arXiv subjects

Yik-Man Chiang

Publications and source records attributed to Yik-Man Chiang.

17 recordsLinked to original sources

arXiv2022

D-module approach to Liouville's Theorem for difference operators

We establish analogues of Liouville's theorem in the complex function theory, with the differential operator replaced by various difference operators. This is done generally by the extraction of (formal) Taylor coefficients using a residue map which measures the obstruction having local "anti-derivative". The residue map is based on a Weyl algebra or $q$-Weyl algebra structure satisfied by each corresponding operator. This explains the different senses of "boundedness" required by the respective analogues of Liouville's theorem in this article.

math.CV↗

arXiv2021

Resolving singularities and monodromy reduction of Fuchsian connections

We study monodromy reduction of Fuchsian connections from a sheave theoretic viewpoint, focusing on the case when a singularity of a special connection with four singularities has been resolved. The main tool of study is {based on} a bundle modification technique due to Drinfeld and Oblezin. This approach via invariant spaces and eigenvalue problems allows us not only to explain Erdélyi's classical infinite hypergeometric expansions of solutions to Heun equations, but also to obtain new expansions not found in his papers. As a consequence, a geometric proof of Takemura's eigenvalues inclusion theorem is obtained. Finally, we observe a precise matching between the monodromy reduction criteria giving those special solutions of Heun equations and that giving classical solutions of the Painlevé VI equation.

math.CA↗

arXiv2020

Wiman-Valiron theory for a polynomial series based on the Askey-Wilson operator

We establish a Wiman-Valiron theory of a polynomial series based on the Askey-Wilson operator $\mathcal{D}_q$, where $q\in(0,1)$. For an entire function $f$ of log-order smaller than $2$, this theory includes (i) an estimate which shows that $f$ behaves locally like a polynomial consisting of the terms near the maximal term of its Askey-Wilson series expansion, and (ii) an estimate of $\mathcal{D}_q^n f$ compared to $f$. We then apply this theory in studying the growth of entire solutions to difference equations involving the Askey-Wilson operator.

math.CV↗

arXiv2019

Invariant subspaces of biconfluent Heun operators and special solutions of Painlevé IV

We show that there is a full correspondence between the parameters space of the degenerate biconfluent Heun connection (BHC) and that of Painlevé IV that admits special solutions. The BHC degenerates when either the Stokes' data for the irregular singularity at $\infty$ degenerates or the regular singular point at the origin becomes an apparent singularity. We show that if the BHC is written as isomonodromy family of biconfluent Heun equations (BHE), then the BHE degenerates precisely when it admits eigen-solutions of the biconfluent Heun operators, after choosing appropriate accessory parameter, of specially constructed invariant subspaces of finite dimensional solution spaces spanned by parabolic cylinder functions. We have found all eigen-solutions over this parameter space apart from three exceptional cases after choosing the right accessory parameters. These eigen-solutions are expressed as certain finite sum of parabolic cylinder functions. We extend the above sum to new convergent series expansion in terms of parabolic cylinder functions to the BHE. The infinite sum solutions of the BHE terminates precisely when the parameters of the BHE assumes the same values as those of the degenerate biconfluent Heun connection except at three instances after choosing the right accessory parameter.

math.CA↗

arXiv2019

Solutions of Darboux Equations, its Degeneration and Painlevé VI Equations

In this paper, we study the Darboux equations in both classical and system form, which give the elliptic Painlevé VI equations by the isomonodromy deformation method. Then we establish the full correspondence between the special Darboux equations and the special Painlevé VI equations. Instead of the system form, we especially focus on the Darboux equation in a scalar form, which is the generalization of the classical Lamé equation. We introduce a new infinite series expansion (in terms of the compositions of hypergeometric functions and Jacobi elliptic functions) %around each of the four regular singular points of the for the solutions of the Darboux equations and regard special solutions of the Darboux equations as those terminating series. The Darboux equations characterized in this manner have an almost (but not completely) full correspondence to the special types of the Painlevé VI equations. Finally, we discuss the convergence of these infinite series expansions.

math.CA↗

arXiv2018

Galoisian approach to complex oscillation theory of some Hill equations

We apply Kovacic's algorithm from differential Galois theory to show that all complex non-oscillatory solutions (finite exponential of convergence of zeros) of certain Hill equations considered by Bank and Laine using Nevanlinna theory must be Liouvillian solutions. That is, solutions are obtainable by suitable differential field extensions construction. In particular, we have established a full correspondence between solutions of non-oscillatory type and Liouvillian solutions for a particular Hill equation. Explicit closed-form solutions are obtained via both methods for this Hill equation whose potential has four exponential functions in the Bank-Laine theory. The differential equation is a periodic form of biconfluent Heun equation. We further show that these Liouvillian solutions exhibit novel single and double orthogonality and a Fredholm integral equation over suitable integration regions in $\mathbf{C}$ that mimic single/double orthogonality for the corresponding Liouvillian solutions of the Lamé and Whittaker-Hill equations, discovered by Whittaker and Ince almost a century ago.

