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Register a new userOn the convergence of generalized power series solutions of $q$-difference equations
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A sufficient condition for the convergence of a generalized formal power series solution to an algebraic $q$-difference equation is provided. The main result leans on a geometric property related to the semi-group of (complex) power exponents of such a series. This property corresponds to the situation in which the small divisors phenomenon does not arise. Some examples illustrating the cases where the obtained sufficient condition can be or cannot be applied are also depicted.
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We propose a sufficient condition of the convergence of a generalized power series formally satisfying an algebraic (polynomial) ordinary differential equation. The proof is based on the majorant method.
Here we present some compliments to theorems of Gerard and Sibuya, on the convergence of multivariate formal power series solutions of nonlinear meromorphic Pfaffian systems. Their the most known results concern completely integrable systems with nondegenerate linear parts, whereas we consider some cases of non-integrability and degeneracy.
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 give a $q$-analog of middle convolution for linear $q$-difference equations with rational coefficients. In the differential case, middle convolution is defined by Katz, and he examined properties of middle convolution in detail. In this paper, we define a $q$-analog of middle convolution. Moreover, we show that it also can be expressed as a $q$-analog of Euler transformation. The $q$-middle convolution transforms Fuchsian type equation to Fuchsian type equation and preserves rigidity index of $q$-difference equations.
We provide an exact infinite power series solution that describes the trajectory of a nonlinear simple pendulum undergoing librating and rotating motion for all time. Although the series coefficients were previously given in [V. Fairen, V. Lopez, and L. Conde, Am. J. Phys 56 (1), (1988), pp. 57-61], the series itself -- as well as the optimal location about which an expansion should be chosen to assure series convergence and maximize the domain of convergence -- was not examined, and is provided here. By virtue of its representation as an elliptic function, the pendulum function has singularities that lie off of the real axis in the complex time plane. This, in turn, imposes a radius of convergence on the physical problem in real time. By choosing the expansion point at the top of the trajectory, the power series converges all the way to the bottom of the trajectory without being affected by these singularities. In constructing the series solution, we re-derive the coefficients using an alternative approach that generalizes to other nonlinear problems of mathematical physics. Additionally, we provide an exact resummation of the pendulum series -- Motivated by the asymptotic approximant method given in [Barlow et al., Q. J. Mech. Appl. Math., 70 (1) (2017), pp. 21-48] -- that accelerates the series convergence uniformly from the top to the bottom of the trajectory. We also provide an accelerated exact resummation of the infinite series representation for the elliptic integral used in calculating the period of a pendulums trajectory. This allows one to preserve analyticity in the use of the period to extend the pendulum series for all time via symmetry.
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