Hypergeometric solutions for third order linear ODEs
aa r X i v : . [ m a t h . C A ] A p r Hypergeometric solutionsfor third order linear ODEs
E.S. Cheb-Terrab a and A.D. Roche a,b a Maplesoft, Waterloo Maple Inc. b Department of MathematicsSimon Fraser University, Vancouver, Canada.
Abstract
In this paper we present a decision procedure for computing p F q hypergeometric solutions for thirdorder linear ODEs, that is, solutions for the classes of hypergeometric equations constructed from the F , F , F and F standard equations using transformations of the form x → F ( x ) , y → P ( x ) y ,where F ( x ) is rational in x and P ( x ) is arbitrary. A computer algebra implementation of this work ispresent in Maple 12. Introduction
Given a third order linear ODE y ′′′ + c y ′′ + c y ′ + c y = 0 (1)where y ≡ y ( x ) is the dependent variable and the c j ≡ c j ( x ) are any functions of x such that the quantities I = c ′ + c − c I = c ′′ − c
27 + c c − c (2)are rational functions of x , the problem under consideration is that of systematically computing solutions for(1) even when no Liouvillian solutions exist . Recalling, Liouvillian solutions can be computed systematically[1] and implementations of the related algorithm exist in various computer algebra systems. The linearODEs involved in mathematical physics formulations, however, frequently admit only non-Liouvillian specialfunction solutions, and for this case the existing algorithms cover a rather restricted portion of the problem.The special functions associated with linear ODEs frequently happen to be particular cases of some gen-eralized hypergeometric p F q functions [2]. One natural approach is thus to directly search for p F q solutionsinstead of special function solutions of one or another kind, and this is the approach discussed here. Relatedcomputer algebra routines were implemented in 2007 and are now at the root of the Maple (release 12) [3]ability for solving non-trivial 3rd order linear ODE problems.The approach used consists of resolving an equivalence problem between a given equation of the form(1) and the four standard p F q differential equations associated to third order linear ODEs, that is, the F , F , F and F equations [4], respectively: I and I are invariant under transformations of the dependent variable of the form y ( x ) → P ( x ) y ( x ), P arbitrary. Expressions that can be expressed in terms of exponentials, integrals and algebraic functions, are called Liouvillian. Thetypical example is exp( R R ( x ) , dx ) where R ( x ) is rational or an algebraic function representing the roots of a polynomial. ′′′ − ( δ + η + 1 − ( α + β + γ + 3) x ) x ( x − y ′′ − ( η δ − (( β + γ + 1) α + ( β + 1) ( γ + 1)) x ) x ( x − y ′ + α β γx ( x − y = 0 y ′′′ − ( x − γ − δ − x y ′′ − (( α + β + 1) x − γ δ ) x y ′ − α βx y = 0 y ′′′ + ( β + γ + 1) x y ′′ − ( x − β γ ) x y ′ − αx y = 0 y ′′′ + ( α + β + 1) x y ′′ + α βx y ′ − x y = 0 (3)where { α, β, γ, δ, η } represent arbitrary expressions constant with respect to x . The equivalence classes areconstructed by applying to these equations the general transformation x → F ( x ) , y → P ( x ) y (4)where P ( x ) is arbitrary, with the only restriction that F ( x ) is rational in x , resulting in rather general ODEfamilies. When the equation being solved belongs to this class, apart from providing the values of F ( x ) and P ( x ) that resolve the problem, the algorithm systematically returns the values of the (five, four, three ortwo) p F q parameters entering each of the three independent solutions.It is important to note that the idea of seeking hypergeometric function solutions for linear ODEs orusing an equivalence approach for that purpose is not new, although in most cases the approaches presentedonly handle second order linear equations [6, 7, 8, 9]. An exception to that situation are the algorithms[10, 11] for computing p F q solutions for third and higher order linear ODEs, and a similar one implementedin Mathematica [12]. It is our understanding, however, that the transformations defining the