Hypergeometric solutions for third order linear ODEs

Hyp ergeometric solutions for third order linear ODEs E.S. Cheb-T errab a and A.D. Ro c he a,b a Maplesof t, Waterlo o Maple Inc. b Dep artment of Ma thematics Simon F r aser Un iv ersity, V anc ouver, Canada. Abstract In this pap er w e present a decision pro cedure for comput i ng p F q hypergeometric solutions for th ir d order linear OD Es , t hat is, solutions for the classes of hyp ergeome tric equations constructed from t he 3 F 2 , 2 F 2 , 1 F 2 and 0 F 2 standard equations using tran s formations 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 is present in Maple 12. In tro duction Given a third order linea r ODE y ′′′ + c 2 y ′′ + c 1 y ′ + c 0 y = 0 (1) where y ≡ y ( x ) is the dependent v ariable and the c j ≡ c j ( x ) a re any functions of x s uc h that the qua n tities 1 I 1 = c 2 ′ + c 2 2 3 − c 1 I 0 = c 2 ′′ 3 − 2 c 2 3 27 + c 1 c 2 3 − c 0 (2) are rationa l functions o f x , the pro blem under considera tion is that o f systematica lly computing solutions for (1) even when no Liouvillia n solutions exist 2 . Reca lling , Lio uvillian solutions can b e computed s y stematically [1] and implemen tations of the rela ted a lgorithm exist in v arious computer a lgebra systems. The linear ODEs involv ed in mathematical physics formulations, how ever, frequently admit only no n-Liouvillian sp ecial function solutions, a nd for this cas e the exis ting algor ithms co ver a rather restr icted po rtion of the pro ble m. The sp ecial functions asso ciated with linear ODEs frequen tly happen to be particular cases of some gen- eralized hyper g eometric p F q functions [2]. One na tural approach is thus to directly search for p F q solutions instead of sp ecial function so lutio ns of one o r a nother kind, and this is the approach discusse d here. Related computer algebra routines were implemented in 20 0 7 and ar e now at the ro ot of the Maple (release 12) [3] ability fo r solving non-trivial 3rd o rder linear ODE pro blems. The approa c h used co nsists of r esolving an eq uiv alence problem b et ween a given equation of the form (1) and the four standar d p F q differential equations asso ciated to third or der linea r ODEs, that is, the 3 F 2 , 2 F 2 , 1 F 2 and 0 F 2 equations [4], r espectively: 1 I 1 and I 0 are inv ariant under transform a tions of the dep e ndent v ariable of the form y ( x ) → P ( x ) y ( x ), P ar bitrary . 2 Expressions that can b e expressed in terms of exponen tials, i n tegrals and algebraic functions, are called Li ou vill ian. The t ypical exa mple i s exp( R R ( x ) , dx ) where R ( x ) is rational or an algebraic f unc tion represen ting the ro ots of a p olynomial. y ′′′ − ( δ + η + 1 − ( α + β + γ + 3) x ) x ( x − 1 ) y ′′ − ( η δ − (( β + γ + 1) α + ( β + 1) ( γ + 1)) x ) x 2 ( x − 1) y ′ + α β γ x 2 ( x − 1) y = 0 y ′′′ − ( x − γ − δ − 1) x y ′′ − (( α + β + 1) x − γ δ ) x 2 y ′ − α β x 2 y = 0 y ′′′ + ( β + γ + 1) x y ′′ − ( x − β γ ) x 2 y ′ − α x 2 y = 0 y ′′′ + ( α + β + 1) x y ′′ + α β x 2 y ′ − 1 x 2 y = 0 (3) where { α, β , γ , δ, η } r epresen t ar bitrary expressio ns c o nstan t with resp ect to x . The equiv alence classes are constructed by applying to these equations the gener al transformation 3 x → F ( x ) , y → P ( x ) y (4) where P ( x ) is arbitrary , with the o nly r estriction that F ( x ) is rational in x , resulting in rather general ODE families. When the equation being solved b elongs to this cla ss, apar t from providing the v a lues of F ( x ) and P ( x ) that reso lv e the pro blem, the algor ith m systematically returns the v alue s o f the (five, four, three or t wo) p F q parameters entering eac h of the three indep enden t solutions . It is imp ortant to no te that the ide a o f seeking hyp ergeometric function solutions for linear ODEs or using an equiv alence approach fo r that purpose is not new, a ltho ugh in most ca ses the approaches presented only handle second o r der linear equations [6, 7, 8, 9]. An exception to that situa tio n are the alg orithms [10, 11] for computing p F q solutions for third and higher order linear ODEs , and a similar one implemented in Mathematica [1 2]. It is o ur understanding , how ever, that the tr ansformations defining the classes of equiv ale nc e that those algorithms can ha ndle are restr icted to x → ax b , y → P ( x ) y , with a and b constan ts, not having the generality of (4) with r ational F ( x ) presented here. Apart fr o m concretely expanding the ability to solve third order linear ODEs, the decis ion pro cedure being presented generalizes previo us work in that: 1. The ide a s presented in [9], useful for decompos ing tw o sets of inv aria n ts into each other, were extended for third o rder equations and elab orated further. 