scholarly journals Generating functions for vector partition functions and a basic recurrence relation

2019 ◽  
Vol 25 (7) ◽  
pp. 1052-1061 ◽  
Author(s):  
Alexander P. Lyapin ◽  
Sreelatha Chandragiri
Keyword(s):  
Recurrence Relation ◽  
Generating Functions ◽  
Partition Functions

scholarly journals Generating functions and congruences for some partition functions related to mock theta functions

2019 ◽  
Vol 16 (02) ◽  
pp. 423-446 ◽  
Author(s):  
Nayandeep Deka Baruah ◽  
Nilufar Mana Begum
Keyword(s):  
Generating Functions ◽  
Theta Functions ◽  
Partition Functions ◽  
Mock Theta Functions ◽  
Third Order

Recently, Andrews, Dixit and Yee introduced partition functions associated with Ramanujan/Watson third-order mock theta functions [Formula: see text] and [Formula: see text]. In this paper, we find several new exact generating functions for those partition functions as well as the associated smallest part functions and deduce several new congruences modulo powers of 5.

scholarly journals Universality and Exact Finite-Size Corrections for Spanning Trees on Cobweb and Fan Networks

Entropy ◽  
2019 ◽  
Vol 21 (9) ◽  
pp. 895
Author(s):  
Nickolay Izmailian ◽  
Ralph Kenna
Keyword(s):  
Generating Functions ◽  
Spanning Trees ◽  
Central Charge ◽  
Finite Size ◽  
Universality Class ◽  
Partition Functions ◽  
Free Energies ◽  
Conformal Theory ◽  
Alternative Means ◽  
The Individual

The concept of universality is a cornerstone of theories of critical phenomena. It is very well understood in most systems, especially in the thermodynamic limit. Finite-size systems present additional challenges. Even in low dimensions, universality of the edge and corner contributions to free energies and response functions is less investigated and less well understood. In particular, the question arises of how universality is maintained in correction-to-scaling in systems of the same universality class but with very different corner geometries. Two-dimensional geometries deliver the simplest such examples that can be constructed with and without corners. To investigate how the presence and absence of corners manifest universality, we analyze the spanning tree generating function on two different finite systems, namely the cobweb and fan networks. The corner free energies of these configurations have stimulated significant interest precisely because of expectations regarding their universal properties and we address how this can be delivered given that the finite-size cobweb has no corners while the fan has four. To answer, we appeal to the Ivashkevich–Izmailian–Hu approach which unifies the generating functions of distinct networks in terms of a single partition function with twisted boundary conditions. This unified approach shows that the contributions to the individual corner free energies of the fan network sum to zero so that it precisely matches that of the web. It therefore also matches conformal theory (in which the central charge is found to be c = − 2 ) and finite-size scaling predictions. Correspondence in each case with results established by alternative means for both networks verifies the soundness of the Ivashkevich–Izmailian–Hu algorithm. Its broad range of usefulness is demonstrated by its application to hitherto unsolved problems—namely the exact asymptotic expansions of the logarithms of the generating functions and the conformal partition functions for fan and cobweb geometries. We also investigate strip geometries, again confirming the predictions of conformal field theory. Thus, the resolution of a universality puzzle demonstrates the power of the algorithm and opens up new applications in the future.

scholarly journals ON GENERALIGATIONS OF PARTITION FUNCTIONS

2015 ◽  
Vol 3 (10) ◽  
pp. 1-29
Author(s):  
Nil Ratan Bhattacharjee ◽  
Sabuj Das
Keyword(s):  
Generating Function ◽  
Generating Functions ◽  
Partition Functions ◽  
Auxiliary Functions ◽  
Numerical Example ◽  
Lost Notebook ◽  
Ramanujan's Lost Notebook ◽  
Leonhard Euler

In 1742, Leonhard Euler invented the generating function for P(n). Godfrey Harold Hardy said Srinivasa Ramanujan was the first, and up to now the only, Mathematician to discover any such properties of P(n). In 1916, Ramanujan defined the generating functions for   X(n),Y(n) . In 2014, Sabuj developed the generating functions for .  In 2005, George E. Andrews found the generating functions for    In 1916, Ramanujan showed the generating functions for  ,  ,   and  . This article shows how to prove the Theorems with the help of various auxiliary functions collected from Ramanujan’s Lost Notebook. In 1967, George E. Andrews defined the generating functions for P1r (n) and P2r (n). In this article these generating functions are discussed elaborately. This article shows how to prove the theorem P2r (n) = P3r (n) with a numerical example when n = 9 and r = 2. In 1995, Fokkink, Fokkink and Wang defined the   in terms of , where   is the smallest part of partition . In 2013, Andrews, Garvan and Liang extended the FFW-function and defined the generating function for FFW (z, n) in differnt way.

