Second order differential equations with hypergeometric solutions of degree three

Abstract

Let L be a second order linear homogeneous differential equation with rational function coefficients. The goal in this paper is to solve L in terms of hypergeometric function 2F1(a, b; c | f ) where f is a rational function of degree 3.

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

“…The maximum number of disappeared singularities for [ 1 2 , 1 3 , 1 7 ] is not ( 1 2 + 1 3 + 1 7 )d f because that contradicts the formula (7). The maximum number consistent with (7) is…”

“…general than in prior work. Less general in the sense that papers [2], [7] considered 3 transformations instead of the 2 in section 2.3 and more general in the sense that prior work was restricted to either a specific number of singularities (4 in [9] and 5 in [6]) or specific degrees (degree 3 in [7] and a degree-2 decomposition in [2]). Moreover, our program can also find algebraic functions f in (1) (although at the moment this requires additional user inputs).…”

2 F 1 -type Solutions of Second Order Differential Equations

“…1 Parameter f of the change of variables transformation. − −− →G is called gauge transformation which is allowed in [1] but not in the Special Case.…”

“…Compared with [1], our algorithm covers only the special case, but it can handle rational functions f of higher degrees. Our future research question is the following: Can the general case be reduced to the special case?…”

“…We are interested in finding 2 F 1 -type solutions of second order differential equations. Paper [1] gives an algorithm to find 2 F 1 -type solutions of second order differential equations when the degree of the pullback function 1 of f is three. In our algorithm, the pullback function f is of arbitrary degree (however, we restrict to a special case defined below).…”