2. The class ification ideas presented in [9] fo r second or der linear equations were extended for thir d order. 3. When the p F q parameters are such that less tha n three independent p F q solutions exis t, instead of int ro ducing integrals [11], MeijerG functions a re used to express the miss ing independent solutions. The combination o f items 1 a nd 2 re sulted in the new ability to solve the p F q ODE classes gener ated by transformatio ns as general as (4) with F ( x ) rational. Item 3 is not new 4 , though w e a re not aw are of literature presenting the related proble m and solution. Altogether, these ideas and its related algor ithm p ermit the systematic computation of three indep enden t so lutio ns for a large set of third order linear equations that w e didn’t know ho w to solve b efore. 1 Computing h yp ergeometric solutions T o co mput e p F q solutions to (1) the idea is to formulate an equiv alence approach to the underlying h yp er- geometric differen tial e q uations, that is, to determine whe ther a giv en linea r O DE can be obtained from one 3 The pr ob lem of equiv alence under transformations { x → F ( x ) , y → P ( x ) y + Q ( x ) } for linear O D Es can alwa ys be mapped int o one with Q ( x ) = 0, see [5]. 4 Mathematica 6 also uses Meij erG f unc tions as describ ed in i te m 3. 2 of the p F q ODEs (3) b y means of a transformation of a certain t yp e. If so , the solution to the given ODE is obtained by applying the sa me transformation to the solution o f the corres p onding p F q equation. The appr oac h also requir es deter mining the v alues of the hyperge ometric par ameters { α, β , γ , δ, η } for which the equiv alence exis ts, and it is clear that the b ottleneck in this approa c h is the g eneralit y of the class of transformations to be cons idered. F or instance, one can v erify that for linear transformations of the form (4) with arbitra ry F ( x ), in the ca se of seco nd order linear ODEs, the problem is to o gener al in that the de ter mination of F ( x ) requir es so lving the given ODE itself [13], making the approach o f no pr actical use. This has to do with the fact that in the second order case, any linear ODE can b e obtained from any other one through a transformation of the fo r m (4). The situation for third order equations is differen t: the transformatio n (4) is not enough to map any equation into a n y o ther one 5 [14], so that its determination when the equiv alence exists is in pr inciple p ossible. By restr icting the form o f F ( x ) entering (4) to b e rationa l in x the problem b ecomes tractable by using a tw o step s trategy: 1. Compute a ra tional transformation R ( x ) mapping the nor mal form of the given equation 6 int o o ne having invariants with m i nimal de gr e es (defined in sec. 3). 2. Reso lve an equiv alence problem b et w een this equation with minimal degr ees and the standa rd p F q equations (3) under transformatio ns 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 } cons tan ts with resp ect to x . In doing so, deter mine a lso the parameters { α, β , γ , δ, η } o f the p F q or MeijerG functions en tering the three indep enden t so lutions. The key obs e r v ation in this “tw o steps” appr oac h is that a transformation of the for m (4) with rational F ( x ) mapping into the p F q equations (3), when it exists, it ca n alwa ys be expre s sed as the comp osition o f tw o transformatio ns, e ac h one related to ea c h of the tw o steps ab o ve (see sec. 3), because (3) hav e in v ariants with minimal degrees. The adv antage of splitting the pro blem in this wa y is that the determina tio n of R ( x ) in step one, and o f the (up to fiv e) p F q parameters in step tw o, as well as of the v a lues of { a, b, c, d, k } entering (5), is sys tematic (see sec. 2 and sec. 3), even when the pro blem is nonlinear in ma ny v ar iables. 2 Equiv alence under x → ( a x k + b ) / ( c x k + d ) , y → P ( x ) y This type o f equiv alence is discuss e d in [9] a nd gener alized here for third order ODE s. Recalling the main po in ts, these transformations , which do not for m a group in the strict sense, can b e obtained b y sequentially comp osing three differen t transfor mations, each o f which do es cons titute a g roup. The s equence starts with linear fractio nal - also called M¨ obius - transfo rmations x → a x + b c x + d , (6) is followed by power tra nsformations x → x k , (7) and ends with linear homogene o us tra ns formations of the dependent v a riable y → P y . (8) 5 Therefore there exist enough absolute inv ariants to f orm ulate the equiv alence problem under (4) - s ee s ec. 3. 