classes ofequivalence that those algorithms can handle are restricted to x → ax b , y → P ( x ) y , with a and b constants,not having the generality of (4) with rational F ( x ) presented here.Apart from concretely expanding the ability to solve third order linear ODEs, the decision procedurebeing presented generalizes previous work in that:1. The ideas presented in [9], useful for decomposing two sets of invariants into each other, were extendedfor third order equations and elaborated further.2. The classification ideas presented in [9] for second order linear equations were extended for third order.3. When the p F q parameters are such that less than three independent p F q solutions exist, instead ofintroducing integrals [11], MeijerG functions are used to express the missing independent solutions.The combination of items 1 and 2 resulted in the new ability to solve the p F q ODE classes generated bytransformations as general as (4) with F ( x ) rational. Item 3 is not new , though we are not aware of literaturepresenting the related problem and solution. Altogether, these ideas and its related algorithm permit thesystematic computation of three independent solutions for a large set of third order linear equations that wedidn’t know how to solve before. To compute p F q solutions to (1) the idea is to formulate an equivalence approach to the underlying hyper-geometric differential equations, that is, to determine whether a given linear ODE can be obtained from one The problem of equivalence under transformations { x → F ( x ) , y → P ( x ) y + Q ( x ) } for linear ODEs can always be mappedinto one with Q ( x ) = 0, see [5]. Mathematica 6 also uses MeijerG functions as described in item 3.
2f the p F q ODEs (3) by means of a transformation of a certain type. If so, the solution to the given ODE isobtained by applying the same transformation to the solution of the corresponding p F q equation.The approach also requires determining the values of the hypergeometric parameters { α, β, γ, δ, η } forwhich the equivalence exists, and it is clear that the bottleneck in this approach is the generality of theclass of transformations to be considered. For instance, one can verify that for linear transformations of theform (4) with arbitrary F ( x ), in the case of second order linear ODEs, the problem is too general in thatthe determination of F ( x ) requires solving the given ODE itself [13], making the approach of no practicaluse. This has to do with the fact that in the second order case, any linear ODE can be obtained from anyother one through a transformation of the form (4). The situation for third order equations is different: thetransformation (4) is not enough to map any equation into any other one [14], so that its determinationwhen the equivalence exists is in principle possible. By restricting the form of F ( x ) entering (4) to be rationalin x the problem becomes tractable by using a two step strategy:1. Compute a rational transformation R ( x ) mapping the normal form of the given equation into onehaving invariants with minimal degrees (defined in sec. 3).2. Resolve an equivalence problem between this equation with minimal degrees and the standard p F q equations (3) under transformations of the form discussed in [9], that is x → ( a x k + b )( c x k + d ) , y → P ( x ) y (5)with P ( x ) arbitrary and { a, b, c, d, k } constants with respect to x . In doing so, determine also theparameters { α, β, γ, δ, η } of the p F q or MeijerG functions entering the three independent solutions.The key observation in this “two steps” approach is that a transformation of the form (4) with rational F ( x )mapping into the p F q equations (3), when it exists, it can always be expressed as the composition of twotransformations, each one related to each of the two steps above (see sec. 3), because (3) have invariants withminimal degrees. The advantage of splitting the problem in this way is that the determination of R ( x ) instep one, and of the (up to five) p F q parameters in step two, as well as of the values of { a, b, c, d, k } entering(5), is systematic (see sec. 2 and sec. 3), even when the problem is nonlinear in many variables. x → ( a x k + b ) / ( c x k + d ) , y → P ( x ) y This type of equivalence is discussed in [9] and generalized here for third order ODEs. Recalling the mainpoints, these transformations, which do not form a group in the strict sense, can be obtained by sequentiallycomposing three different transformations, each of which does constitute a group. The sequence starts withlinear fractional - also called M¨obius - transformations x → a x + bc x + d , (6)is followed by power transformations x → x k , (7)and ends with linear homogeneous transformations of the dependent variable y → P y. (8) Therefore there exist enough absolute invariants to formulate the equivalence problem under (4) - see sec. 3. The coefficients of y ′ and y in the normalized equation are the invariants I and I defined in (2), assumed to be rational. .1 Equivalence under transformations of the dependent variable y → P ( x ) y Transformations of the form (8) can easily be factored out of the problem: if two equations of the form (1)can be obtained from each other by means of (8), the transformation relating them is computable directlyfrom these coefficients. For that purpose first rewrite both equations in normal form using y → y e − R c ( x ) / dx (9)and the transformation relating the two hypothetical ODEs - say with coefficients c j and ˜ c k , when it exists,is given by y → y e R ( c ( x ) − ˜ c ( x )) / dx . M¨obius transformations preserve the structure of the singularities of (1). For example, all of the F , F and F hypergeometric equations in (3) have one regular singularity at the origin and one irregular singularityat infinity, and after transforming them using the M¨obius transformations (6), they continue having oneregular singularity and one irregular singularity, now respectively located at − b/a and − d/c .In the case of the F differential equation (the first listed in (3)), under (6) the three regular singularitiesmove from { , , ∞} to {− b/a, − d/c, ( d − b ) / ( a − c ) } . So from the singularities of an ODE, not only one cantell with respect to which of the four differential equations (3) could the equivalence under (6) be resolved,but also one can extract the values of the parameters { a, b, c, d } entering the transformation (6).More generally, through M¨obius transformations one can formulate a classification of singularities of thelinear ODEs “equivalent” to the third order p F q equations (3) as done in [9] for second order p F q equations.So, for each p F q family obtained from (3) using (6), a classification table can be constructed based only on: • the degrees of the numerators and denominators of the invariants (2); • the presence of roots with multiplicity in the denominators; • the possible cancellation of factors between the numerator and denominator of each invariant.With this classification in hands, from the knowledge of the degrees with respect to x of the numeratorand denominator of the invariants (2) of a given third order linear ODE, one can determine systematicallywhether or not the equation could be obtained from the F , F , F or F equations (3) using (6). x → F ( x ) and equivalence under x → x k Changing x → F ( x ) in (1), the new invariants ˜ I j can be expressed in terms of the invariants (2) of (1) by˜ I ( x ) = F ′ I ( F ) − S ( F )˜ I ( x ) = F ′ F ′′ I ( F ) + F ′ I ( F ) − S ( F ) ′ (10)where S ( F ) is the Schwarzian [15] S ( F ) = F ′′′ F ′ − (cid:18) F ′′ F ′ (cid:19) . (11)The form of S ( F ) is particularly simple when F ( x ) is a M¨obius transformation, in which case S ( F ) = 0.Regarding power transformations F ( x ) = x k , unlike M¨obius transformations, they do not preserve thestructure of singularities; the Schwarzian (11) is: S ( x k ) = 1 − k x . (12)From (10) and (12), for instance the transformation rule for I ( x ) becomes When either a or c are equal to zero, the corresponding singularity is located at ∞ ˜ I ( x ) + 1 = k (cid:0) ( x k ) I ( x k ) + 1 (cid:1) . (13)Generalizing to third order the presentation of shifted invariants in [9], we define here J ( x ) = x I ( x ) + 1 ,J ( x ) = x I ( x ) + x I ( x ) . (14)From (10) rewritten in terms of these J n ( x ), their transformation rule under x → x k is given by˜ J ( x ) = k J ( x k ) , ˜ J ( x ) = k J ( x k ) . (15)The equivalence of two linear ODEs A and B under x → x k can then be formulated as follows: Giventhe shifted invariants ˜ J n,A ( x ) and ˜ J n,B ( x ), computed using their definition (14) in terms of ˜ I n ( x ) definedin (2), compute k A and k B entering (15) such that the degrees of J n,A ( x ) and J n,B ( x ) are minimal. Fromthe knowledge of x → x k A and x → x k B , respectively leading to J n,A and J n,B with minimized degrees,equations A and B are related through power transformations only when J n,A = J n,B and, if so, the mappingrelating A and B is x → x k A − k B . Finally, the computation of k simultaneously minimizing the degrees ofthe two J n ( x ) in (15) is performed as explained in section 3 of [9]. The decision procedure presented in the previous section serves for systematically solving well defined familiesof p F q F ( x ) entering (4) to the composition of M¨obius with power transfor-mations is unsatisfactory: for linear equations of order higher than two, (4) does not map any linear equationinto any other one of the same order and so the problem is already restricted .As shown in what follows, one possible extension of the algorithm is thus to consider the general trans-formations (4) restricting F ( x ) to be a rational function of x . For that purpose, instead of working withinvariants I j under y → P ( x ) y we introduce absolute invariants L i under { x → F ( x ) , y → P ( x ) y } : L = (6 rr ′′ + 9 I r − r ′ ) r , L = (27 I ′ r − I r r ′ + 56 r ′ − r ′′ r ′ r + 18 r ′′′ r ) r ; (16)where r = I ′ − I is a relative invariant of weight 3 [16]. Under (4), L i transforms as L i ( x ) → L i ( F ( x )) and(16) can be inverted using as intermediate variables the relative invariants s = ( L L ) /L ′ , and t = L /s : I = st − t ′′ t + 7 t ′ t , I = ( s ′ − t + t ′ st − t ′′′ t + 20 t ′′ t ′ t − t ′ t . (17)Thus, any canonical form for the L i that can be achieved using (4) automatically implies on a canonicalform for the I i and so for the ODE (1). The canonical form we propose here is one where the L i have minimal degrees , that is, where the maximum of the degrees of the numerator and denominator in each ofthe L i ( x ) is the minimal one that can be obtained using a rational transformation x → F ( x ). This canonicalform is not unique in that it is still possible to perform a M¨obius transformation (6), that changes the L i but not their degrees.The equivalence of two linear ODEs A and B under (4) with rational F ( x ) can then be formulatedby rewriting both equations in this canonical form, where the invariants L i of each equation have minimaldegrees, followed by determining whether these canonical forms are related through a M¨obius transformation. The equivalence problem for linear equations of order n involves a system of n − I j ( x ), thatincludes the equivalence function F ( x ). When n >
2, eliminating F ( x ) from the problem results in an interrelation betweenthe I j so that the equivalence is only possible when these relationships between the I j hold [14].
5n the framework of this paper, B is one of the hypergeometric equations (3), all of them already incanonical form in that the corresponding L i already have minimal degrees. Hence, the equivalence of A, ofthe form (1), and any of B of the form (3) requires determining only a canonical form for A (the rationalfunction F ( x ) minimizing the degrees of the L i of A), followed by resolving an equivalence under M¨obiustransformations between this canonical form and any of the equations (3), done as explained in sec. 2.2.The key computation in this formulation of the equivalence problem under (4) is thus the computationof a rational F ( x ) that minimizes the degrees of the L i of A. The computation of F ( x ) can