math.CA↗

arXiv2018

Nevanlinna theory of the Askey-Wilson divided difference operator

This paper establishes a version of Nevanlinna theory based on Askey-Wilson divided difference operator for meromorphic functions of finite logarithmic order in the complex plane $\mathbb{C}$. A second main theorem that we have derived allows us to define an Askey-Wilson type Nevanlinna deficiency which gives a new interpretation that one should regard many important infinite products arising from the study of basic hypergeometric series as zero/pole-scarce. That is, their zeros/poles are indeed deficient in the sense of difference Nevanlinna theory. A natural consequence is a version of Askey-Wilosn type Picard theorem. We also give an alternative and self-contained characterisation of the kernel functions of the Askey-Wilson operator. In addition we have established a version of unicity theorem in the sense of Askey-Wilson. This paper concludes with an application to difference equations generalising the Askey-Wilson second-order divided difference equation.

math.CV↗

arXiv2017

Symmetries of the Darboux equation

This paper establishes the symmetries of Darboux's equations (1882) on tori. We extend Ince's work (1940) by developing new infinite series expansions in terms of Jacobi elliptic functions around each of the four regular singular points of the Darboux equation which are located at the four half-periods of the torus. The symmetry group of the Darboux equation is given by the Coxeter group $B_4\cong G_{\mathrm{I}}\rtimes_Γ G_{\mathrm{II}}$ where the actions of $G_{\mathrm{I}}$ correspond to the sign changes of the parameters in the solutions of the equations, which have no effective changes on the equation itself, while the actions of $G_{\mathrm{II}}$ permute the four half-periods of the torus, which are described by a short-exact sequence. We are able to clarify the symmetries of the equation when ordered bases of the underlying torus change. Our results show that it is much more transparent to consider the symmetries of the Darboux equation on a torus than on the Riemann sphere $\mathbb{CP}^1$ as was usually considered by earlier researchers such as Maier (2007). We list the symmetry tables of the Darboux equation in both the Jacobian form and Weierstrass form. A consolidated list of 192 solutions of the Darboux equation is also given. We then consider the symmetries of several classical equations (e.g. Lamé equation) all of which are special cases of the Darboux equation. Those terminating solutions of the Darboux equation generalize the classical Lamé polynomials.

math.CA↗

arXiv2017

Difference Nevanlinna theories with vanishing and infinite periods

By extending the idea of a difference operator with a fixed step to varying-steps difference operators, we have established a difference Nevanlinna theory for meromorphic functions with the steps tending to zero (vanishing period) and a difference Nevanlinna theory for finite order meromorphic functions with the steps tending to infinity (infinite period) in this paper. We can recover the classical little Picard theorem from the vanishing period theory, but we require additional finite order growth restriction for meromorphic functions from the infinite period theory. Then we give some applications of our theories to exhibit connections between discrete equations and and their continuous analogues.

math.CV↗

arXiv2017

Nevanlinna Theory of the Wilson Divided-difference Operator

Sitting at the top level of the Askey-scheme, Wilson polynomials are regarded as the most general hypergeometric orthogonal polynomials. Instead of a differential equation, they satisfy a second order Sturm-Liouville type difference equation in terms of the Wilson divided-difference operator. This suggests that in order to better understand the distinctive properties of Wilson polynomials and related topics, one should use a function theory that is more natural with respect to the Wilson operator. Inspired by the recent work of Halburd and Korhonen, we establish a full-fledged Nevanlinna theory of the Wilson operator for meromorphic functions of finite order. In particular, we prove a Wilson analogue of the lemma on logarithmic derivatives, which helps us to derive Wilson operator versions of Nevanlinna's Second Fundamental Theorem, some defect relations and Picard's Theorem. These allow us to gain new insights on the distributions of zeros and poles of functions related to the Wilson operator, which is different from the classical viewpoint. We have also obtained a relevant five-value theorem and Clunie type theorem as applications of our theory, as well as a pointwise estimate of the logarithmic Wilson difference, which yields new estimates to the growth of meromorphic solutions to some Wilson difference equations and Wilson interpolation equations.

math.CV↗

arXiv2016

On complex oscillation theory, quasi-exact solvability and Fredholm Integral Equations