6 The coefficients of y ′ and y in the normalized equation are the i nv ari an ts I 1 and I 0 defined in (2), assumed to b e rational. 3 2.1 Equiv alence under transformations of the dep enden t v ariable y → P ( x ) y T r ansformations of the form (8) can eas ily be factored o ut of the problem: if tw o equations of the form (1) can b e o btained fro m each o ther by means of (8), the transformatio n relating them is c o mputable dir e c tly from these co efficients. F or that purp ose first r ewrite both e q uations in normal form using y → y e − R c 2 ( x ) / 3 dx (9) and the transforma tion r e la ting the tw o h yp othetical ODEs - s a y with coefficie nts c j and ˜ c k , when it exists, is given b y y → y e R ( c 2 ( x ) − ˜ c 2 ( x )) / 3 dx . 2.2 Equiv alence under M¨ obius transformations, singularities and classification M¨ obius transfor mations preserve the structure of the singular ities of (1). F or example, a ll of the 0 F 2 , 1 F 2 and 2 F 2 hypergeometr ic equations in (3) hav e one regular singularity a t the orig in and one irr egular singularity at infinity , a nd after tra nsforming them using the M¨ obius transfor mations (6), they co n tin ue having one regular singula rit y and one irr e gular singula rit y , no w resp ectively lo cated at 7 − b/a and − d/c . In the case of the 3 F 2 differential equation (the fir st lis ted in (3)), under (6) the three regular singula r ities mov e from { 0 , 1 , ∞} to {− b /a, − d/c, ( d − b ) / ( a − c ) } . So from the singularities of a n ODE, no t only one can tell with resp ect to whic h of the four differential equatio ns (3) could the equiv alence under (6) be reso lv ed, but also o ne can extract the v alues of the parameters { a, b, c, d } ent ering the transforma tion (6). More generally , through M¨ obius tra ns formations one can formulate a classifica tion of singularities of the linear ODEs “equiv alent” to the third order p F q equations (3) as do ne in [9] for second o rder p F q equations. So, for each p F q family obtained fr om (3) using (6), a classific a tion table can b e constructed based only on: • the degrees of the numerators and denomina to rs o f the in v ariants (2); • the presence of ro ots with multiplicit y in the denominators; • the poss ible c a ncellation of factors b et ween the n umerator and denominator of ea c h in v ariant. With this classificatio n in ha nds, fro m the knowledge o f the degrees with resp ect to x of the numerator and denominator o f the inv ariants (2) o f a given third order linea r ODE, one can deter mine systematically whether or not the equation co uld b e obtained from the 3 F 2 , 2 F 2 , 1 F 2 or 0 F 2 equations (3) using (6). 2.3 T ransformations x → F ( x ) and equiv alence under x → x k Changing x → F ( x ) in (1 ), the new inv a rian ts ˜ I j can b e expressed in terms o f the in v ariants (2 ) of (1) by ˜ I 1 ( x ) = F ′ 2 I 1 ( F ) − 2 S ( F ) ˜ I 0 ( x ) = F ′ F ′′ I 1 ( F ) + F ′ 3 I 0 ( F ) − S ( F ) ′ (10) where S ( F ) is the Sch warzian [15] S ( F ) = F ′′′ F ′ − 3 2  F ′′ F ′  2 . (11) The form o f S ( F ) is particular ly simple when F ( x ) is a M¨ obius transformation, in which case S ( F ) = 0. Regarding p ow er tra nsformations F ( x ) = x k , unlike M¨ obius tra nsformations, they do not pr eserve the structure of singularities; the Sch warzian (11) is: S ( x k ) = 1 − k 2 2 x 2 . (12) F r om (10) and (12), for insta nce the transfor mation rule fo r I 1 ( x ) b e c omes 7 When either a or c are equal to zero, the co rr e sp onding si ng ularity is lo cated at ∞ 4 x 2 ˜ I 1 ( x ) + 1 = k 2  ( x k ) 2 I 1 ( x k ) + 1  . (13) Generalizing to thir d order the presentation of shifted inv ariants in [9], we define here J 1 ( x ) = x 2 I 1 ( x ) + 1 , J 2 ( x ) = x 3 I 0 ( x ) + x 2 I 1 ( x ) . (14) F r om (10) rewritten in ter ms of these J n ( x ), their transformatio n rule under x → x k is given b y ˜ J 1 ( x ) = k 2 J 1 ( x k ) , ˜ J 2 ( x ) = k 3 J 2 ( x k ) . (15) The equiv alence of tw o linea r O DEs A a nd B under x → x k can then b e formulated as follows: Given the shifted inv ariants ˜ J n,A ( x ) and ˜ J n,B ( x ), co mput ed using their definition (14) in terms of ˜ I n ( x ) defined in (2), compute k A and k B ent ering (15) such that the degrees of J n,A ( x ) and J n,B ( x ) are minimal. F rom the knowledge of x → x k A and x → x k B , res pectively leading to J n,A and J n,B with minimized degrees, equations A and B are related thro ugh p o wer transformatio ns only when J n,A = J n,B and, if so , the mapping relating A and B is x → x k A − k B . Finally , the computation of k simultaneously minimizing the degrees of the tw o J n ( x ) in (15) is p erformed as explained in section 3 of [9]. 