Solutions of linear ordinary differential equations in terms of special functions

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1 Solutions of linear ordinary differential equations in terms of special functions Manuel Bronstein INRIA Projet Café 004, Route des Lucioles, BP 93 F-0690 Sophia Antipolis Cedex, France Sébastien Lafaille ABSTRACT We describe a new algorithm for computing special function solutions of the form yx) = mx)f x)) of second order linear ordinary differential equations, where mx) is an arbitrary Liouvillian function, x) is an arbitrary rational function, and F satisfies a given second order linear ordinary differential equation Our algorithm, which is based on finding an appropriate point transformation between the equation defining F and the one to solve, is able to find all rational transformations for a large class of functions F, in particular but not only) the 0F and F special functions of mathematical physics, such as Airy, Bessel, Kummer and Whittaker functions It is also able to identify the values of the parameters entering those special functions, and can be generalized to equations of higher order INTRODUCTION Algorithms and software for computing closed form solutions of linear ordinary differential equations have improved significantly in the past decade, but mostly in the direction of computing their Liouvillian solutions see eg [, 8, 9]) In particular, computing the Liouvillian solutions of second order linear ordinary differential equations has become a routine task in recent versions of several computer algebra systems The situation is different with respect to solving such equations in terms of non Liouvillian special functions While it is possible to detect whether the solutions of an equation can be expressed in terms of the solutions of equations of second order [7], there is no complete algorithm for deciding whether such solutions can be expressed in terms of the solutions of specific equations, usually the ones defining known special functions This is a restricted instance of the equivalence problem for secondorder linear ODEs [3]: given a target equation y = uy with u Cx) and a known fundamental solution set {F, F } for example the Airy equation y = xy), and an arbitrary input equation y = vy with v Cx), to find functions mx) and x) such that {mx)f x)), mx)f x))} is a fundamental solution set of y = vy This is the equivalent to looking for a point transformation of the form x x) y mx)y ) that transforms y = uy into y = vy It is classically known that all second-order linear ODEs are equivalent under the group of transformations of the form ), hence that an appropriate transformation always exists [4] However, the functions x) and mx) are given implicitely by differential equations themselves, so this does not provide explicit solutions in terms of F We are interested in this paper in determining whether an explicit transformation of the form ) exists, with Cx) and m a Liouvillian function, and to compute it when it exists Applying the transformation ) to y = uy and matching the coefficients of the resulting equation with y = vy or equivalently, substituting y = mx)f x)) in y = vy) one obtains the equations / and 3 + 4u)4 4v = 0, ) so the remaining problem is to solve the above equation explicitely Methods using that approach have appeared, in particular [0], who proceeds heuristically by trying various candidates functions with undetermined constants parameters in ) Each attempt yields systems of algebraic equations for the undetermined constants and parameters of the special functions), and those equations can then be solved by existing computer algebra systems Our main contribution in this paper is an algorithm for computing all the solutions Cx) of ) Our algorithm is