arXiv subjects

Zhi-Guo Liu

Publications and source records attributed to Zhi-Guo Liu.

At least 19 recordsLinked to original sources

arXiv2026

On Schultz's generalization of Borweins' cubic identity

In 1991, the Borweins established a cubic analogue of Jacobi's identity for theta functions, which is used by B.C. Berndt, S. Bhargava, and F.G. Garvan in the development of Ramanujan's cubic theory of elliptic functions. In 2013, D. Schultz discovered an identity for theta series in three variables which generalizes the Borweins' identity. In this article, we revisit Schultz's identity and present two distinct approaches to its derivation. Our investigation not only provides new proofs but also yields several new Schultz-type identities.

math.NT↗

arXiv2024

On a Transformation of Triple $q$-Series and Rogers-Hecke Type Series

Using the method of the $q$-exponential differential operator, we give an extension of the Sears $_4ϕ_3$ transformation formula. Based on this extended formula and a $q$-series expansion formula for an analytic function around the origin, we present a transformation formula for triple $q$-series, which includes several interesting special cases, especially a double $q$-series summation formula. Some applications of this transformation formula to Rogers-Hecke type series are discussed. More than 100 Rogers-Hecke type identities including Andrews' identities for the sums of three squares and the sums of three triangular numbers are obtained.

math.CV↗

arXiv2022

Exploration of the computational model and the focusing process with a Flat Multi-channel Plate and a Curved Multi-channel Plate in the MATLAB

By simulating the X-ray paths and the Chapman Model of a flat multi-channel plate and a curved multi-channel plate in the MATLAB, the field of view, local reflection efficiency, spherical aberration, point-spread function, collection efficiency of incident X-ray and peak-to-background ratio on the focal plane of the two devices were compared. At the same time, the advantages and disadvantages of the flat multi-channel plate and the curved multi-channel plate were compared.

physics.optics↗

arXiv2022

Theoretical Simulation and Experiment Investigation of X-ray transmission characteristics though Square Polycapillary Slice Lens with quadratic curve

The x-ray polycapillary lens is an optical device with good optic performance. Similar to the traditional X-ray polycapillary lens, square polycapillary slice lens was regulated on X-ray based on the full reflection principle of X-ray in the capillaries surfaces. According to its geometrical structure model and the X-ray tracing principle, a set of X-ray transmission procedures was established. A complete square polycapillary slice lens with quadratic curve was produced and the optical performance was tested

physics.optics↗

arXiv2022

Recent neutron focusing experiments using polycapillary lens in CSNS

Higher neutron current densities can provide convenience for neutron experiments. Using neutron optical focusing elements, large flux beams transported to sample can be achieved. As one kind of focusing elements, polycapillary lens is very suitable for neutron absorption experiments such as PGAA and NDP technology. At present, a Neutron Physics and Application Spectrometer was in construction in CSNS, which is the first pulsed neutron source in China. To provide some suggestions and ideas for the following design of enhanced PGAA or NDP instrument with polycapillary lens in CSNS, a first neutron focusing experiment using polycapillary lens in CSNS was conducted. For 0.5-12.6 polychromatic beam, a focal spot with FWHM of 800 was obtained. As the value of wavelength increased, the beam size, transmission efficiency and gain increased. For cold neutron, the gain maintained in a level of 7.

physics.ins-det↗

arXiv2021

A universal identity for theta functions of degree eight and applications

Previously, we proved an identity for theta functions of degree eight, and several applications of it were also discussed. This identity is a natural extension of the addition formula for the Weierstrass sigma-function. In this paper we will use this identity to reexamine our work in theta function identities in the past two decades. Hundreds of results about elliptic modular functions, both classical and new, are derived from this identity with ease. Essentially, this general theta function identity is a theta identities generating machine. Our investigation shows that many well-known results about elliptic modular functions with different appearances due to Jacobi, Kiepert, Ramanujan and Weierstrass among others, actually share a common source. This paper can also be seen as a summary of my past work on theta function identities. A conjecture is also proposed.

math.NT↗

arXiv2021

Uniform asymptotic formulas for the Fourier coefficients of the inverse of theta functions

In this paper, we use basic asymptotic analysis to establish some uniform asymptotic formulas for the Fourier coefficients of the inverse of Jacobi theta functions. In particular, we answer and improve some problems suggested and investigated by Bringmann, Manschot, and Dousse. As applications, we establish the asymptotic monotonicity properties for the rank and crank of the integer partitions introduced and investigated by Dyson, Andrews, and Garvan.

math.NT↗

arXiv2021

Infections Forecasting and Intervention Effect Evaluation for COVID-19 via a Data-Driven Markov Process and Heterogeneous Simulation

The Coronavirus Disease 2019 (COVID-19) pandemic has caused tremendous amount of deaths and a devastating impact on the economic development all over the world. Thus, it is paramount to control its further transmission, for which purpose it is necessary to find the mechanism of its transmission process and evaluate the effect of different control strategies. To deal with these issues, we describe the transmission of COVID-19 as an explosive Markov process with four parameters. The state transitions of the proposed Markov process can clearly disclose the terrible explosion and complex heterogeneity of COVID-19. Based on this, we further propose a simulation approach with heterogeneous infections. Experimentations show that our approach can closely track the real transmission process of COVID-19, disclose its transmission mechanism, and forecast the transmission under different non-drug intervention strategies. More importantly, our approach can helpfully develop effective strategies for controlling COVID-19 and appropriately compare their control effect in different countries/cities.

