scholarly journals Obtaining Explicit Formulas and Identities for Polynomials Defined by Generating Functions of the Form F(t)x · G(t)α

2019 ◽  
Author(s):  
Dmitry Kruchinin ◽  
Vladimir Kruchinin ◽  
Yuriy Shablya
Keyword(s):  
Generating Functions ◽  
Explicit Formulas

New family of Whitney numbers

Filomat ◽  
2017 ◽  
Vol 31 (2) ◽  
pp. 309-320 ◽  
Author(s):  
B.S. El-Desouky ◽  
Nenad Cakic ◽  
F.A. Shiha
Keyword(s):  
Generating Functions ◽  
Recurrence Relations ◽  
Stirling Numbers ◽  
Explicit Formulas ◽  
Basic Properties ◽  
New Family ◽  
Whitney Numbers ◽  
Special Cases

In this paper we give a new family of numbers, called ??-Whitney numbers, which gives generalization of many types of Whitney numbers and Stirling numbers. Some basic properties of these numbers such as recurrence relations, explicit formulas and generating functions are given. Finally many interesting special cases are derived.

scholarly journals Liouville quantum gravity — holography, JT and matrices

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Thomas G. Mertens ◽  
Gustavo J. Turiaci
Keyword(s):  
String Theory ◽  
Generating Functions ◽  
Preliminary Evidence ◽  
Quantum Deformation ◽  
Dilaton Gravity ◽  
Continuum Approach ◽  
Genus Zero ◽  
Explicit Formulas ◽  
Liouville Gravity ◽  
The Continuum

Abstract We study two-dimensional Liouville gravity and minimal string theory on spaces with fixed length boundaries. We find explicit formulas describing the gravitational dressing of bulk and boundary correlators in the disk. Their structure has a striking resemblance with observables in 2d BF (plus a boundary term), associated to a quantum deformation of SL(2, ℝ), a connection we develop in some detail. For the case of the (2, p) minimal string theory, we compare and match the results from the continuum approach with a matrix model calculation, and verify that in the large p limit the correlators match with Jackiw-Teitelboim gravity. We consider multi-boundary amplitudes that we write in terms of gluing bulk one-point functions using a quantum deformation of the Weil-Petersson volumes and gluing measures. Generating functions for genus zero Weil-Petersson volumes are derived, taking the large p limit. Finally, we present preliminary evidence that the bulk theory can be interpreted as a 2d dilaton gravity model with a sinh Φ dilaton potential.

Coordination sequences and layer-by-layer growth of periodic structures

2019 ◽  
Vol 234 (5) ◽  
pp. 291-299
Author(s):  
Anton Shutov ◽  
Andrey Maleev
Keyword(s):  
Generating Functions ◽  
Diophantine Equations ◽  
Periodic Structures ◽  
Layer Growth ◽  
Layer By Layer ◽  
Explicit Formulas ◽  
New Approach ◽  
Coordination Numbers ◽  
Linear Diophantine Equations ◽  
Recurrent Relations

Abstract A new approach to the problem of coordination sequences of periodic structures is proposed. It is based on the concept of layer-by-layer growth and on the study of geodesics in periodic graphs. We represent coordination numbers as sums of so called sector coordination numbers arising from the growth polygon of the graph. In each sector we obtain a canonical form of the geodesic chains and reduce the calculation of the sector coordination numbers to solution of the linear Diophantine equations. The approach is illustrated by the example of the 2-homogeneous kra graph. We obtain three alternative descriptions of the coordination sequences: explicit formulas, generating functions and recurrent relations.

Explicit formulas for the characteristics of the M/M/2/2 queue with repeated attempts

1987 ◽  
Vol 24 (2) ◽  
pp. 486-494 ◽  
Author(s):  
Thomas Hanschke
Keyword(s):  
Steady State ◽  
Generating Functions ◽  
Explicit Formulas ◽  
Steady State Probabilities

In this paper we study the M/M/2/2 queue with repeated attempts. It is shown that the part generating functions of the steady state probabilities can be expressed in of generalized hypergeometric unctions.

scholarly journals Some Properties of the Hermite Polynomials and Their Squares and Generating Functions

2016 ◽  
Author(s):  
Feng Qi ◽  
Bai-Ni Guo
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Generating Functions ◽  
Recurrence Relations ◽  
Hermite Polynomials ◽  
Higher Order ◽  
Explicit Formulas ◽  
Derivatives Of

In the paper, the authors consider the generating functions of the Hermite polynomials and their squares, present explicit formulas for higher order derivatives of the generating functions of the Hermite polynomials and their squares, which can be viewed as ordinary differential equations or derivative polynomials, find differential equations that the generating functions of the Hermite polynomials and their squares satisfy, and derive explicit formulas and recurrence relations for the Hermite polynomials and their squares.

scholarly journals The multiparameter r-Whitney numbers

Filomat ◽  
2019 ◽  
Vol 33 (3) ◽  
pp. 931-943 ◽  
Author(s):  
B. El-Desouky ◽  
F.A. Shiha ◽  
Ethar Shokr
Keyword(s):  
Generating Functions ◽  
Recurrence Relations ◽  
Matrix Representation ◽  
Stirling Numbers ◽  
Combinatorial Identities ◽  
Harmonic Numbers ◽  
Explicit Formulas ◽  
Whitney Numbers ◽  
Special Cases ◽  
Generalized Stirling Numbers

