x P2n+1 cylinder with a vacancy on the boundary obeys the same recurrence relation as the sequence of the numbers of perfect matchings on Cm x P2n. Further more, we prove that for all odd values of m denominator of the generating function for the total number of the near-perfect matchings on Cm x P2n+1 graph is always the square of denominator of generating function for the sequence of the numbers of perfect matchings on Cm x P2n graph, as recently conjectured by Perepechko.

scholarly journals On a quaternionic sequence with Vietoris’ numbers

Filomat â—˝  
10.2298/fil2104065c â—˝  
2021 â—˝  
Vol 35 (4) â—˝  
pp. 1065-1086
Author(s):  
P. Catarino â—˝  
Almeida de
Keyword(s):  
Recurrence Relation â—˝  
Generating Function â—˝  
Type Formula â—˝  
Tridiagonal Matrices â—˝  
Three Term Recurrence Relation â—˝  
Rational Sequence

Special integers sequences have been the center of attention for many researchers, as well as the sequences of quaternions where its components are the elements of these sequences. Motivated by a rational sequence, we consider the quaternions with components Vietoris? numbers and investigate some of its properties. For this sequence a two and three term recurrence relation is established, as well as a Binet?s type formula. Moreover the generating function for this sequence is introduced and also the determinant of some tridiagonal matrices are used in order to find elements of this sequence.

Linear change of variable in normal systems

Celestial Mechanics â—˝  
10.1007/bf01227858 â—˝  
1973 â—˝  
Vol 7 (3) â—˝  
pp. 384-387
Author(s):  
J. Derral Mulholland
Keyword(s):  
Linear Change â—˝  
Change Of Variable

scholarly journals Linear change of variable in normal systems

Celestial Mechanics â—˝  
10.1007/bf01228398 â—˝  
1973 â—˝  
Vol 8 (1) â—˝  
pp. 152-152
Author(s):  
J. Derral Mulholland
Keyword(s):  
Linear Change â—˝  
Change Of Variable

scholarly journals ON THE BINOMIAL TRANSFORMS OF THE HORADAM QUATERNION SEQUENCES

2020 â—˝  
Author(s):  
Faruk Kaplan â—˝  
Arzu Özkoç Öztürk
Keyword(s):  
Recurrence Relation â—˝  
General Formula â—˝  
Generating Function â—˝  
Second Order â—˝  
Binet Formula â—˝  
New Type â—˝  
Binomial Transform

The main object of the present paper is to consider the binomial transforms for Horadam quaternion sequences. We gave new formulas for recurrence relation, generating function, Binet formula and some basic identities for the binomial sequence of Horadam quaternions. Working with Horadam quaternions, we have found the most general formula that includes all binomial transforms with recurrence relation from the second order. In the last part, we determined the recurrence relation for this new type of quaternion by working with the iterated binomial transform, which is a dierent type of binomial transform.

scholarly journals On Generating Functions

1930 â—˝  
Vol 2 (2) â—˝  
pp. 71-82 â—˝  
Author(s):  
W. L. Ferrar
Keyword(s):  
Differential Equation â—˝  
Recurrence Relation â—˝  
Generating Function â—˝  
Generating Functions â—˝  
Order Differential Equation â—˝  
Second Order â—˝  
Second Order Differential Equation â—˝  
Image Position â—˝  
Three Term Recurrence Relation â—˝  
Sequence Of Functions

It is well known that the polynomial in x,has the following properties:—(A) it is the coefficient of tn in the expansion of (1–2xt+t2)–½;(B) it satisfies the three-term recurrence relation(C) it is the solution of the second order differential equation(D) the sequence Pn(x) is orthogonal for the interval (— 1, 1),i.e. whenSeveral other familiar polynomials, e.g., those of Laguerre Hermite, Tschebyscheff, have properties similar to some or all of the above. The aim of the present paper is to examine whether, given a sequence of functions (polynomials or not) which has one of these properties, the others follow from it : in other words we propose to examine the inter-relation of the four properties. Actually we relate each property to the generating function.

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