stat.AP↗

arXiv2020

Kronecker theta function and a decomposition theorem for theta functions I

The Kronecker theta function is a quotient of the Jacobi theta functions, which is also a special case of Ramanujan's $_1ψ_1$ summation. Using the Kronecker theta function as building blocks, we prove a decomposition theorem for theta functions. This decomposition theorem is the common source of a large number of theta function identities. Many striking theta function identities, both classical and new, are derived from this decomposition theorem with ease. A new addition formula for theta functions is established. Several known results in the theory of elliptic theta functions due to Ramanujan, Weierstrass, Kiepert, Winquist and Shen among others are revisited. A curious trigonometric identities is proved.

math.CV↗

arXiv2019

On the complex Hermite polynomials and partial differential equations

In this paper we use a set of partial differential equations to prove an expansion theorem for multiple complex Hermite polynomials. This expansion theorem allows us to develop a systematic and completely new approach to the complex Hermite polynomials. Using this expansion, we derive the Poisson Kernel, the Nielsen type formula, the addition formula for the complex Hermite polynomials with ease. A multilinear generating function for the complex Hermite polynomials is proved.

math.CV↗

arXiv2018

Askey--Wilson polynomials and a double $q$-series transformation formula with twelve parameters

The Askey--Wilson polynomials are the most general classical orthogonal polynomials that are known and the Nassrallah--Rahman integral is a very general extension of Euler's integral representation of the classical $_2F_1$ function. Based on a $q$-series transformation formula and the Nassrallah--Rahman integral we prove a $q$--beta integral which has twelve parameters, with several other results, both classical and new, included as special cases. This $q$-beta integral also allows us to derive a curious double $q$--series transformation formula, which includes one formula of Al--Salam and Ismail as a special case

math.CO↗

arXiv2018

Extensions of Ramanujan's reciprocity theorem and the Andrews--Askey integral

Ramanujan's reciprocity theorem may be considered as a three-variable extension of Jacobi's triple product identity. Using the method of $q$-partial differential equations, we extend Ramanujan's reciprocity theorem to a seven-variable reciprocity formula. The Andrews--Askey integral is a $q$-integral having four parameters with base $q$. Using the same method we extend the Andrews--Askey integral formula to a $q$-integral formula which has seven parameters with base $q$.

math.CO↗

arXiv2018

Gauss summation and Ramanujan type series for $1/π$

Using some properties of the gamma function and the well-known Gauss summation formula for the classical hypergeometric series, we prove a four-parameter series expansion formula, which can produce infinitely many Ramanujan type series for $1/π$.

math.CV↗

arXiv2018

A $q$-extension of a partial differential equation and the Hahn polynomials

Using the theory of analytic functions of several complex variables, we prove that if an analytic function in several variables satisfies a system of $q$-partial differential equations, then, it can be expanded in terms of the product of the homogeneous Hahn polynomials. Some non-trivial applications of this expansion theorem to $q$-series are discussed.

math.CO↗

arXiv2018

A $q$-summation and the orthogonality relations for the $q$-Hahn polynomials and the big $q$-Jacobi polynomials

Using a general $q$-summation formula, we derive a generating function for the $q$-Hahn polynomials, which is used to give a complete proof of the orthogonality relation for the $q$-Hahn polynomials. A new proof of the orthogonality relation for the big $q$-Jacobi polynomials is also given. A simple evaluation of the Nassrallah-Rahman integral is derived by using this summation formula. A new $q$-beta integral formula is established, which includes the Nassrallah-Rahman integral as a special case. The $q$-summation formula also allows us to recover several strange $q$-series identities.

math.CO↗

arXiv2018

On the $q$-derivative and $q$-series expansions

Using a general $q$-series expansion, we derive some nontrivial $q$-formulas involving many infinite products. A multitude of Hecke--type series identities are derived. Some general formulas for sums of any number of squares are given. A new representation for the generating function for sums of three triangular numbers is derived, which is slightly different from that of Andrews, also implies the famous result of Gauss where every integer is the sum of three triangular numbers.

math.NT↗

arXiv2018

On the $q$-partial differential equations and $q$-series

Using the theory of functions of several complex variables, we prove that if an analytic function in several variables satisfies a system of $q$-partial differential equations, then, it can be expanded in terms of the product of the Rogers-Szegő polynomials. This expansion theorem allows us to develop a general method for proving $q$-identities. A general $q$-transformation formula is derived, which implies Watson's $q$-analog of Whipple's theorem as a special case. A multilinear generating function for the Rogers-Szegő polynomials is given. The theory of $q$-exponential operator is revisited.

math.AP↗

arXiv2017

On a system of $q$-partial differential equations with applications to $q$-series

Using the theory of functions of several variables and $q$-calculus, we prove an expansion theorem for the analytic function in several variables which satisfies a system of $q$-partial differential equations. Some curious applications of this expansion theorem to $q$-series are discussed. In particular, an extension of Andrews' transformation formula for the $q$-Lauricella function is given.

math.CV↗