scholarly journals HECKE-TYPE CONGRUENCES FOR ANDREWS' SPT-FUNCTION MODULO 16 AND 32

2014 ◽  
Vol 10 (02) ◽  
pp. 375-390 ◽  
Author(s):  
FRANK G. GARVAN ◽  
CHRIS JENNINGS-SHAFFER
Keyword(s):  
Generating Function ◽  
Hecke Operators ◽  
Powers Of 2

Inspired by recent congruences by Andersen with varying powers of 2 in the modulus for partition related functions, we extend the modulo 32760 congruences of the first author for the function spt (n). We show that a normalized form of the generating function of spt (n) is an eigenform modulo 32 for the Hecke operators T(ℓ2) for primes ℓ ≥ 5 with ℓ ≡ 1, 11, 17, 19 (mod 24), and an eigenform modulo 16 for ℓ ≡ 13, 23 (mod 24).

scholarly journals Overpartitions with Restricted Odd Differences

10.37236/5248 ◽  
2015 ◽  
Vol 22 (3) ◽  
Author(s):  
Kathrin Bringmann ◽  
Jehanne Dousse ◽  
Jeremy Lovejoy ◽  
Karl Mahlburg
Keyword(s):  
Modular Form ◽  
Generating Function ◽  
Difference Equations ◽  
Circle Method ◽  
Hecke Operators ◽  
Asymptotic Formulas ◽  
Linear Congruences ◽  
The Difference ◽  
Mock Modular Form

We use $q$-difference equations to compute a two-variable $q$-hypergeometric generating function for overpartitions where the difference between two successive parts may be odd only if the larger part is overlined. This generating function specializes in one case to a modular form, and in another to a mixed mock modular form. We also establish a two-variable generating function for the same overpartitions with odd smallest part, and again find modular and mixed mock modular specializations. Applications include linear congruences arising from eigenforms for $3$-adic Hecke operators, as well as asymptotic formulas for the enumeration functions. The latter are proven using Wright's variation of the circle method.

PARITY OF THE PARTITION FUNCTION AND TRACES OF SINGULAR MODULI

2012 ◽  
Vol 08 (02) ◽  
pp. 395-409 ◽  
Author(s):  
PAK-HIN LEE ◽  
ALEXANDR ZAMORZAEV
Keyword(s):  
Partition Function ◽  
Generating Function ◽  
Logarithmic Derivative ◽  
Complex Multiplication ◽  
Hecke Operators ◽  
Singular Moduli ◽  
Class Polynomials

We prove that the parity of the partition function is given by the "trace" of the Hauptmodul [Formula: see text] for [Formula: see text] at points of complex multiplication. Using Hecke operators, we generalize this to relate the Hecke traces of [Formula: see text] to the partition function modulo 2. We then prove that the generating function for these Hecke traces is equal to the logarithmic derivative of the level 6 Hilbert class polynomial. Finally, we give a procedure involving Hilbert class polynomials for computing the parity of the partition function, and make some speculations about the distribution of these universal polynomials modulo class polynomials.

scholarly journals CONGRUENCES MODULO SQUARES OF PRIMES FOR FU'S k DOTS BRACELET PARTITIONS

2013 ◽  
Vol 09 (04) ◽  
pp. 939-943 ◽  
Author(s):  
CRISTIAN-SILVIU RADU ◽  
JAMES A. SELLERS
Keyword(s):  
Positive Integer ◽  
Generating Function ◽  
Infinite Family ◽  
Combinatorial Proof ◽  
Combinatorial Approach ◽  
Powers Of 2 ◽  
The Family ◽  
Fixed Positive Integer ◽  
Congruence Properties

In 2007, Andrews and Paule introduced the family of functions Δk(n) which enumerate the number of broken k-diamond partitions for a fixed positive integer k. In that paper, Andrews and Paule proved that, for all n ≥ 0, Δ1(2n+1) ≡ 0 (mod 3) using a standard generating function argument. Soon after, Shishuo Fu provided a combinatorial proof of this same congruence. Fu also utilized this combinatorial approach to naturally define a generalization of broken k-diamond partitions which he called k dots bracelet partitions. He denoted the number of k dots bracelet partitions of n by 𝔅k(n) and proved various congruence properties for these functions modulo primes and modulo powers of 2. In this note, we extend the set of congruences proven by Fu by proving the following congruences: For all n ≥ 0, [Formula: see text] We also conjecture an infinite family of congruences modulo powers of 7 which are satisfied by the function 𝔅7.