clearly beformulated as a rational function decomposition problem subject to constraints: “given two rational functions L i ( x ) , i = 1 .. , find rational functions ˜ L i ( x ) and F ( x ) satisfying L i = ˜ L i ◦ F and such that the rationaldegree of F is maximized” (and therefore the degrees of the canonical invariants ˜ L i are minimized). In turn,this type of function decomposition associated to “minimizing the degrees” of the L i can be interpretedas the reparametrization, in terms of polynomials of lower degree, of a rational curve that is improperlyparameterized, as discussed in [17], where an algorithm to perform this reparametrization is presented.One key feature of the algorithm presented in [17] is that it reduces the computation of F ( x ) to a sequenceof univariate GCD computations, avoiding the expensive computation of bivariate GCD. However, it is notclear for us whether the prescriptions in [17] (at page 71) for mapping the bivariate GCDs into univariateones is complete. We also failed in obtaining a copy of the computer algebra packages
FRAC [18] or
Cadecom [19] that contain an implementation of the algorithm presented in [17]. Mainly for these reasons, and withoutthe intention of being original, we describe here a slightly modified version of the algorithm presented in [17]. x → F ( x ) minimizing the degrees of the L i ( x ) Let L i ( x ) = ˜ L i ( F ( x )) = N i ( x ) /D i ( x ) , i = 1 ..n , and F ( x ) = p ( x ) /q ( x ), where the ˜ L i have minimal degrees, N i is relatively prime to D i and p is relatively prime to q . Construct polynomials Q i ( x, t ) = numerator( L i ( x ) − L i ( t )) = N i ( x ) D i ( t ) − N i ( t ) D i ( x ) , (18)and let P ( x, t ) be the bivariate GCD of these Q i ( x, t ). Consequently P ( x, t ) = numerator( F ( x ) − F ( t )) = X i P i ( x ) t i = p ( x ) q ( t ) − p ( t ) q ( x ) , (19)The coefficient P i ( x ) of each power of t in P ( x, t ) is a linear combination of p ( x ) and q ( x ), and becausethe quotient of any two relatively prime of these linear combinations is fractional linear in F ( x ), so is thequotient of any two relatively prime P i ( x ). Finally, because F ( x ) is defined up to a M¨obius transformationwe can take that quotient itself - say, P i ( x ) /P j ( x ) - as the solution F ( x ).The slowest step of this algorithm is the computation of the bivariate GCD between the Q i ( x, t ) thatdetermines the function P ( x, t ) from which the P i ( x ) are computed. It is possible however to avoid computingthat bivariate GCD, using a small number of univariate GCD computations instead.For that purpose, notice first that what is relevant in the P i ( x ) is that they are linear combinations of p ( x )and q ( x ). Now, we can also obtain linear combinations of p ( x ) and q ( x ) by directly substituting numericalvalues t k for t into P ( x, t ), and from there compute F ( x ) as the quotient, e.g., of P ( x, t ) /P ( x, t ). The keyobservation here is that these P ( x, t k ) can also be obtained by substituting t = t k directly into the Q i ( x, t )followed by computing the univariate GCD of Q ( x, t k ) and Q ( x, t k ) , avoiding in this way the computationof the expensive bivariate GCD leading to P ( x, t ).Repeating this process with another t -value gives a second, in general different, such linear combinationof p ( x ) and q ( x ), with F being the resulting quotient of two of these linear combinations obtained usingdifferent values of t . The rest of the algorithm entails avoiding invalid t -values at the time of substituting t = t k and this is accomplished by considering different t k until the following conditions are both satisfied: For example, suppose the x -solutions of P ( x, t ) = 0 are x = X j ( t ) , j = 1 ..m , i.e., P ( x, t ) = − P m ( t ) Q j x − X j ( t ). Theneach x = X j ( t ) is a solution of both Q ( x, t ) = 0 and Q ( x, t ) = 0. For most values of t (all but a finite set in fact) these X j ( t ) will be the only such common solutions, and therefore the GCD of Q ( x, t ) and Q ( x, t ) is in fact P ( x, t ).
