VANISHING COEFFICIENTS IN QUOTIENTS OF THETA FUNCTIONS OF MODULUS FIVE

2020 ◽  
Vol 102 (3) ◽  
pp. 387-398
Author(s):  
SHANE CHERN ◽  
DAZHAO TANG
Keyword(s):  
Infinite Product ◽  
Theta Functions ◽  
Infinite Products ◽  
General Family

Following recent investigations of vanishing coefficients in infinite products, we show that such instances are very rare when the infinite product is among a family of theta-quotients of modulus five. We also prove that a general family of products of theta functions of modulus five can always be effectively 5-dissected.

scholarly journals Vanishing coefficients in infinite product expansions

1979 ◽  
Vol 27 (2) ◽  
pp. 199-202 ◽  
Author(s):  
George E. Andrews ◽  
David M. Bressoud
Keyword(s):  
Power Series ◽  
Infinite Product ◽  
Series Expansions ◽  
Infinite Products ◽  
General Family ◽  
Power Series Expansions

AbstractRichmond and Szekeres (1977) have conjecturned that certain of the coefficients in the power series expansions of certain infinite products vanish. In this paper, we prove a general family of results of this nature which includes the above conjectures.

scholarly journals Wiener-Hopf Factorization for a Family of Lévy Processes Related to Theta Functions

2010 ◽  
Vol 47 (04) ◽  
pp. 1023-1033 ◽  
Author(s):  
A. Kuznetsov
Keyword(s):  
Lévy Processes ◽  
Series Representation ◽  
Theta Functions ◽  
Characteristic Exponent ◽  
Levy Processes ◽  
Transcendental Equation ◽  
Simple Expression ◽  
Lévy Measure ◽  
Infinite Products ◽  
Wide Range

In this paper we study the Wiener-Hopf factorization for a class of Lévy processes with double-sided jumps, characterized by the fact that the density of the Lévy measure is given by an infinite series of exponential functions with positive coefficients. We express the Wiener-Hopf factors as infinite products over roots of a certain transcendental equation, and provide a series representation for the distribution of the supremum/infimum process evaluated at an independent exponential time. We also introduce five eight-parameter families of Lévy processes, defined by the fact that the density of the Lévy measure is a (fractional) derivative of the theta function, and we show that these processes can have a wide range of behavior of small jumps. These families of processes are of particular interest for applications, since the characteristic exponent has a simple expression, which allows efficient numerical computation of the Wiener-Hopf factors and distributions of various functionals of the process.

Radial and circular slit maps of unbounded multiply connected circle domains

2008 ◽  
Vol 464 (2095) ◽  
pp. 1719-1737 ◽  
Author(s):  
T.K DeLillo ◽  
T.A Driscoll ◽  
A.R Elcrat ◽  
J.A Pfaltzgraff
Keyword(s):  
Analytic Continuation ◽  
Recent Progress ◽  
Infinite Product ◽  
Reflection Principle ◽  
Numerical Implementation ◽  
Infinite Products ◽  
Multiply Connected ◽  
Slit Domain ◽  
Circular Slit ◽  
Product Mapping

Infinite product formulae for conformally mapping an unbounded multiply connected circle domain to an unbounded canonical radial or circular slit domain, or to domains with both radial and circular slit boundary components are derived and implemented numerically and graphically. The formulae are generated by analytic continuation with the reflection principle. Convergence of the infinite products is proved for domains with sufficiently well-separated boundary components. Some recent progress in the numerical implementation of infinite product mapping formulae is presented.

scholarly journals CONGRUENCE PROPERTIES OF BORCHERDS PRODUCT EXPONENTS

2013 ◽  
Vol 09 (06) ◽  
pp. 1563-1578
Author(s):  
KEENAN MONKS ◽  
SARAH PELUSE ◽  
LYNNELLE YE
Keyword(s):  
Modular Forms ◽  
Logarithmic Derivative ◽  
Infinite Product ◽  
Modular Representation ◽  
Infinite Products ◽  
Analogous Statement ◽  
Half Integral Weight ◽  
Integral Weight ◽  
Congruence Properties ◽  
Invent Math

In his striking 1995 paper, Borcherds [Automorphic forms on Os+2,2(ℝ) and infinite products, Invent. Math.120 (1995) 161–213] found an infinite product expansion for certain modular forms with CM divisors. In particular, this applies to the Hilbert class polynomial of discriminant -d evaluated at the modular j-function. Among a number of powerful generalizations of Borcherds' work, Zagier made an analogous statement for twisted versions of this polynomial. He proves that the exponents of these product expansions, A(n,d), are the coefficients of certain special half-integral weight modular forms. We study the congruence properties of A(n,d) modulo a prime ℓ by relating it to a modular representation of the logarithmic derivative of the Hilbert class polynomial.

scholarly journals Generalized Bilateral Eighth Order Mock Theta Functions and Continued Fractions