Proof of the Churchhouse conjecture concerning binary partitions

1971 ◽  
Vol 70 (1) ◽  
pp. 53-56 ◽  
Author(s):  
Hansraj Gupta
Keyword(s):  
Generating Function ◽  
Powers Of 2 ◽  
Image Position

In a recent paper, R. F. Churchhouse has studied the function b(n) giving the number of partitions of n into powers of 2. The generating function of b(n) isso thatUsing this relation, Churchhouse has shown thatandwherewithMoreoverand b(n) is even for n ≥ 2, while b(2n) ≡ (mod 4), for n = 22m−l(2k + 1); and, it is not ≡ (mod 8) for any n. He has conjectured that for k ≥ 1 and n odd,andThe object of this note is to prove these conjectures.

scholarly journals GENERATING FUNCTIONS FOR HECKE OPERATORS

2009 ◽  
Vol 05 (01) ◽  
pp. 125-140 ◽  
Author(s):  
HALA AL HAJJ SHEHADEH ◽  
SAMAR JAAFAR ◽  
KAMAL KHURI-MAKDISI
Keyword(s):  
Rational Function ◽  
Generating Function ◽  
Modular Forms ◽  
Generating Functions ◽  
Hecke Operators ◽  
Systematic Method ◽  
Hecke Operator ◽  
The Matrix ◽  
Arbitrary Level

Fix a prime N, and consider the action of the Hecke operator TN on the space [Formula: see text] of modular forms of full level and varying weight κ. The coefficients of the matrix of TN with respect to the basis {E4i E6j | 4i + 6j = κ} for [Formula: see text] can be combined for varying κ into a generating function FN. We show that this generating function is a rational function for all N, and present a systematic method for computing FN. We carry out the computations for N = 2, 3, 5, and indicate and discuss generalizations to spaces of modular forms of arbitrary level.

System of thermodynamic quantities in the McMillan-Mayer solution theory

1985 ◽  
Vol 50 (4) ◽  
pp. 791-798 ◽  
Author(s):  
Vilém Kodýtek
Keyword(s):  
Free Energy ◽  
Generating Function ◽  
Simple Form ◽  
Unit Volume ◽  
Thermodynamic Quantities ◽  
Solution Theory ◽  
Pure Solvent ◽  
Second Derivatives ◽  
Definition Of ◽  
Derivatives Of

The McMillan-Mayer (MM) free energy per unit volume of solution AMM, is employed as a generating function of the MM system of thermodynamic quantities for solutions in the state of osmotic equilibrium with pure solvent. This system can be defined by replacing the quantities G, T, P, and m in the definition of the Lewis-Randall (LR) system by AMM, T, P0, and c (P0 being the pure solvent pressure). Following this way the LR to MM conversion relations for the first derivatives of the free energy are obtained in a simple form. New relations are derived for its second derivatives.

scholarly journals A General Family of q-Hypergeometric Polynomials and Associated Generating Functions

Mathematics ◽  
2021 ◽  
Vol 9 (11) ◽  
pp. 1161
Author(s):  
Hari Mohan Srivastava ◽  
Sama Arjika
Keyword(s):  
Generating Function ◽  
Generating Functions ◽  
Hypergeometric Functions ◽  
Additional Parameter ◽  
Physical Sciences ◽  
General Family ◽  
Hypergeometric Polynomials

Basic (or q-) series and basic (or q-) polynomials, especially the basic (or q-) hypergeometric functions and the basic (or q-) hypergeometric polynomials are studied extensively and widely due mainly to their potential for applications in many areas of mathematical and physical sciences. Here, in this paper, we introduce a general family of q-hypergeometric polynomials and investigate several q-series identities such as an extended generating function and a Srivastava-Agarwal type bilinear generating function for this family of q-hypergeometric polynomials. We give a transformational identity involving generating functions for the generalized q-hypergeometric polynomials which we have introduced here. We also point out relevant connections of the various q-results, which we investigate here, with those in several related earlier works on this subject. We conclude this paper by remarking that it will be a rather trivial and inconsequential exercise to give the so-called (p,q)-variations of the q-results, which we have investigated here, because the additional parameter p is obviously redundant.

On the absorption probabilities and mean time for absorption for discrete Markov chains

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Nikolaos Halidias
Keyword(s):  
Markov Chain ◽  
Random Walk ◽  
Markov Chains ◽  
Generating Function ◽  
Discrete Time ◽  
Probability Generating Function ◽  
The Mean ◽  
Mean Time ◽  
Absorption Probabilities

Abstract In this note we study the probability and the mean time for absorption for discrete time Markov chains. In particular, we are interested in estimating the mean time for absorption when absorption is not certain and connect it with some other known results. Computing a suitable probability generating function, we are able to estimate the mean time for absorption when absorption is not certain giving some applications concerning the random walk. Furthermore, we investigate the probability for a Markov chain to reach a set A before reach B generalizing this result for a sequence of sets A 1 , A 2 , … , A k {A_{1},A_{2},\dots,A_{k}} .

The number of meetings in free Poisson traffic

1974 ◽  
Vol 11 (2) ◽  
pp. 320-331
Author(s):  
Hans D. Unkelbach ◽  
Helmut Wegmann
Keyword(s):  
Differential Equation ◽  
Generating Function ◽  
Practical Interest ◽  
Time Interval ◽  
Integro Differential Equation

Using Rényi's model of free Poisson traffic the distribution of the number of meetings of vehicles on a highway section during a given time interval is investigated. An integro-differential equation for the generating function of that variable is deduced and the first moments are calculated. The generating function is given explicitly in simple cases and approximately in cases of practical interest.

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