applicable whenever the target equation y = uy has an irregular singularity at infinity, in addition to any number of affine singularities of arbitrary type This allows our algorithm to handle the 0F and F special functions of mathematical physics eg the Airy, Bessel, Kummer and Whittaker functions) as well as non-hypergeometric ones We also show that if the input equation has no Liouvillian solution, then our algorithm decides whether there is any solution of the form mx)f x)) for F any solution of the target equation Our algorithm has been implemented in the computer algebra system Maple and our implementation can be tried interactively on the web While the abilities cathode/kovacic_demohtml

2 of the Maple 7 differential equations solver have also been improved regarding solutions in terms of special functions our algorithm is able to solve a larger class of examples, eg 4x ) 8 d y dx = 3 50x+6x 60x 3 +45x 4 8x 5 +3x 6 )yx), whose solutions can be expressed in terms of Airy functions with rational functions as arguments see examples below) We would like to thank the referees for their numerous comments, in particular for pointing out the link with the equivalence problem FORMAL CHANGE OF VARIABLE The differential equations for and m that result from having ) map a given operator to another given one can always be obtained by substituting y = mx)f x)) in the corresponding differential equation, and this is a classic construction We describe it in this section using differential polynomials and linear algebra, in a way that is easily performed in a computer algebra system for linear operators of arbitrary order Let k, ) be a differential field, k[d; ] be the ring of differential operators with coefficients in k, and L = D n + n i=0 aidi k[d; ] be an operator of order n > 0 Let M, Z be differential indeterminates over k, G 0,, G n be algebraic indeterminates over k M, Z and extend the derivation to k {M, Z} [G 0,, G n ] via G i = Z G i+ for 0 i < n and G n = Z n i=0 aigi Let y = MG0 y is a linear form in G 0,, G n and preserves the total degree in G 0,, G n, the successive derivatives of y are all linear forms in G 0,, G n, so y, y,, y n) are linearly dependent over k M, Z for i < n, G i appears with the nonzero coefficient MZ i in y i) but does not appear in y i ), the elements y, y,, y n ) must be linearly independent over k M, Z, so there is a unique linear dependence of the form y n) + n i=0 biyi) = 0, which can be computed by linear algebra over k M, Z Define then n L M,Z = D n + b id i k M, Z [D; ] i=0 to be the generic M Z associate of L Given a differential extension K of k and any m, K such that m 0, we can specialize L M,Z at M = m and Z =, and we denote the resulting operator L m, If k contains an element x such that x = and if the elements of k can be viewed as functions 3 in x, then for any f k, we write f) for the result of evaluating f at x = Replacing each a i by a i) in L m,, we obtain a new operator, which we denote L x,y my By construction, it has the following property: if Ly) = 0 for some y in a differential extension of k, then L x,y my my)) = 0 So if F,, F n is a fundamental solution set of L, then mf ),, mf n) are solutions of See eg where the candidate = ax k + b)/cx k + d) is tried 3 This is obviously the case when k = Cx) for some constant field C, and Seidenberg s Embedding Theorem [5, 6] implies that is is also the case when k is a finitely generated differential extension of Qx) L x,y my, WrmF ),, mf n)) = m n N WrF,, F n)) for some integer N > 0, it follows that mf ),, mf n) is a fundamental solution set of L x,y my in other words, the transformation ) sends L into L x,y my ) Let now R = D n + n i=0 cidi k[d; ] be another operator and suppose that there exist m, in a differential extension of