6. The two P ( x, t ), P ( x, t ) whose quotient gives the solution F ( x ) must be relatively prime.2. The degree of F must divide the degrees of each L i , i = 1 ..n . p F q approach for third order linear ODEs The idea consists of assuming that the given linear ODE is one of p F q equations (3) transformed using (4)for some F ( x ) rational in x and P ( x ) arbitrary and for some values of the pFq parameters. Resolving theequivalence is about determining the F ( x ), P ( x ) and the values of the p F q parameters { α, β, γ, δ, η } suchthat the equivalence exists. An itemized description of the decision procedure to resolve this equivalence,following the presentation the previous sections, is as follows.1. Rewrite the given equation (1) we want to solve, in normal form y ′′′ = ˜ I ( x ) y ′ + ˜ I ( x ) y (20)where the invariants ˜ I n ( x ) are constructed using the formulas (2).2. Verify whether an equivalence of the form { x → ( a x k + b ) / ( c x k + d ) , y → P ( x ) y } exists:(a) Compute ˜ J n ( x ), the shifted invariants (14), and use transformations x → x k to reduce to theinteger minimal values the powers entering the numerator and denominator; i.e., compute k and J n ( x ) in (15).(b) Determine the singularities of the J n ( x ) and use the classification of singularities mentioned insection 2 to tell whether an equivalence under M¨obius transformations to any of the F , F , F or F equations (3) exists.(c) When the equivalence exists, from the singularities of the two J n ( x ) compute the parameters { a, b, c, d } entering the M¨obius transformation (6) as well as the hypergeometric parameters { α, β, γ, δ, η } entering the p F q equation (3).(d) Compose the three transformations to obtain one of the form x → αx k + βγx k + δ , y → P ( x ) y mapping the p F q equation involved into the ODE being solved.3. When the equivalence of the previous step does not exist, perform step 1 in the itemization of section 1,that is, compute the absolute invariants L i (16) and compute a rational transformation R ( x ) minimizingthe degrees of the invariants (16) of the given equation(a) When R ( x ) is not of M¨obius form, change x → R ( x ) rewriting the given equation in canonical formand re-enter step (2) with it, to resolve the remaining M¨obius transformation and determiningthe values of the p F q parameters.4. When either of the equivalences considered in steps (2) or (3) exist, compose all the transformationsused and apply the composition to the known solution of the p F q equation to which the equivalencewas resolved, obtaining the solution to the given ODE.7 Special cases and MeijerG functions
Giving a look at the series expansion of any of the F , F , F or F functions one can see that thereare some different situations that require special attention at the time of constructing the three independentsolutions to (1). Consider for instance the standard F equation and its three independent solutions, y ′′′ + ( α + β + 1) x y ′′ + α βx y ′ − x y = 0 y = F ( ; α, β ; x ) C + x − β F ( ; 2 − β, α − β ; x ) C + x − α F ( ; 2 − α, − α + β ; x ) C (21)where the C i are arbitrary constants. Expanding in series the first F function entering this solution we get1 + 1 α β x + 12 α β ( α + 1) (1 + β ) x + 16 α β ( α + 1) (1 + β ) ( α + 2) ( β + 2) x + O (cid:0) x (cid:1) (22)This series does not exist when α or β are zero or negative integers, and the same happens when the p F q parameters entering any of the other two independent solutions is a non-positive integer. By inspection,however, one of the three p F q functions entering the solution in (21) always exists, because there are no α and β such that the three F functions simultaneously contain non-positive integer parameters.Consider now the second independent solution, x − β F ( ; 2 − β, α − β ; x ): when β = 1 it becomesequal to the first one and so we have only two independent p F q solutions. In the same way, when α = 1the first and third solutions entering (21) are the same