3 Mapping in to equations ha ving in v arian ts with minimal degrees The decis io n pro cedure presented in the previous s ection serves for sy s tematically solving well defined fa milies of p F q 3rd order equatio ns for whic h no solv ing algor ithm was av ailable before to the b est of o ur knowledge. How ever, the restriction in the form of F ( x ) entering (4) to the compos ition of M¨ obius with power transfor - mations is unsatisfactor y: fo r linear equations of order higher than t wo, (4) do es not map any linear equation int o any other o ne of the same or der and so the pr o blem is already restricted 8 . As s ho wn in what follows, one p ossible extensio n of the algor ithm is thus to consider the gener al trans- formations (4) restricting F ( x ) to b e a rationa l function o f x . F or that pur pose, instead of working with inv a rian ts I j under y → P ( x ) y we int ro duce absolute inv ariants L i under { x → F ( x ) , y → P ( x ) y } : L 1 = (6 rr ′′ + 9 I 1 r 2 − 7 r ′ 2 ) 3 r 8 , L 2 = (27 I 1 ′ r 3 − 18 I 1 r 2 r ′ + 56 r ′ 3 − 72 r ′′ r ′ r + 1 8 r ′′′ r 2 ) r 4 ; (16) where r = I 1 ′ − 2 I 0 is a re la tiv e inv ariant o f weigh t 3 [16]. Under (4), L i transforms as L i ( x ) → L i ( F ( x )) and (16) can be in verted using as intermediate v aria bles the relative in v ar ian ts s = ( L 2 L 1 ) /L 1 ′ , and t = L 1 /s 3 : I 1 = st 3 − 6 t ′′ t + 7 t ′ 2 9 t 2 , I 0 = ( s ′ − 9) t 4 + t ′ st 3 − 6 t ′′′ t 2 + 20 t ′′ t ′ t − 14 t ′ 3 18 t 3 . (17) Thu s, any ca nonical fo r m for the L i that can be a chieved using (4) automatica lly implies on a canonical form for the I i and so for the ODE (1 ). The canonical for m we prop ose here is one wher e the L i hav e minimal de gr e es , that is, where the maximu m of the degrees of the numerator and denomina to r in e ac h o f the L i ( x ) is the minima l one that can b e obtained using a ra tio nal transfor mation x → F ( x ). This canonica l form is not unique in that it is still p ossible to p e rform a M¨ obius transforma tion (6), that changes the L i but not their degrees. The equiv alence of tw o linea r O DEs A and B under (4) with rational F ( x ) ca n then be formulated by rewr iting b oth eq uations in this ca no nical form, wher e the inv ariants L i of each equatio n hav e minimal degrees, followed b y determining whether these canonical forms are related through a M¨ obius tr ansformation. 8 The equiv alence problem for linear equations of order n inv olves a system of n − 1 equations and inv ariant s I j ( x ), that includes the equiv alence function F ( x ). When n > 2, eliminating F ( x ) from the problem results in an inte rr e lation b et ween the I j so that the equiv alence is only poss i ble when these relationships b et we en the I j hold [14]. 5 In the framework of this pap er, B is one of the hype rgeometric eq ua tions (3), a ll o f them already in canonical form in that the corr esponding L i already have minimal degree s . Hence, the equiv alence o f A, of the fo rm (1), and a n y of B of the for m (3) r equires determining only a canonica l form for A (the rationa l function F ( x ) minimizing the deg r ees of the L i of A), follow ed by resolving an equiv alence under M¨ obius transformatio ns b et ween this canonical form a nd a n y of the eq uations (3), done as explained in sec. 2.2. The k ey computation in this formulation o f the equiv alence problem under (4) is thus the computatio n of a rational F ( x ) that minimizes the degre e s o f the L i of A. The computation of F ( x ) can clearly b e formulated as a rational function decomp osition pro blem sub ject to constraints: “given two r ational functions L i ( x ) , i = 1 .. 2 , find r ational functions ˜ L i ( x ) and F ( x ) satisfying L i = ˜ L i ◦ F and such that the r ational de gr e e of F is maximize d” (and therefor e the degrees of the canonical in v aria n ts ˜ L i are minimized). In turn, this t yp e of function decomp osition asso ciated to “minimizing the degrees” of the L i can b e interpreted as the repa rametrization, in terms of p olynomials o f low er degr ee, of a ra tional curve that is improp erly parameterize d, as disc us sed in [17], where an algorithm to p erform this repa rametrization is presented. One key feature of the algo r ithm presented in [17] is that it reduces the computation of F ( x ) to a sequence of univariate GCD computations, avoiding the exp ensiv e computation o f biv ariate GCD. Ho wev er, it is not clear for us whether the pr escriptions in [17] (at page 71) for mapping the biv aria te GCDs in to univ aria te ones is complete. W e als o failed in obtaining a copy of the computer algebra pack ages FRAC [18] o r Cade c om [19] that contain an implementation of the algor ithm presented in [1 7 ]. Mainly for these r easons, and without the inten tion o f be ing o riginal, we descr ibe her e a s lig h tly mo dified version o f the a lgorithm presented in [17]. 