k such that m 0 and mf ),, mf n) are solutions of R Then, mf ),, mf n) is a fundamental solution set of both R and of L x,y my they are both monic and of order n, we must have R = L x,y my Equating the coefficients of the same powers of D in R and L x,y my yield a system of n nonlinear ordinary differential equations that m and must satisfy Finding a fundamental solution set of the form mf ),, mf n) of R is thus reduced to solving those equations We can also ask a weaker question, namely does R admit some solution of the form mf ) where F is a nonzero solution of L and m 0 In that case, we can only say that R and L x,y my have a nontrivial right factor in k m, [D; ], so we cannot generate equations for m and However, if we request in addition that R be irreducible in k m, [D; ], then the existence of such a solution implies that R = L x,y my, hence that m and satisfy the n equations generated In particular, a second order equation with no Liouvillian solution over k must be irreducible over any Liouvillian extension of k, so if such an equation has a solution of the form mf ) with m 0 and m and Liouvillian over k, then R = L x,y my 3 SECOND ORDER EQUATIONS We carry out explicitly in this section the derivation of the above nonlinear differential equations in the case of secondorder operators Computing the generic M Z associate of L = D + a D + a 0 k[d; ], we get and y = MG 0, y = M G 0 + MG 0 = M G 0 + MZ G, y = M G 0 + M G 0 + M Z G + MZ G + MZ G = M G 0 + M Z + MZ )G MZ a 0G 0 + a G ) = M a 0MZ )G 0 + M Z + MZ a MZ )G A calculation of the linear dependence between y, y and y shows that L M,Z = D M ) M + Z a Z D 3) Z ) M M M M M ) Z + a Z M M Z M a0z Let now v k be given As explained in the previous section, if there are m and in a differential extension of k such that m 0 and either mf ) and mf ) are solutions of y = vy, where and F, F is a fundamental solution set of L, or mf ) is a solution of y = vy, where F is some solution of L, m and are Liouvillian over k and y = vy has no Liouvillian solution,

3 then D v = L x,y my Using 3) and equating the coefficients of D and D 0 on both sides, we get and m m m m + a ) = 0 4) ) m m m + a ) m m m a0) = v 5) Equation 4) implies that m ) a ) and using that to eliminate m /m from 5) we obtain 3 + a ) + a ) 4a 0) ) 4 4v = 0, 7) which is equation ) when a = 0 and a 0 = u 4 RATIONAL SOLUTIONS FOR We now proceed to show that for a large class of target operators L, there is an algorithm for computing all the rational solutions of 7) Suppose from now on that our differential field k is a rational function field k = Cx) where x = and c = 0 for all c C Recall that the order at is the function ν q) = degq) for q C[x] \ {0}, and given an irreducible p C[x], the order at p is the function ν pq) = max{n Z such that p n q} for q C[x] \ {0} Both functions are extended to fractions via ν a/b) = ν a) ν b) and ν pa/b) = ν pa) ν pb) By convention, ν 0) = ν p0) = + Furthermore, for a, b Cx), they satisfy the following properties where ν stands for either ν or ν p): νab) = νa) + νb), νa + b) minνa), νb)) νa) νb) = νa + b) = minνa), νb)), νa) < 0 = νba)) = ν b)νa), ν a) < 0 = ν a ) = ν a) +, ν pa) < 0 = ν pa ) = ν pa) Given an hypothesis on the pair a 0, a ), the following gives an ansatz with a finite number of undetermined constants for the rational solutions of 7) Theorem Let i Qi i be the squarefree decomposition of the denominator of v Cx) If ν a + a 4a 0) <, then any solution Cx) of 7) can be written as = P/Q where 6) Q i ν a +a 4a 0))i+ C[x], 8) and P C[x] is such that either degp ) degq) + or degp ) = degq) + ν v) ν a + a 4a0) 9) Proof Write = a + a 4a 0, δ = ν ) and suppose that δ < The solution = 0 can certainly be written in the above form, so let Cx) be a nonzero solution of 7), and p