and when α = β the second and third solutions arethe same. And when the two conditions hold, that is α = β = 1, actually the three solutions are the same.Notwithstanding, in these cases too one of the three F solutions always exists.The same two type of special cases exist for the F , F and F function solutions and the problemat hand consists of having a way to represent the three independent solutions to (1) without introducingintegrals or iterating reductions of order . For this purpose, we use a set of 3 MeijerG functions for eachof the four p F q families that can be used to replace the missing p F q solutions in these special cases. Thekey observation is that at these special values of the last two parameters of the p F q functions the MeijerGreplacements exist, satisfy the same differential equation and are independent of the available p F q functionsolutions. A table with these 3 x 4 = 12 MeijerG function replacements is as follows:Table 1: MeijerG alternative solutions to the p F q equations p F q family MeijerG functions F ( ; α, β ; x ) G , , (cid:16) x, (cid:12)(cid:12)(cid:12) , − α, − β (cid:17) G , , (cid:16) − x, (cid:12)(cid:12)(cid:12) , − α, − β (cid:17) G , , (cid:16) x, (cid:12)(cid:12)(cid:12) − α, − β, (cid:17) F ( α ; β, γ ; x ) G , , (cid:16) x, (cid:12)(cid:12)(cid:12) − α , − β, − γ (cid:17) G , , (cid:16) − x, (cid:12)(cid:12)(cid:12) − α , − γ, − β (cid:17) G , , (cid:16) x, (cid:12)(cid:12)(cid:12) − α − γ, − β, (cid:17) F ( α, β ; δ, γ ; x ) G , , (cid:16) x, (cid:12)(cid:12)(cid:12) − β, − α , − γ, − δ (cid:17) G , , (cid:16) − x, (cid:12)(cid:12)(cid:12) − β, − α , − γ, − δ (cid:17) G , , (cid:16) x, (cid:12)(cid:12)(cid:12) − β, − α − γ, − δ, (cid:17) F ( α, β, γ ; δ, η ; x ) G , , (cid:16) x, (cid:12)(cid:12)(cid:12) − β, − α, − γ , − δ, − η (cid:17) G , , (cid:16) − x, (cid:12)(cid:12)(cid:12) − β, − α, − γ , − δ, − η (cid:17) G , , (cid:16) x, (cid:12)(cid:12)(cid:12) − β, − α, − γ − δ, − η, (cid:17) Equivalence under power composed with M¨obius transformations for the F class Consider the third order linear ODE Recall that given two independent solutions, it is always possible to write the third one in terms of integrals constructedwith the two existing solutions, and in the case of a single solution it is still possible to reduce the order to a second order linearequation that may or not be solvable. ′′′ = (cid:0)
37 + 2 µ + 6 ν − x (cid:1) x ( x + 1) ( x − y ′′ (23)+ (cid:0) ν + 6) (11 / − µ ) + (36 ν + 294 + 12 µ ) x − x (cid:1) x ( x + 1) ( x − y ′ − x ( x + 1) ( x − y This equation has two regular singularities at { , ∞} and two irregular singularities at {− , } . Followingthe steps mentioned in the Summary, we rewrite the equation in normal form and, in step 2.(a), computethe value of k leading to an equation with minimal degrees entering J n ( x ) in (15). The value of k found is k = 2 so the equation from which (23) is derived changing x → x is y ′′′ = (6 ν + 2 µ + 73 − x )24 x ( x − y ′′ (24) − (cid:0) ν + 8) ( µ + 1 / − (48 ν + 16 µ + 584) x + 576 x (cid:1) x ( x − y ′ − x ( x − y and has invariants with minimal degrees with respect to power transformations. In step 2.(b), analyzingthe structure of singularities of (24) we find one regular singularity at the origin and one irregular at ∞ .Using the classification discussed in section 3.2 based on the degrees with respect to x of the numeratorsand denominators of the invariants of (24) as well as the factors entering these denominators the equationis identified as equivalent to the F class under M¨obius transformations (6). So we proceed with step 2.(c),constructing the M¨obius transformation and computing the values of the hypergeometric parameters { µ, ν } entering the F equation in (3) such that the equivalence under M¨obius exists, obtaining: α = ν/ , β = µ/
12 + 1 / , M := x → xx − M above with the power transformation used to obtain (24) and using the values above for α and β , in step 4 we obtain the solution of (23) y ( x ) = F ( ; ν/ , µ/