scholarly journals Solutions to the T-Systems with Principal Coefficients

10.37236/5698 ◽  
2016 ◽  
Vol 23 (2) ◽  
Author(s):  
Panupong Vichitkunakorn
Keyword(s):  
Recurrence Relation ◽  
Cluster Algebra ◽  
Perfect Matchings ◽  
Partition Functions ◽  
Initial Condition ◽  
Free Cluster ◽  
Octahedron Recurrence ◽  
T System

The $A_\infty$ T-system, also called the octahedron recurrence, is a dynamical recurrence relation. It can be realized as mutation in a coefficient-free cluster algebra (Kedem 2008, Di Francesco and Kedem 2009). We define T-systems with principal coefficients from cluster algebra aspect, and give combinatorial solutions with respect to any valid initial condition in terms of partition functions of perfect matchings, non-intersecting paths and networks. This also provides a solution to other systems with various choices of coefficients on T-systems including Speyer's octahedron recurrence (Speyer 2007), generalized lambda-determinants (Di Francesco 2013) and (higher) pentagram maps (Schwartz 1992, Ovsienko et al. 2010, Glick 2011, Gekhtman et al. 2016).

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 Identities and derivative formulas for the combinatorial and Apostol-Euler type numbers by their generating functions

Filomat ◽  
2018 ◽  
Vol 32 (20) ◽  
pp. 6879-6891
Author(s):  
Irem Kucukoglu ◽  
Yilmaz Simsek
Keyword(s):  
Recurrence Relation ◽  
Generating Functions ◽  
Stirling Numbers ◽  
Bell Numbers ◽  
New Family ◽  
Derivative Formulas ◽  
Combinatorial Sums ◽  
Computation Algorithm ◽  
Fractional Derivative Operators ◽  
And Inversion

The first aim of this paper is to give identities and relations for a new family of the combinatorial numbers and the Apostol-Euler type numbers of the second kind, the Stirling numbers, the Apostol-Bernoulli type numbers, the Bell numbers and the numbers of the Lyndon words by using some techniques including generating functions, functional equations and inversion formulas. The second aim is to derive some derivative formulas and combinatorial sums by applying derivative operators including the Caputo fractional derivative operators. Moreover, we give a recurrence relation for the Apostol-Euler type numbers of the second kind. By using this recurrence relation, we construct a computation algorithm for these numbers. In addition, we derive some novel formulas including the Stirling numbers and other special numbers. Finally, we also some remarks, comments and observations related to our results.

A General Asymptotic Result for Partitions

1975 ◽  
Vol 27 (5) ◽  
pp. 1083-1091 ◽  
Author(s):  
Bruce Richmond
Keyword(s):  
Generating Functions ◽  
Asymptotic Relation ◽  
Asymptotic Result ◽  
Partition Functions ◽  
Image Position

In this paper we are concerned with partition functions pϒ(n) that have generating functions of the formwhere γ(n) ≧ 0. We shall obtain an asymptotic relation for pϒ(n) under suitable restrictions on ϒ (see Theorem 1.1). These restrictions are weaker than those of Brigham [2] who considered this problem previously.

scholarly journals Product partitions and recursion formulae

2004 ◽  
Vol 2004 (33) ◽  
pp. 1725-1735 ◽  
Author(s):  
M. V. Subbarao
Keyword(s):  
Generating Functions ◽  
Partition Functions

Utilizing a method briefly hinted in the author's paper written in 1991 jointly with V. C. Harris, we derive here a number of unpublished recursion formulae for a variety of product partition functions which we believe have not been considered before in the literature. These include the functionsp*(n;k,h)(which stands for the number of product partitions ofn>1intokparts of whichhare distinct), andp(d)*(n;m)(which stands for the number of product partitions ofninto exactlymparts with at mostdrepetitions of any part). We also derive recursion formulae for certain product partition functions without the use of generating functions.

scholarly journals The Kostant partition functions for twisted Kac-Moody algebras

2000 ◽  
Vol 24 (1) ◽  
pp. 11-27
Author(s):  
Ranabir Chakrabarti ◽  
Thalanayar S. Santhanam
Keyword(s):  
Generating Functions ◽  
Infinite Product ◽  
Triple Product ◽  
Partition Functions ◽  
Recursion Relations

Employing the method of generating functions and making use of some infinite product identities like Euler, Jacobi's triple product and pentagon identities we derive recursion relations for Kostant's partition functions for the twisted Kac-Moody algebras.

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