C[x] be an irreducible such that ν p) < 0 Then, ν p ) = ν p ) = ν p) 4 and ν p4 ) = 4ν p) 4 In addition, so ν pa ) + a ) 4a 0)) = ν p )) = δν p), ν p a) + a ) 4a 0))4) = 4 δ)ν p) 4 δ <, 4 δ)ν p) 4 < ν p) 4, so ν p 3 + a ) + a ) 4a 0))4) = 4 δ)ν p) 4 Thus, we must have ν p4v ) = 4 δ)ν p) 4 ν p4v ) = ν pv) + ν p), we get ν pv) = δ)ν p) 3 This implies that the affine poles of are among the poles of v of multiplicity 3 or more Furthermore, ν p) = νpv) + δ so must be of the form = P/Q where P C[x] and Q i δ)i+ 0) Suppose now that degp ) > degq)+ Then, ν ) <, so ν 4 ) = 4ν ) + 4 and ν a ) 4a 0)) = ν )) = δν ), which implies that ν a) + a ) 4a 0))4) = 4 δ)ν ) + 4 In addition, ν ) = ν ) + 4 and either ν ) = ν ) + 4 when ν ) <, or ν ) when ν ) = δ <, 4 δ)ν ) + 4 < ν ) + 4, and 4 δ)ν ) + 4 = δ 4 < when ν ) =, so ν 3 + a ) + a ) 4a 0))4) = 4 δ)ν ) + 4 in any case We must then have ν 4v ) = 4 δ)ν ) + 4 ν 4v ) = ν v) + ν ) +, we get and the theorem follows ν v) = δ)ν ) + ) We note that the upper bound degp ) degq) + can be improved when δ < 0 In that case, if degp ) = degq) +, then ν ) =, so an argument similar to the above shows that ν 3 + a ) + a ) 4a 0))4) = δ < 0 We must then have ν 4v ) = δ, so ν v) = δ and 9) holds Therefore, when ν a + a 4a 0) < 0, either degp ) degq) or degp ) is given by 9)

4 When it is applicable, Theorem yields an immediate algorithm for computing all the solutions Cx) of 7) given v Cx) as input: we substitute n j=0 cjxj /Q for in 7), where Q is given by 8), n is the upper bound on degp ) given by Theorem and the c j are undetermined constants This yields a nonlinear system Σ of algebraic equations for the c j, whose solutions correspond to all the solutions Cx) of 7) any constant satisfies 7), Σ always has the line of solutions c 0,, c n) = λq 0,, q n) where Q = q 0 + q x + + q nx n note that n is always at least degq)) Those solutions do not satisfy the condition m 0, so we adjoin to Σ the additional equation n q jc N q N c j)w j = ) j=0 where w 0,, w n are new indeterminates and N is chosen such that q N 0 Any solution of this augmented system must satisfy q jc N q N c j for some j, which implies that the corresponding Cx) is a nonconstant solution of 7) In addition, when a 0 and a contain parameters as in the case of families of special functions, eg Bessel functions), considering them as unknowns in Σ allows the values of those parameters to be found also this is illustrated in the examples below) Essentially all the computation time of our algorithm is spent finding a solution of Σ, a problem whose complexity is exponential in degp ) Our approach can obviously be used to find all the rational solutions Cx) of 7), it just means searching for solutions of Σ in C rather than C Of more interest, it can also be used to find some algebraic function solutions of 7) Indeed, equations 0) and ) provide the ramifications of at the singularities of the equation and at infinity, so it is natural to look for solutions of the form = P x / ν)) i> Q i )/ ν) i 3) where ν = ν a + a 4a 0) and P C[x] To bound degp ), we note that ) is valid for ν ) only, so either or degp ) < ν a + a 4a 0))degQ) + ) degp ) = ν a + a 4a 0)) degq) + ν v) As for rational functions, substituting a candidate with undetermined constant coefficients for yields a nonlinear algebraic system for those coefficients This method does not yield all the algebraic functions solutions of 7) however Once a nonconstant solution is found rational or otherwise), the corresponding m is given by 6), which can be integrated yielding e a ) 4) 5 CLASSICAL SPECIAL FUNCTIONS We now apply the algorithm of the previous section to classical classes of 0F and F special functions, all satisfying the hypothesis of Theorem Although Kummer and Whittaker functions are rationally