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Register a new userSolutions to difference equations have few defects
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Authors
Patrick Ingram
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We demonstrate a strong form of Nevanlinnas Second Main Theorem for solutions to difference equations f(z+1)=R(z, f(z)), with the coefficients of R growing slowly relative to f, and R of degree at least 2 in the second coordinate.
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For R(z, w) rational with complex coefficients, of degree at least 2 in w, we show that the number of rational functions f(z) solving the difference equation f(z+1)=R(z, f(z)) is finite and bounded just in terms of the degrees of R in the two variables. This complements a result of Yanagihara, who showed that any finite-order meromorphic solution to this sort of difference equation must be a rational function. We prove a similar result for the differential equation f(z)=R(z, f(z)), building on a result of Eremenko.
We investigate and derive second solutions to linear homogeneous second-order difference equations using a variety of methods, in each case going beyond the purely formal solution and giving explicit expressions for the second solution. We present a new implementation of dAlemberts reduction of order method, applying it to linear second-order recursion equations. Further, we introduce an iterative method to obtain a general solution, giving two linearly independent polynomial solutions to the recurrence relation. In the case of a particular confluent hypergeometric function for which the standard second solution is not independent of the first, i.e. the solutions are degenerate, we use the corresponding differential equation and apply the extended Cauchy-integral method to find a polynomial second solution for the difference equation. We show that the standard dAlembert method also generates this polynomial solution.
We prove that the main examples in the theory of algebraic differential equations possess a remarkable total differential overconvergence property. This allows one to consider solutions to these equations with coordinates in algebraically closed fields.
We study the relation between Hecke groups and the modular equations in Ramanujans theories of signature 2, 3 and 4. The solution $(alpha,beta)$ to the generalized modular equation satisfies a polynomial equation $P(alpha,beta)=0$ and we determine the degree in each of $alpha$ and $beta$ of the polynomial $P(alpha,beta)$ explicitly. We establish some mutually equivalent statements related to Hecke subgroups and modular equations, and prove that $(1-beta, 1-alpha)$ is also a solution to the generalized modular equation and $P(1-beta, 1-alpha)=0$.
We develop the theory of $p$-adic confluence of $q$-difference equations. The main result is the surprising fact that, in the $p$-adic framework, a function is solution of a differential equation if and only if it is solution of a $q$-difference equation. This fact implies an equivalence, called ``Confluence, between the category of differential equations and those of $q$-difference equations. We obtain this result by introducing a category of ``sheaves on the disk $mathrm{D}^-(1,1)$, whose stalk at 1 is a differential equation, the stalk at $q$ is a $q$-difference equation if $q$ is not a root of unity $xi$, and the stalk at a root of unity is a mixed object, formed by a differential equation and an action of $sigma_xi$.
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