In this paper, we define the multiparameter r-Whitney numbers of the first and second kind. The recurrence relations, generating functions , explicit formulas of these numbers and some combinatorial identities are derived. Some relations between these numbers and generalized Stirling numbers of the first and second kind, Lah numbers, C-numbers and harmonic numbers are deduced. Furthermore, some interesting special cases are given. Finally matrix representation for these relations are given.

scholarly journals Explicit Formulas for the First Form (q,r)-Dowling Numbers and (q,r)-Whitney-Lah Numbers

2021 ◽  
Vol 14 (1) ◽  
pp. 65-81
Author(s):  
Roberto Bagsarsa Corcino ◽  
Jay Ontolan ◽  
Maria Rowena Lobrigas
Keyword(s):  
Recurrence Relation ◽  
Explicit Formula ◽  
Taylor Series ◽  
Generating Functions ◽  
Recurrence Relations ◽  
Difference Operator ◽  
Interpolation Formula ◽  
Explicit Formulas ◽  
Fundamental Properties ◽  
Whitney Numbers

In this paper, a q-analogue of r-Whitney-Lah numbers, also known as (q,r)-Whitney-Lah number, denoted by $L_{m,r} [n, k]_q$ is defined using the triangular recurrence relation. Several fundamental properties for the q-analogue are established such as vertical and horizontal recurrence relations, horizontal and exponential generating functions. Moreover, an explicit formula for (q, r)-Whitney-Lah number is derived using the concept of q-difference operator, particularly, the q-analogue of Newton’s Interpolation Formula (the umbral version of Taylor series). Furthermore, an explicit formula for the first form (q, r)-Dowling numbers is obtained which is expressed in terms of (q,r)-Whitney-Lah numbers and (q,r)-Whitney numbers of the second kind.

scholarly journals Counting water cells in bargraphs of compositions and set partitions

2018 ◽  
Vol 12 (2) ◽  
pp. 413-438 ◽  
Author(s):  
Toufik Mansour ◽  
Mark Shattuck
Keyword(s):  
Generating Function ◽  
Generating Functions ◽  
Horizontal Line ◽  
Explicit Formulas ◽  
Set Partitions ◽  
Unit Square ◽  
Joint Distributions ◽  
Water Cell ◽  
First Moment

In this paper, we consider statistics on compositions and set partitions represented geometrically as bargraphs. By a water cell, we mean a unit square exterior to a bargraph that lies along a horizontal line between any two squares contained within the area subtended by the bargraph. That is, if a large amount of a liquid were poured onto the bargraph from above and allowed to drain freely, then the water cells are precisely those cells where the liquid would collect. In this paper, we count both compositions and set partitions according to the number of descents and water cells in their bargraph representations and determine generating function formulas for the joint distributions on the respective structures. Comparable generating functions that count non-crossing and non-nesting partitions are also found. Finally, we determine explicit formulas for the sign balance and for the first moment of the water cell statistic on set partitions, providing both algebraic and combinatorial proofs.

scholarly journals Enumerating set partitions by the number of positions between adjacent occurrences of a letter

2010 ◽  
Vol 4 (2) ◽  
pp. 284-308 ◽  
Author(s):  
Toufik Mansour ◽  
Mark Shattuck ◽  
Stephan Wagner
Keyword(s):  
Generating Functions ◽  
Stirling Numbers ◽  
Asymptotic Estimates ◽  
Explicit Formulas ◽  
Set Partitions ◽  
Minimal Elements

A partition ? of the set [n] = {1, 2,...,n} is a collection {B1,...,Bk} of nonempty disjoint subsets of [n] (called blocks) whose union equals [n]. Suppose that the subsets Bi are listed in increasing order of their minimal elements and ? = ?1, ?2...?n denotes the canonical sequential form of a partition of [n] in which iEB?i for each i. In this paper, we study the generating functions corresponding to statistics on the set of partitions of [n] with k blocks which record the total number of positions of ? between adjacent occurrences of a letter. Among our results are explicit formulas for the total value of the statistics over all the partitions in question, for which we provide both algebraic and combinatorial proofs. In addition, we supply asymptotic estimates of these formulas, the proofs of which entail approximating the size of certain sums involving the Stirling numbers. Finally, we obtain comparable results for statistics on partitions which record the total number of positions of ? of the same letter lying between two letters which are strictly larger.

On some statistics connected with runs in Markov chains

1988 ◽  
Vol 25 (04) ◽  
pp. 815-821 ◽  
Author(s):  
Dragan Banjević
Keyword(s):  
Markov Chain ◽  
Markov Chains ◽  
Generating Functions ◽  
Random Variables ◽  
Explicit Formulas ◽  
First Occurrence ◽  
History Of

We consider a class of statistics connected with the history of a Markov chain before the first occurrence of certain runs. Methods for deriving generating functions, distributions and moments of the statistics considered are given. In the case of i.i.d. random variables, explicit formulas are derived.

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