Computing Hypergeometric Solutions of Second Order Linear Differential Equations using Quotients of Formal Solutions

Abstract

Let L be a second order differential equation with coefficients in C(x). The goal of this paper is to find solutions of L in the formwhere r, f ∈ Q(x), and a1, a2, b1 ∈ Q.

The art of algorithmic guessing in gfun

“…For example, highly efficient algorithms for computing analytic continuation and evaluation of D-finite functions, based on ideas of the Chudnovsky brothers [CC90] were created by van der Hoeven [vdH99, vdH01, vdH07] and Mezzarobba [Mez10]. Notably, as we will see later, very useful in practice is the work by van Hoeij and collaborators [KvH13,vHV15,IvH15] on explicit solutions of differential equations in terms of known special functions. The famous method of creative telescoping propagated by Zeilberger [Zei91], and constantly improved in the last decades, allows for finding and proving linear recurrences/differential equations for sequences given as explicit sums or functions given as integrals; see for example [Chy14] for a great exposition of many achievements in this field and [CK17] for open problems.…”

Computing hypergeometric solutions of second order linear differential equations using quotients of formal solutions and integral bases

“…Section 3.1 of Imamoglu and van Hoeij (2015) gives an a priori bound for d f , d f ≤ 6(n true − 2), logarithmic case, 36 n true − 7 3 , non-logarithmic case.…”

“…The parts Q1,...,Q5 are already in our ISSAC 2015 paper Imamoglu and van Hoeij (2015). Algorithm 4.1, which finds solutions of form (2), is new compared to Imamoglu and van Hoeij (2015). So Q6 is the main new part in this paper.…”

Some open problems related to creative telescoping

“…In recent years there has been some activity by van Hoeij and collaborators concerning solutions of recurrences or differential equations in terms of hypergeometric series [93,23,22,67,57]. In a way, these algorithms solve only special cases of the inverse problem for creative telescoping, thus indicating perhaps that the general problem may be very difficult.…”