3.1 An algorithm for computing 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 hav e minimal degrees, N i is rela tiv ely prime to D i and p is r elativ ely prime to q . Construc t p olynomials 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 ) b e the biv ariate GCD of these Q i ( x, t ). Co nsequen tly 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 co efficien t P i ( x ) of e a c h p ower of t in P ( x, t ) is a linear combination of p ( x ) and q ( x ), and b ecause the q uotien t of any t wo relatively prime of these linear combinations is fractional linear in F ( x ), so is the quotient of any tw o relatively prime P i ( x ). Finally , b ecause F ( x ) is defined up to a M¨ obius transfor ma tion we ca n tak e that quotient itself - s a y , P i ( x ) /P j ( x ) - as the solution F ( x ). The slow est step of this a lgorithm is the co mput ation of the biv a riate GCD b et w een the Q i ( x, t ) that determines the function P ( x, t ) from which the P i ( x ) a r e computed. It is p ossible how ev er to av oid computing that biv ariate GCD, using a small num b er of univ ar iate GCD computations instead. F o r that purp ose, notice first that wha t is relev ant in the P i ( x ) is that they a re linear combinations of p ( x ) and q ( x ). Now, we can also obtain linear co m binations of p ( x ) and q ( x ) b y dir ectly substituting numerical v alues t k for t into P ( x, t ), and from there compute F ( x ) as the quotient , e.g., of P ( x, t 0 ) /P ( x, t 1 ). The key observ ation here is that these P ( x, t k ) can also b e obtained b y substituting t = t k directly into the Q i ( x, t ) follow ed by computing the univ a riate GCD of Q 1 ( x, t k ) a nd Q 2 ( x, t k ) 9 , av oiding in this way the computatio n of the e xpensive biv ariate GCD leading to P ( x, t ). Repe a ting this pro cess with another t -v alue gives a second, in general different, such linear combination of p ( x ) and q ( x ), with F b eing the resulting quotient of tw o of these linear combinations obta ined using different v alues of t . The re s t of the a lgorithm en tails avoiding inv alid t - v alues a t the time of s ubs tit uting t = t k and this is accomplished by considering different t k un til the following co nditions are bo th satisfied: 9 F or example, supp ose 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 ). Then eac h x = X j ( t 0 ) is a solution of both Q 1 ( x, t 0 ) = 0 and Q 2 ( x, t 0 ) = 0. F or most v alues of t 0 (all but a finite set in f ac t) these X j ( t 0 ) will b e the only such common s olut ions, and therefore the GCD of Q 1 ( x, t 0 ) and Q 2 ( x, t 0 ) is in fact P ( x, t 0 ). 6 1. The tw o P ( x, t 0 ), P ( x, t 1 ) whose quotient gives the solution F ( x ) must b e relatively pr ime. 2. The degr ee of F must divide the degrees of each L i , i = 1 ..n . 4 Summary of the p F q approac h for third order linear ODEs The idea consists of assuming that the given linear ODE is o ne of p F q equations (3) transformed using (4) for some F ( x ) r ational in x and P ( x ) arbitra ry and for some v alues of the pFq parameters. Reso lv ing the equiv ale nc e is ab out determining the F ( x ), P ( x ) a nd the v a lues o f the p F q parameters { α, β , γ , δ, η } such that the equiv alence exists. An itemized description of the decision pro cedure to r esolv e this equiv alence , following the presentation the previous sec tions, is as follows. 1. Rewrite the given equation (1) we w ant to solve, in normal form y ′′′ = ˜ I 1 ( x ) y ′ + ˜ I 0 ( x ) y (20) where the inv aria n ts ˜ I n ( x ) are cons tructed us ing the formulas (2). 