equivalent, we explicit the solving algorithm for both of them, allowing users to choose one over the other 5 Airy functions The operator defining the Airy functions is L = D x, so a = 0, a 0 = x and equation 7) becomes v = 0 5) a + a 4a 0 = 4x, ν a + a 4a 0) = < 0, so by Theorem and the remark following it, any solution of 5) must be of the form = P/Q where Q i 3i+ C[x], and P C[x] is such that either degp ) degq) or degp ) = degq) + ν v) 3 Finally, since a = 0, equation 4) becomes 6) 5 Bessel functions The operator defining the Bessel and modified Bessel functions is L = D + x D + ɛ ν x where ɛ = for the Bessel functions and ɛ = for the modified Bessel functions Therefore, a = /x and a 0 = ɛ ν /x, so equation 7) becomes 3 + 4ν ) 4 4ɛ4 4v = 0 7) a + a 4a 0 = 4ν x 4ɛ, ν a + a 4a 0) = 0 <, so by Theorem, any solution of 7) must be of the form = P/Q where Q i i+ C[x], and P C[x] is such that either degp ) degq) + or degp ) = degq) + ν v) Finally, since a = /x, equation 4) becomes 8) 53 Kummer functions The operator defining the Kummer functions is L = D ν ) + x D µ x, so a = ν/x, a 0 = µ/x and equation 7) becomes 3 + ν ν) µ ν) + 4 4v = 0 9) a + a 4a 0 = + 4µ ν x + ν ν x,

5 ν a + a 4a 0) = 0 <, so by Theorem, any solution of 9) must be of the form = P/Q where Q i i+ C[x], and P C[x] is such that either degp ) degq) + or degp ) = degq) + ν v) Finally, since a = ν/x, equation 4) becomes e ν 54 Whittaker functions The operator defining the Whittaker functions is L = D 4 µ ) /4 ν, x x so a = 0, a 0 = /4+µ/x+/4 ν )/x and equation 7) becomes ) 3 4µ 4ν + 4 4v = 0 0) a + a 4a 0 = 4µ x 4ν x, ν a + a 4a 0) = 0 <, so by Theorem, any solution of 7) must be of the form = P/Q where Q i i+ C[x], and P C[x] is such that either degp ) degq) + or degp ) = degq) + ν v) Finally, since a = 0, equation 4) becomes ) as in the case of Airy functions 6 EXAMPLES 6 Airy functions We start by solving the equation given at the end of the introduction in terms of Airy functions The equation is y = vy with v = 3 50x + 6x 60x x 4 8x 5 + 3x 6 4x ) 8, so ν v) = and its denominator is 4x ) 8 Therefore, any solution Cx) of 7) must be of the form = P x ) 3 where P C[x] is of degree 0,, or 3 Substituting = c 0 + c x + c x + c 3x 3 )/x ) 3 in 5) yields a system of 4 algebraic equations The nonconstant condition ) becomes 3c 0 + c )w + 3c 0 + c )w + c 0 + c 3)w 3 = 0 and solving the resulting system for c 0, c, c, c 3, w, w and w 3 yields the 3 solutions = xx ) and = ± 3 ) xx ) x ) x ) Using 6) we compute = cx )3/ for some constant c Therefore, a basis of the solutions of y = vy is given by ) ) xx ) xx ) x ) 3/ Ai and x ) 3/ Bi x ) x ) where Ai and Bi are Airy functions 6 Bessel functions We now look for solutions in terms of modified Bessel functions of y v 0 + v x) n y = 0 where n > 0 and v 0 ) Letting v = v 0 + v x) n, ν v) = n and its denominator is, so any solution Cx) of 7) must be a polynomial of degree 0,, or + n/ Substituting = c 0 + c x in 7) yields c 4 + 4ɛc 0 4ν ) + 8ɛc 0c 5 x + 4ɛc 6 x c 0 + c x) = 4c v 0 + v x) n, whose only solution for n > 0 and v 0 is c = 0 Therefore, any nonconstant solution must be a polynomial of degree exactly +n/, which implies that there can be such solutions only when n is even We proceed with n = 4, which is the smallest even value for which Maple 7 is unable to solve the above equation Substituting = c 0 + c x + c x + c 3x 3 in 7) yields a system of 5 algebraic equations The nonconstant condition ) becomes c w + c w + c 3w 3 + = 0 and solving the resulting system for c 0, c, c, c 3, w, w, w 3 and ν with parameters v 0 and v and ɛ = yields the 4 solutions ν = ± 6, = ± v 0 + v x) 3 3 v Using 8) we compute = c v 0 + v x for some constant c Therefore, a basis of the solutions of ) for n = 4 is given by ) v 0 + v x) 3 v0 + v x I /6 3 and v ) v 0 + v x) 3 v0 + v x K /6 3 where I ν and K ν are the modified Bessel functions of the first and second kinds v