Biconfluent Heun equation (BHE) is a confluent case of the general Heun equation which has one more regular singular points than the Gauss hypergeometric equation on the Riemann sphere $\hat{\mathbb{C}}$. Motivated by a Nevanlinna theory (complex oscillation theory) approach, we have established a theory of \textit{periodic} BHE (PBHE) in parallel with the Lamé equation verses the Heun equation, and the Mathieu equation verses the confluent Heun equation. We have established condition that lead to explicit construction of eigen-solutions of PBHE, and their single and double orthogonality, and a related first-order Fredholm-type integral equation for which the corresponding eigen-solutions must satisfy. We have also established a Bessel polynomials analogue at the BHE level which is based on the observation that both the Bessel equation and the BHE have a regular singular point at the origin and an irregular singular point at infinity on the Riemann sphere $\hat{\mathbb{C}}$, and that the former equation has orthogonal polynomial solutions with respect to a complex weight. Finally, we relate our results to an equation considered by Turbiner, Bender and Dunne, etc concerning a quasi-exact solvable Schrödinger equation generated by first order operators such that the second order operators possess a finite-dimensional invariant subspace in a Lie algebra of $SL_2(\mathbb{C})$

math.CA↗

arXiv2016

On the growth of logarithmic difference of meromorphic functions and a Wiman-Valiron estimate

The paper gives a precise asymptotic relation between higher order logarithmic difference and logarithmic derivatives for meromorphic functions with order strictly less then one. This allows us to formulate a useful Wiman-Valiron type estimate for logarithmic difference of meromorphic functions of small order. We then apply this estimate to prove a classical analogue of Valiron about entire solutions to linear differential equations with polynomials coefficients for linear difference equations.

math.CV↗

arXiv2015

Two integrable differential-difference equations derived from NLS-type equation

Two integrable differential-difference equations are derived from a (2+1)-dimensional modified Heisenberg ferromagnetic equation and a resonant nonlinear Schröinger equation respectively. Multi-soliton solutions of the resulted semi-discrete systems are given through Hirota's bilinear method. Elastic and inelastic interaction behavior between two solitons are studied through the asymptotic analysis. Dynamics of two-soliton solutions are shown with graphs.

nlin.SI↗

arXiv2013

Estimates on the Growth of Meromorphic Solutions of Linear Differential Equations with density conditions

We give an alternative and simpler method for getting pointwise estimate of meromorphic solutions of homogeneous linear differential equations with coefficients meromorphic in a finite disk or in the open plane originally obtained by Hayman and the author. In particular, our estimates generally give better upper bounds for higher order derivatives of the meromorphic solutions under consideration, are valid, however, outside an exceptional set of finite logarithmic density. The estimates again show that the growth of meromorphic solutions with a positive deficiency at infinity can be estimated in terms of initial conditions of the solution at or near the origin and the characteristic functions of the coefficients.

math.CV↗

arXiv2011

On complex oscillation, function-theoretic quantization of non-homogeneous periodic ODEs and special functions

New necessary and sufficient conditions are given for the quantization of a class of periodic second order non-homogeneous ordinary differential equations in the complex plane in this paper. The problem is studied from the viewpoint of complex oscillation theory first developed by Bank and Laine (1982, 1983) and Gundersen and Steinbart (1994). We show that when a solution is complex non-oscillatory (finite exponent of convergence of zeros) then the solution, which can be written as special functions, must degenerate. This gives a necessary and sufficient condition when the Lommel function has finitely many zeros in every branch and this is a type of quantization for the non-homogeneous differential equation. The degenerate solutions are of polynomial/rational-type functions which are of independent interest. In particular, this shows that complex non-oscillatory solutions of this class of differential equations are equivalent to the subnormal solutions considered in a previous paper of the authors (to appear). In addition to the asymptotics of special functions, the other main idea we apply in our proof is a classical result of E. M. Wright which gives precise asymptotic locations of large zeros of a functional equation.

math.CA↗

arXiv2008

On the growth of logarithmic differences, difference quotients and logarithmic derivatives of meromorphic functions

A crucial ingredient in the recent discovery by Ablowitz, Halburd, Herbst and Korhonen \cite{AHH}, \cite {HK-2} that a connection exists between discrete Painlevé equations and (finite order) Nevanlinna theory is an estimate of the integrated average of $\log^+|f(z+1)/f(z)|$ on $|z|=r$. We obtained essentially the same estimate in our previous paper \cite{CF} independent of Halburd et al \cite{HK-1}. We continue our study in this paper by establishing complete asymptotic relations amongst the logarithmic differences, difference quotients and logarithmic derivatives for finite order meromorphic functions. In addition to the potential applications of our new estimates in integrable systems, they are also of independent interest. In particular, our findings show that there are marked differences between the growth of meromorphic functions with Nevanlinna order smaller and greater than one. We have established a "difference" analogue of the classical Wiman-Valiron type estimates for meromorphic functions with order less than one, which allow us to prove all entire solutions of linear difference equations (with polynomial coefficients) of order less than one must have positive rational order of growth. We have also established that any entire solution to a first order algebraic difference equation (with polynomial coefficients) must have a positive order of growth, which is a "difference" analogue of a classical result of Pólya.

math.CV↗

arXiv2006

Difference independence of the Riemann zeta function

It is proved that the Riemann zeta function does not satisfy any nontrivial algebraic difference equation whose coefficients are meromorphic functions $ϕ$ with Nevanlinna characteristic satisfying $T(r, ϕ)=o(r)$ as $r\to \infty$

math.CV↗