2019 ◽  
Vol 53 (2) ◽  
pp. 185-193
Author(s):  
Bhaskar Srivastava
Keyword(s):  
Continued Fractions ◽  
Continued Fraction ◽  
Infinite Product ◽  
Theta Functions ◽  
Generalized Function ◽  
Mock Theta Functions ◽  
Eighth Order ◽  
Independent Variable

We give a two independent variable generalization of bilateral eighth order mock theta functions and expressed them as infinite product. On specializing parameters, we have given a continued fraction representation for the generalized function, which I think is a new representation.

scholarly journals On uniform convergence of some classes of infinite products

1987 ◽  
pp. 107
Author(s):  
K.M. Slepenchuk
Keyword(s):  
Uniform Convergence ◽  
Infinite Product ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Infinite Products ◽  
Necessary And Sufficient

We find necessary and sufficient conditions $\{ \alpha_k(x) \}$ must satisfy for the infinite product$$\prod\limits_{k=1}^{\infty} \bigl[ 1 + \alpha_k(x) u_k(x) \bigr]$$to converge uniformly under the condition that:1) the series $\sum\limits_{k=1}^{\infty} |\Delta u_k(x)|$ converges uniformly; 2) $\sum\limits_{k=1}^{\infty} |\Delta u_k(x)| = O(1)$.

scholarly journals Digit Sums and Infinite Products

2020 ◽  
Author(s):  
Samin Riasat
Keyword(s):  
Closed Form ◽  
Large Class ◽  
Rational Function ◽  
Systematic Approach ◽  
Infinite Product ◽  
Infinite Products

Consider the sequence un defined as follows: un=+1 if the sum of the base b digits of n is even, and un=−1 otherwise, where we take b=2. Recall that the Woods-Robbins infinite product involves a rational function in n and the sequence un. Although several generalizations of the Woods-Robbins product are known in the literature, no other infinite product involving a rational function in n and the sequence un was known in closed form until recently. In this chapter we introduce a systematic approach to these products, which may be generalized to other values of b. We illustrate the approach by evaluating a large class of similar infinite products.

scholarly journals A Note on Certain Modular Equations about Infinite Products of Ramanujan

2013 ◽  
Vol 2013 ◽  
pp. 1-5
Author(s):  
Hong-Cun Zhai
Keyword(s):  
Theta Functions ◽  
Modular Equations ◽  
Infinite Products

Ramanujan proposed additive formulae of theta functions that are related to modular equations about infinite products. Employing these formulaes, we derived some identities on infinite products. In the same spirit, we also could present elementary and simple proofs of certain Ramanujan's modular equations on infinite products.

STOCHASTIC EQUATIONS AND DIRICHLET OPERATORS ON INFINITE PRODUCT MANIFOLDS

2003 ◽  
Vol 06 (03) ◽  
pp. 455-488 ◽  
Author(s):  
SERGIO ALBEVERIO ◽  
ALEXEI DALETSKII ◽  
YURI KONDRATIEV
Keyword(s):  
Riemannian Manifolds ◽  
Existence And Uniqueness ◽  
Infinite Product ◽  
Lattice Models ◽  
Glauber Dynamics ◽  
Stochastic Equations ◽  
Existence And Uniqueness Theorem ◽  
Classical Lattice ◽  
Infinite Products ◽  
Dirichlet Operators

We discuss elements of stochastic analysis on product manifolds (infinite products of compact Riemannian manifolds). We introduce differentiable structures on product manifolds and prove the existence and uniqueness theorem for stochastic differential equations on them. This result is applied to the construction of Glauber dynamics for classical lattice models with compact spin spaces.

scholarly journals The Mean Value for Infinite Volume Measures, Infinite Products, and Heuristic Infinite Dimensional Lebesgue Measures

2017 ◽  
Vol 2017 ◽  
pp. 1-14 ◽  
Author(s):  
Jean-Pierre Magnot
Keyword(s):  
Lebesgue Measure ◽  
Infinite Product ◽  
Mean Value ◽  
Uniform Limit ◽  
Infinite Products ◽  
Infinite Dimensional ◽  
The Mean ◽  
Invariant Means ◽  
Lebesgue Measures ◽  
Dimensional Lebesgue Measure

One of the goals of this article is to describe a setting adapted to the description of means (normalized integrals or invariant means) on an infinite product of measured spaces with infinite measure and of the concentration property on metric measured spaces, inspired from classical examples of means. In some cases, we get a linear extension of the limit at infinity. Then, the mean value on an infinite product is defined, first for cylindrical functions and secondly taking the uniform limit. Finally, the mean value for the heuristic Lebesgue measure on a separable infinite dimensional topological vector space (e.g., on a Hilbert space) is defined. This last object, which is not the classical infinite dimensional Lebesgue measure but its “normalized” version, is shown to be invariant under translation, scaling, and restriction.

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