6 63 Whittaker functions For an example with two parameters to identify, we look for solutions in terms of Whittaker functions of y + ax 4 + bx)y = 0 where a 0, 3) which is Kamke s example 6 [] with a specific integer choice for c Letting v = ax 4 + bx), ν v) = 4 and its denominator is, so any solution Cx) of 0) must be a polynomial of degree 0,, or 3 Substituting = c 0 + c x in 0) yields 4ac 4 x 6 + lower terms = 0, which implies c = 0 whenever a 0 Therefore, any nonconstant solution must be a polynomial of degree exactly 3 Substituting = c 0+c x+c x +c 3x 3 in 0) yields a system of 5 algebraic equations The nonconstant condition ) becomes c w + c w + c 3w 3 + = 0 and solving the resulting system for c 0, c, c, c 3, w, w, w 3, ν and µ with parameters a and b yields the solutions µ = b, ν = ± 6 a 6, = 3 x3 a Using ) we compute = c x for some constant c Therefore, a basis of the solutions of 3) is given by ) x M b 6 a, 6 3 x3 a and ) x W b 6 a, 6 3 x3 a where M µ,ν and W µ,ν are Whittaker functions 64 An algebraic transformation We illustrate the use of the algebraic candidate 3) by solving the Airy equation y = xy in terms of modified Bessel functions, thereby recovering classical expressions of Airy functions as Bessel functions Letting v = x, ν v) = and its denominator is, so any solution Cx) of 7) must be a polynomial of degree 0 or Substituting = c 0 + c x in 7) yields 4c 4 x 3 + lower terms = 0, which implies c = 0, hence that 7) has no nonconstant rational solution However, formula 3) yields the algebraic candidate = P x) where P is a polynomial of degree 0,, or 3 Substituting = c 0+c x / +c x+c 3x 3/ in 7) yields a system of 7 algebraic equations The nonconstant condition ) becomes c w + c w + +c 3w 3 + = 0 and solving the resulting system for c 0, c, c, c 3, w, w, w 3 and ν with ɛ = yields the 4 solutions for some constant c Therefore, a basis of the solutions of the Airy equation y = xy is given by ) ) x I/3 3 x3/ and x K/3 3 x3/ where I ν and K ν are the modified Bessel functions of the first and second kinds It follows that the Airy functions Ai and Bi can be expressed as linear combinations ) )) x c I /3 3 x3/ + c K /3 3 x3/ and the constants c and c can be found by looking at their values at two points 7 REFERENCES [] Bronstein, M Computer algebra algorithms for linear ordinary differential and difference equations In Proceedings of the third European congress of mathematics, vol II 00), vol 0 of Progress in Mathematics, Birkhäuser, Basel, pp 05 9 [] Kamke, E Differentialgleichungen: Lösungsmethoden und Lösungen Akademische Verlagsgesellschaft, Leipzig, 956 [3] Kamran, N, and Olver, P Equivalence of differential operators SIAM Journal of Mathematical Analysis 0 Sept 989), 7 85 [4] Schlesinger, L Handbuch der Theorie der linearen Differentialgleichungen Teubner, Leipzig, 895 [5] Seidenberg, A Abstract differential algebra and the analytic case Proceedings of the American Mathematical Society 9 958), [6] Seidenberg, A Abstract differential algebra and the analytic case II Proceedings of the American Mathematical Society 3 969), [7] Singer, M Solving homogeneous linear differential equations in terms of second order linear differential equations American Journal of Mathematics ), [8] Ulmer, F, and Weil, J-A Note on Kovacic s algorithm J Symbolic Computation, August 996), [9] Van Hoeij, M, Ragot, J-F, Ulmer, F, and Weil, J-A Liouvillian solutions of linear differential equations of order three and higher J Symbolic Computation 8, 4 and 5 October/November 999), [0] Willis, B An extensible differential equation solver SIGSAM Bulletin 35, 00), 3 7 ν = ± 3, = ± 3 x3/ Using 8) we compute = c x