scholarly journals Automated Proofs for Some Stirling Number Identities

10.37236/726 ◽  
2008 ◽  
Vol 15 (1) ◽  
Author(s):  
Manuel Kauers ◽  
Carsten Schneider
Keyword(s):  
Stirling Number ◽  
Eulerian Numbers ◽  
Difference Fields

We present computer-generated proofs for some summation identities for ($q$-)Stirling and ($q$-)Eulerian numbers that were obtained by combining a recent summation algorithm for Stirling number identities with a recurrence solver for difference fields.

On a Convolution of Eulerian Numbers: 10609

1999 ◽  
Vol 106 (7) ◽  
pp. 690
Author(s):  
Donald E. Knuth ◽  
David Callan
Keyword(s):  
Eulerian Numbers

Index for Model theory of difference fields

2005 ◽  
pp. 103-104
Keyword(s):  
Model Theory ◽  
Difference Fields

scholarly journals New Eulerian Numbers of Type $D$

10.37236/5514 ◽  
2016 ◽  
Vol 23 (1) ◽  
Author(s):  
Anna Borowiec ◽  
Wojciech Młotkowski
Keyword(s):  
Recurrence Relation ◽  
Generating Function ◽  
Probability Distributions ◽  
Type A ◽  
Eulerian Numbers ◽  
Eulerian Polynomials ◽  
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 On $xD$-Generalizations of Stirling Numbers and Lah Numbers via Graphs and Rooks

10.37236/6699 ◽  
2017 ◽  
Vol 24 (2) ◽  
Author(s):  
Sen-Peng Eu ◽  
Tung-Shan Fu ◽  
Yu-Chang Liang ◽  
Tsai-Lien Wong
Keyword(s):  
Weyl Algebra ◽  
Stirling Numbers ◽  
Stirling Number ◽  
Combinatorial Structures ◽  
Normal Ordering ◽  
Rook Placements ◽  
Threshold Graphs ◽  
Combinatorial Interpretations ◽  
Ferrers Boards

This paper studies the generalizations of the Stirling numbers of both kinds and the Lah numbers in association with the normal ordering problem in the Weyl algebra $W=\langle x,D|Dx-xD=1\rangle$. Any word $\omega\in W$ with $m$ $x$'s and $n$ $D$'s can be expressed in the normally ordered form $\omega=x^{m-n}\sum_{k\ge 0} {{\omega}\brace {k}} x^{k}D^{k}$, where ${{\omega}\brace {k}}$ is known as the Stirling number of the second kind for the word $\omega$. This study considers the expansions of restricted words $\omega$ in $W$ over the sequences $\{(xD)^{k}\}_{k\ge 0}$ and $\{xD^{k}x^{k-1}\}_{k\ge 0}$. Interestingly, the coefficients in individual expansions turn out to be generalizations of the Stirling numbers of the first kind and the Lah numbers. The coefficients will be determined through enumerations of some combinatorial structures linked to the words $\omega$, involving decreasing forest decompositions of quasi-threshold graphs and non-attacking rook placements on Ferrers boards. Extended to $q$-analogues, weighted refinements of the combinatorial interpretations are also investigated for words in the $q$-deformed Weyl algebra.

Beyond the Euler summation formula

2004 ◽  
Vol 88 (513) ◽  
pp. 432-440 ◽  
Author(s):  
Barry Lewis
Keyword(s):  
Summation Formula ◽  
Eulerian Numbers ◽  
The Subject ◽  
Established Technique

The Eulerian numbers are not strangers to readers of the Gazette but they are not normally associated with the subject of this article, despite the similarity of their names. This article seeks to use Eulerian numbers in generalised telescoping sums, a role that is a powerful extension of an established technique – the Euler summation formula.

scholarly journals Pseudofinite difference fields

2019 ◽  
Vol 19 (02) ◽  
pp. 1950011
Author(s):  
Tingxiang Zou
Keyword(s):  
Finite Fields ◽  
Transcendence Degree ◽  
Order Property ◽  
Frobenius Automorphism ◽  
Difference Fields ◽  
Definable Sets ◽  
Partial Connection

We study a family of ultraproducts of finite fields with the Frobenius automorphism in this paper. Their theories have the strict order property and TP2. But the coarse pseudofinite dimension of the definable sets is definable and integer-valued. Moreover, we establish a partial connection between coarse dimension and transformal transcendence degree in these difference fields.

scholarly journals Model theory of difference fields

2017 ◽  
pp. 45-94 ◽  
Author(s):  
Zoé Chatzidakis
Keyword(s):  
Model Theory ◽  
Difference Fields

scholarly journals On Fully Degenerate Daehee Numbers and Polynomials of the Second Kind

2020 ◽  
Vol 2020 ◽  
pp. 1-9
Author(s):  
Sang Jo Yun ◽  
Jin-Woo Park
Keyword(s):  
Exponential Function ◽  
Special Functions ◽  
Eulerian Numbers

In a study, Carlitz introduced the degenerate exponential function and applied that function to Bernoulli and Eulerian numbers and degenerate special functions have been studied by many researchers. In this paper, we define the fully degenerate Daehee polynomials of the second kind which are different from other degenerate Daehee polynomials and derive some new and interesting identities and properties of those polynomials.

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