2. V erify whether an equiv alence o f the form { x → ( a x k + b ) / ( c x k + d ) , y → P ( x ) y } exists: (a) Compute ˜ J n ( x ), the s hif ted in v a rian ts (14), a nd use trans formations x → x k to reduce to the int eger minimal v alues the p o wers ent ering the numerator and denominator; i.e., compute k and J n ( x ) in (15). (b) Determine the singula r ities of the J n ( x ) and use the cla s sification of singularities mentioned in section 2 to tell whether an equiv alence under M¨ obius transforma tio ns to any of the 3 F 2 , 2 F 2 , 1 F 2 or 0 F 2 equations (3) exists . (c) When the equiv alence exists, from the singularities of the t wo J n ( x ) compute the parameters { a, b, c, d } entering the M¨ obius tra nsformation (6) as well as the hypergeometr ic parameter s { α, β , γ , δ, η } entering the p F q equation (3). (d) Comp ose the three transforma tions to o bta in o ne o f the form x → αx k + β γ x k + δ , y → P ( x ) y mapping the p F q equation inv olved into the ODE being solved. 3. When the equiv alence of the pr evious s tep do es not exist, p erform step 1 in the itemizatio n of s e c tion 1 , that is, compute the absolute inv ariants L i (16) and co mpute a rational transfor mation R ( x ) minimizing the degre e s of the inv ariants (16 ) o f the given equa tion (a) When R ( x ) is not o f M¨ obius form, change x → R ( x ) rewriting the given equation in ca nonical form and r e-en ter step (2) with it, to r esolv e the r emaining M¨ obius tra nsformation a nd determining the v a lues of the p F q parameters . 4. When either of the equiv alences consider ed in steps (2) or (3) e x ist, compo se all the transformatio ns used and apply the comp osition to the known solution of the p F q equation to which the equiv alence was re s olv ed, obtaining the so lutio n to the given ODE. 7 5 Sp ecial cases and MeijerG functions Giving a lo ok at the s eries expa nsion of any of the 3 F 2 , 2 F 2 , 1 F 2 or 0 F 2 functions one can see that there are so me different situations that requir e sp ecial attention at the time o f constructing the three indep enden t solutions to (1). Co nsider for instance the standard 0 F 2 equation and its three indepe nden t solutions , y ′′′ + ( α + β + 1) x y ′′ + α β x 2 y ′ − 1 x 2 y = 0 y = 0 F 2 ( ; α, β ; x ) C 1 + x 1 − β 0 F 2 ( ; 2 − β , 1 + α − β ; x ) C 2 + x 1 − α 0 F 2 ( ; 2 − α, 1 − α + β ; x ) C 3 (21) where the C i are arbitra ry cons tan ts. Expanding in ser ies the first 0 F 2 function entering this so lution we g et 1 + 1 α β x + 1 2 α β ( α + 1 ) (1 + β ) x 2 + 1 6 α β ( α + 1 ) (1 + β ) ( α + 2) ( β + 2) x 3 + O  x 4  (22) This series do es not exist when α or β are zero or negative integers, and the same happ ens when the p F q parameters en tering a n y of the other tw o indep endent so lutions is a non-p ositiv e integer. B y insp ection, how ever, one of the three p F q functions entering the solution in (21) alwa ys exists, bec a use ther e ar e no α and β such that the three 0 F 2 functions simultaneously cont ain non- positive in teger parameter s. Consider now the second independent so lution, x 1 − β 0 F 2 ( ; 2 − β , 1 + α − β ; x ): w he n β = 1 it be comes equal to the first o ne a nd so we hav e only tw o independent p F q solutions. In the sa me way , when α = 1 the fir st and third solutio ns entering (21) are the sa me and when α = β the seco nd and third solutions are the s a me. And when the tw o co ndit ions hold, that is α = β = 1, actually the three solutions are the sa me. Notwit hstanding, in these cas es too one o f the three 0 F 2 solutions always exists . The same t wo type of sp ecial cases exist for the 1 F 2 , 2 F 2 and 3 F 2 function solutions and the problem at hand consists of having a wa y to represent the three indep endent solutions to (1) without introducing int egr als o r iter ating reductions of order 10 . F or this purp ose, we use a s et of 3 MeijerG functions for each of the four p F q families tha t can b e used to r eplace the missing p F q solutions in these sp ecial cases. The key o bs erv ation is that at these spe c ial v alues of the last tw o parameters of the p F q functions the MeijerG replacements exist, satisfy the same differe ntial equation and a re indepe nden t o f the av ailable p F q function solutions. A table with these 3 x 4 = 12 MeijerG function replacements is as follows: T a ble 1: MeijerG a lter nativ e solutions to the p F q equations p F q family MeijerG functions 0 F 2 ( ; α, β ; x ) G 2 , 0 0 , 3  x,    0 , 1 − α, 1 − β  G 3 , 0 0 , 3  − x,    0 , 1 − α, 1 − β  G 2 , 0 0 , 3  x,    1 − α, 1 − β , 0  1 F 2 ( α ; β , γ ; x ) G 2 , 1 1 , 3  x,    1 − α 0 , 1 − β , 1 − γ  G 3 , 1 1 , 3  − x,    1 − α 0 , 1 − γ , 1 − β  G 2 , 1 1 , 3  x,    1 − α 1 − γ , 1 − β , 0  2 F 2 ( α, β ; δ, γ ; x ) G 2 , 2 2 , 3  x,    1 − β , 1 − α 0 , 1 − γ , 1 − δ  G 3 , 2 2 , 3  − x,    1 − β , 1 − α 0 , 1 − γ , 1 − δ  G 2 , 2 2 , 3  x,    1 − β , 1 − α 1 − γ , 1 − δ, 0  3 F 2 ( α, β , γ ; δ, η ; x ) G 2 , 3 3 , 3  x,    1 − β , 1 − α, 1 − γ 0 , 1 − δ, 1 − η  G 3 , 3 3 , 3  − x,    1 − β , 1 − α, 1 − γ 0 , 1 − δ, 1 − η  G 2 , 3 3 , 3  x,    1 − β , 1 − α, 1 − γ 1 − δ, 1 − η, 0  6 Examples Equiv alence under p o w er comp osed with M¨ obius transformations for the 0 F 2 class Consider the third order linear ODE 10 Recall that giv en tw o indep e ndent solutions, it is alwa ys possible to write the third one in terms of inte grals constructed with the tw o existing solutions, and in the case of a si ng le solution it is still p ossible to reduce the order to a second order li ne ar equation that may or not b e solv able. 8 y ′′′ =  37 + 2 µ + 6 ν − 108 x 2  12 x ( x + 1) ( x − 1) y ′′ (23) +  2 ( ν + 6) (11 / 2 − µ ) + (36 ν + 294 + 12 µ ) x 2 − 360 x 4  24 x 2 ( x + 1) 2 ( x − 1) 2 y ′ − 16 x ( x + 1 ) 4 ( x − 1) 4 y This equa tion has tw o reg ular sing ularities at { 0 , ∞} and tw o irreg ular sing ularities at {− 1 , 1 } . F o llo wing the s teps men tioned in the Summary , we rewr ite the equation in nor mal for m and, in step 2.(a), compute the v alue of k leading to an equation with minimal deg rees en tering J n ( x ) in (15). The v alue of k found is k = 2 s o the equation from which (23) is derived changing x → x 2 is y ′′′ = (6 ν + 2 µ + 73 − 144 x ) 24 x ( x − 1) y ′′ (24) −  2 ( ν + 8) ( µ + 1 / 2) − (48 ν + 16 µ + 584) x + 576 x 2  96 x 2 ( x − 1) 2 y ′ − 2 x 2 ( x − 1) 4 y and ha s in v ariants with minimal degr ees with r e spect to p o wer transfor ma tions. In step 2.(b), analyzing the structur e of singularities of (2 4 ) we find one regular singular it y a t the origin and one irre gular at ∞ . Using the clas s ification discuss e d in section 3.2 based on the degr ees with resp ect to x of the numerators and denominators of the inv ariants of (24) a s well a s the factors entering these deno minators the equation is iden tified as eq uiv alent to the 0 F 2 class under M¨ obius transformations (6). So w e pro ceed with step 2.(c), constructing the M¨ obius transformatio n and computing the v alues of the hypergeometr ic para meters { µ, ν } ent ering the 0 F 2 equation in (3 ) suc h that the equiv alence under M¨ o bius exists, obtaining: α = ν / 4 + 2 , β = µ/ 12 + 1 / 24 , M := x → 2 x x − 1 (25) Comp osing M ab ov e with the p o wer transformation used to obtain (24) and using the v alues ab o ve for α and β , in step 4 we obtain the s olution of (23) y ( x ) = 0 F 2 ( ; ν / 4 + 2 , µ/ 1 2 + 1 / 24 ; 2 x 2 x 2 − 1 ) C 1 + x − (2+ ν / 2)  x 2 − 1  (1+ ν / 4) 0 F 2 ( ; − ν / 4 , µ/ 12 − ν / 4 − 23 / 2 4; 2 x 2 x 2 − 1 ) C 2 (26) + x (23 / 12 − µ/ 6)  x 2 − 1  ( µ/ 12 − 23 / 24) 0 F 2 ( ; 47 / 24 − µ/ 12 , 7 1 / 2 4 − µ/ 1 2 + ν / 4; 2 x 2 x 2 − 1 ) C 3 Meijerg functions and equi v alence under rational transformations for the 1 F 2 class Consider the following equation, with no symbolic para meter s and only integer powers y ′′′ = −  6 + 12 x − 15 x 2 − 6 x 3  x (1 + x − x 2 ) ( x + 2 ) y ′′ (27) +  16 + 48 x + 36 x 2 − 20 x 3 + 9 x 4 + 81 x 5 − 20 x 6 − 30 x 7 − 6 x 8  x 4 ( x + 2) 2 (1 + x − x 2 ) 2 y ′ − ( x + 2) 3 (1 + x − x 2 ) 2 x 5 y 9 F o llo wing steps 1 a nd 2 in the Summary , we co nfirm that there exists no equiv alence under (5), so in step 3 we se arc h for a r ational transformation minimizing the deg rees of the inv ariants (1 6), finding R ( x ) = x 2 / (1 + x ) (28) Therefore (27 ) ca n b e obtained by changing v ariables x → R ( x ) in y ′′′ = −  3 − 9 x + 6 x 2  x ( x − 1) 2 y ′′ +  1 − 2 x + 6 x 2 − 6 x 3  x 3 ( x − 1) 2 y ′ − 1 ( x − 1) 2 x 4 y (29) This equation 11 th us has in v ariants with minimal degrees, and has one regula r singula rit y at 1 and one irregula r at the orig in. According to the classification in terms of singularities (29) a dmits an equiv a lence under M¨ obius transformations to the p F q equations ( 1 F 2 case) and hence is solved in the iteration step 3.(a) men tioned in the s umm ar y . Wh en constructing the p F q solutions to (29), how ever, we find tha t the 1 F 2 parameters in the second list are b oth equal to 1, so only o ne 1 F 2 solution is av aila ble, and he nce tw o of the MeijerG alter nativ e solutions presented in the table (5) are necessa ry , resulting in y = 0 F 1 ( ; 1 ; 1 + x − x 2 x 2 ) C 1 + G 2 , 0 0 , 2  1 + x − x 2 x 2 ,    0 , 0  C 2 + G 3 , 1 1 , 3  x 2 − x − 1 x 2 ,    0 0 , 0 , 0  C 3 (30) Note that the first p F q function is a 0 F 1 . This is due to the automatic s implification of or der that ha ppens when identical parameters ar e present in b oth lists of a 1 F 2 function; this 0 F 1 can also b e express ed in ter ms of Bess e l functions. Conclusions In this w ork we pr esen ted a decis ion pr ocedure for third order linear