12 + 1 /
24; 2 x x − C + x − (2+ ν/ (cid:0) x − (cid:1) (1+ ν/
4) 0 F ( ; − ν/ , µ/ − ν/ − /
24; 2 x x − C (26)+ x (23 / − µ/ (cid:0) x − (cid:1) ( µ/ − /
24) 0 F ( ; 47 / − µ/ , / − µ/
12 + ν/
4; 2 x x − C Meijerg functions and equivalence under rational transformations for the F class Consider the following equation, with no symbolic parameters and only integer powers y ′′′ = − (cid:0) x − x − x (cid:1) x (1 + x − x ) ( x + 2) y ′′ (27)+ (cid:0)
16 + 48 x + 36 x − x + 9 x + 81 x − x − x − x (cid:1) x ( x + 2) (1 + x − x ) y ′ − ( x + 2) (1 + x − x ) x y R ( x ) = x / (1 + x ) (28)Therefore (27) can be obtained by changing variables x → R ( x ) in y ′′′ = − (cid:0) − x + 6 x (cid:1) x ( x − y ′′ + (cid:0) − x + 6 x − x (cid:1) x ( x − y ′ − x − x y (29)This equation thus has invariants with minimal degrees, and has one regular singularity at 1 and oneirregular at the origin. According to the classification in terms of singularities (29) admits an equivalenceunder M¨obius transformations to the p F q equations ( F case) and hence is solved in the iteration step 3.(a)mentioned in the summary. When constructing the p F q solutions to (29), however, we find that the F parameters in the second list are both equal to 1, so only one F solution is available, and hence two of theMeijerG alternative solutions presented in the table (5) are necessary, resulting in y = F ( ; 1; 1 + x − x x ) C + G , , (cid:18) x − x x , (cid:12)(cid:12)(cid:12) , (cid:19) C + G , , (cid:18) x − x − x , (cid:12)(cid:12)(cid:12) , , (cid:19) C (30)Note that the first p F q function is a F . This is due to the automatic simplification of order that happenswhen identical parameters are present in both lists of a F function; this F can also be expressed in termsof Bessel functions. Conclusions
In this work we presented a decision procedure for third order linear ODEs for computing three independentsolutions even when they are not Liouvillian or when the hypergeometric parameters involved are such thatonly two or one p F q solution around the origin exists. This algorithm solves complete ODE families we didn’tknow how to solve before.The strategy used is that of resolving an equivalence problem to the F , F , F and F equations,and in doing so, two important generalizations of the algorithm presented in [9] were developed. First,the classification according to singularities and the use of power composed with M¨obius transformations,presented in [9] for 2nd order equations, was generalized for third order ones. Second, the idea of resolving theequivalence mapping into an equation with invariants with “minimal degrees under power transformations”was generalized by determining a transformation mapping into an equation having invariants with “minimaldegrees under general rational transformations”. This permits resolving a much larger class of p F q equations,defined by changing variables in (3) using { x → R ( x ) , y → P ( x ) y } where R ( x ) is a rational function.Symbolic computation routines implementing this algorithm were integrated into the Maple system in 2007.Since at the core of the algorithm being presented there is the concept of singularities, two naturalextensions of this work consist of applying the same ideas to compute solutions for linear ODEs of arbitraryorder, where the equivalence can be solved exactly [14], and for second order equations under rationaltransformations, perhaps generalizing the work by M.Bronstein [8] with regards to F solutions to computealso F solutions. Related work is in progress. References [1] M. van Hoeij, J. F. Ragot, F. Ulmer and J. A. Weil. “Liouvillian solutions of linear differential equationsof order three and higher”. J. Symb. Comp., 28(4-5):589–609, 1999.[2] Seaborn J.B., “Hypergeometric Functions and Their Applications”, Text in Applied Mathematics, 8,Springer-Verlag (1991). The c i entering (29) are computed from the minimized L j by inverting (2) and using (17).by inverting (2) and using (17).