ODEs for computing thr ee indep enden t solutions even when they are not Liouvillian or when the hyper geometric par ameters inv olved ar e such that only tw o or one p F q solution aro und the o rigin exis ts. This alg orithm so lves complete ODE fa milies we didn’t know ho w to solve befo re. The stra tegy used is that o f res o lving an equiv alence problem to the 3 F 2 , 2 F 2 , 1 F 2 and 0 F 2 equations, and in doing so, tw o impor tan t generaliza tions of the alg orithm prese n ted in [9] were dev elop ed. First, the classificatio n acco rding to s ingularities and the use of p o wer comp osed with M¨ obius transfor mations, presented in [9 ] for 2 nd order e q uations, was gener alized for third order ones. Second, the idea of resolving the equiv ale nc e mapping in to an equation with inv aria n ts with “ minimal degr ees under p ow er transformations” was g eneralized by determining a transfor mation mapping into an equation having inv aria n ts with “minimal degrees under general ra tional transfor mations”. This p ermits res olving a muc h la r ger clas s of p F q equations, defined by changing v ariables in (3) using { x → R ( x ) , y → P ( x ) y } wher e R ( x ) is a rational function. Symbolic computation routines implemen ting this a lgorithm were integrated into the Ma ple system in 20 07. Since a t the c ore of the algor ithm b eing presented there is the concept of singularities , t wo natural extensions of this work consist of applying the same ideas to compute solutions fo r linear ODEs o f arbitrar y order, where the equiv alence can be solved exactly [14], and for second order equatio ns under ra tional transformatio ns, p erhaps ge ner alizing the w ork by M.Bronstein [8] with regards to 1 F 1 solutions to compute also 2 F 1 solutions. Related work is in prog r ess. References [1] M. v a n Ho eij, J. F. Rag ot, F. Ulmer and J. A. W eil. “Lio uvillian so lutio ns o f linea r differential e q uations of order thre e a nd hig her”. J. Sym b. Comp., 28(4- 5):589–609, 1999 . [2] Seab orn J.B., “Hyp ergeometric F unctio ns and Their Applications”, T ext in Applied Mathematics, 8, Springer-V erlag (1991). 11 The c i en tering (29) are computed fr om the minim ized L j b y inv erting (2) and using (17). 10 [3] Maples o ft, a division of W aterlo o Maple Inc., “Ma ple 12” (2008). [4] M. Abramowitz and I. A. Stegun, “Ha ndbo ok of mathematical functions” , Dover (1964). [5] E.L. Inc e , “Ordina r y Differential Equations”, Dov er Publications (1 9 56). [6] N. Kamra n and P .J. Olver. “Equiv alence of Differential Op erators.” SIAM J. Math. Anal. 2 0, no. 5, 1172 (198 9). [7] B. Willis, “An extensible different ial equa tion solv er for computer algebra”, SIGSAM, Mar c h (2001). [8] M. Brons tein and S. La faille, “Solutions of linear ordinary differe ntial equations in terms of spe c ial functions”, Pro ceedings of ISSAC’02, Lille, ACM Press, 23-28 (200 2). [9] L.Chan and E.S. Cheb-T errab, “Non-Liouvillian solutions for s e c ond order linear ODEs”, Pro ceedings of ISSA C’04 , Sa n tander, Spain, ACM Press, 80 -86 (20 0 4). [10] M. Petk ovsek and B. Salvy . “Finding All Hyper geometric So lutions of Linear Differential Equa tions.” Pro ceedings of ISSA C ’9 3, Edited b y M. Bronstein. A CM Press, 27-33 (19 93). [11] G.La bahn, “ Recognizing MeijerG ODEs for hig he r or der ”, glabahn@ uwaterloo.ca , unpublished work (2001). [12] W olfram Resea rc h, Inc., “Mathematica” , V ersion 6.0 .1 (200 7). [13] K . von B ¨ ulow, “Eq uiv alence metho ds for second order linear differential equations”, M.Sc. Thesis, F a cult y of Mathematics, University o f W aterlo o (2000 ). [14] E .S. Cheb- T errab, “ ODE trends in co mput er algebra : four linear and nonlinear challenges”, pr o ceedings of the Ma ple Summer W o rkshop, W ater loo, Canada (2 0 02). [15] E .W. W eisstein, “ Concise Encyclop edia o f Mathematics”, second edition, CRC P r ess (1 999). [16] E .J. Wilczynski, “Inv aria n ts of Systems of Linear Different ial Eq ua tions”, T ransa ctions of the America n Mathematical So ciet y , V o l. 2, No. 1 ., 1-24 (1901). [17] T.W. Sederb erg, “Improp erly pa rameterized r a tional cur v es”, C o mputer Aided Geometric Design 3 , 67-75 (1986). [18] C. Alonso , J. Gutierrez, T. Recio, “ FRA C: A Maple pack age for computing in the rational function field K(x)”, Pro ceedings of the Maple Summer W o rkshop and Symp osium, T r o y , New Y ork (1994). [19] J . Gutierrez, R. Rubio, “ Cadecom: Co mput er Algebra soft ware for functional DECOMp osition”, Com- puter Algebra in Scien tific C o mputing CASC’0 0, Samark and, Spring er-V erla g, 233- 248 (200 0). 11