1 Factorization of second-order linear di®erential equations and Liouville-Neumann expansions Esther Garc¶³a1, Lance Littlejohn2, Jos¶eL. L¶opez3 and Ester P¶erezSinus¶³a4 1 Departamento Matem¶aticas,IES Miguel Catal¶ande Zaragoza, Spain. e-mail: [email protected]. 2 Department of Mathematics, Baylor University, One Bear Place # 97328, Waco, Texas 96798-7328. e-mail: Lance¡[email protected]. 3 Departamento de Ingenier¶³aMatem¶aticae Inform¶atica,Universidad P¶ublicade Navarra, 31006-Pamplona, Spain. e-mail: [email protected]. 4 Departamento Matem¶aticaAplicada, IUMA, Universidad de Zaragoza, 50018-Zaragoza, Spain. e-mail: [email protected]. ABSTRACT. We introduce a quasi-factorization technique for second-order linear di®erential equations that brings together three rather diverse subjects of the ¯eld of di®erential equations: di®er- ential Galois theory, Liouville-Neumann approximation, and the Frobenius theory. The factorization of a linear di®erential equation (di®erential Galois theory) is a theoretical tool used to solve the equation exactly, although its range of applicability is quite reduced. On the other hand, the Liouville-Neumann algorithm is a practical tool that approximates a solution of the equation, it is based on a certain inte- gral equation equivalent to the di®erential equation and its range of applicability is extraordinary large. In this paper we combine both procedures. We use the ideas of the factorization to ¯nd families of integral equations equivalent to the di®erential equation. From those families of integral equations we propose new Liouville-Neumann algorithms that approximate the solutions of the equation. The method is valid for either regular or regular singular equations. We discuss the convergence properties of the algorithms and illustrate them with examples of special functions. These families of integral equations are parametrized by a certain regular function, so they constitute indeed a function-parametric family of integral equations. A special choice of that function corresponds with a factorization of the di®eren- tial equation. In this respect, other choices of that function may be considered as quasi-factorizations of the di®erential equation. The quasi-factorization technique let us ¯nd also a relation between the Liouville-Neuman expansion and the Theory of Frobenius and show that the Liouville-Neumann method may be viewed as a generalization of the Theory of Frobenius. 2000 AMS Mathematics Subject Classi¯cation: 34A05; 34A30; 33B15; 33C05; 33C10; 33C15; 33C45. Keywords & Phrases: Factorization of di®erential equations. Integral equations. Liouville-Neuman expansions. Approximation of special functions. 2 1. Introduction Factorization of (linear or non-linear) di®erential equations is a largely studied topic because of its implications in the resolution of di®erential equations. It is, at the level of resolution of di®erential equations, equivalent to the factorization of polynomials at the level of resolution of algebraic equations. The problem of the searching of algebraic algorithms for the solutions of algebraic equations is based on the Galois theory [19]; the problem of the searching of closed algebraic formulas for the solutions of di®erential equations is based on the di®erential Galois theory [16], [23]. There exists an obvious analogy between the algebraic and di®erential theories. In the algebraic case the problem is equivalent to the problem of a factorization of a polynomial using only admissible numbers (elements of the algebraic ¯eld under consideration). The most important Galois theory theorem establishes that "an algebraic equation is solvable by radicals if and only if the corresponding Galois group is solvable". In the di®erential case the problem is equivalent to the problem of a factorization of the dif- ferential equation in terms of a certain set of admissible (Liouvillian) functions [10]. Roughly speaking, this set of admissible functions consists of elementary functions, inde¯nite integrals of elementary functions and exponentials of these inde¯nite integrals (Kovacic's algorithm). The pre- cise de¯nition of admissible or Liouvillian function may be found in [10], although for the purposes of this paper we may content ourselves by considering a vague de¯nition of admissible function: a function written in an algebraically closed way. Analogously to the Galois theorem for alge- braic equations, in the di®erential Galois theory, the following statement is established: "A linear di®erential equation system is solvable by quadratures (Liouvillian extensions) if and only if the connected component of the corresponding Galois group is triangularizable" [2]. In 1986, Jerald Kovacic [10] presents an algorithm to solve second order linear di®erential equations with rational coe±cient functions over C= in terms of admissible functions. However, when the reduced linear di®erential equation Galois group is exactly SL(2,C=), the Kovacic algorithm does not work and the reduced linear di®erential equation has no Liouvillian (admissible) solutions. Even though Kovacic's algorithm is an elegant and precise tool to exactly solve the di®erential equation, the problem is that, in most of the cases, the Galois group of the reduced linear di®erential equation is SL(2,C=) and the algorithm cannot be applied. To go deep inside the theory of factorization of di®erential equations, the Galois Di®erential Theory, or the Kovacic's algorithm, we recommend [3], [5], [10], [16], [18], [20], [22], [23], [24] and references therein. In [4], [7], [8], [17] we can ¯nd applications of the factorization of di®erential equations to the computation of special functions. We give a very introduction to the idea of factorization of regular second order linear di®erential 2 equations. Consider a polynomial of degree two in a real variable x, p2(x) = x + ax + b. It is factorizable in the form p2(x) = (x + x1)(x + x2), where its roots x1 and x2 are the solutions of the non-linear algebraic system ( a = x1 + x2; (1) b = x1x2: Consider now a regular second order linear di®erential operator in the unknown y, L[y] := y00 + f(x)y0 + g(x)y; f 0; g 2 C[0;X] (2) and the corresponding homogeneous equation L[y] = 0. For this di®erential equation, an analogous factorization to the above factorization of the polynomial p2(x) is the following one: L[y] = [y0 + B(x)y]0 + A(x)[y0 + B(x)y]; 3 where the functions A(x) and B(x) are solutions of the ¯rst order nonlinear di®erential system ( f = A + B; (3) g = AB + B0: If we can ¯nd a solution (A; B) of the above system, then we have found two independent solutions of the equation L[y] = 0. To see this, we de¯ne 0 0 LA[y] := y + Ay; LB[y] := y + By: Then we can write L[y] = LA[LB[y]]. (In the following, for simplicity in the notation, we will omit brackets and write only L[y] = LALB[y].) Then, an obvious solution y1(x) of the second order equation L[y] = 0 is a solution of the ¯rst order equation LB[y] = 0, that is, µ Z x ¶ y1(x) = exp ¡ B(t)dt : If u(x) is a solution of the ¯rst order equation LA[u] = 0, that is, µ Z x ¶ u(x) = exp ¡ A(t)dt ; ¡1 then another solution of the second order equation L[y] = 0 is y2(x) = LB [u], because L[y2] = ¡1 ¡1 LALBLB [u] = LA[u] = 0. The inverse operator LB may be constructed using the Green function of LB [11], Z x ¡1 y1(x) LB [y] = GB(x; t)y(t)dt; GB(x; t) = : (4) y1(t) Therefore, a second solution of the second order equation L[y] = 0 is Z µ Z ¶ Z µZ ¶ x u(t) x x t y2(x) = y1(x) dt = exp ¡ B(t)dt exp [B(s) ¡ A(s)]ds dt: y1(t) The solutions y1 and y2 are linearly independent [11] and then fy1; y2g form a basis of the space of solutions of L[y] = 0. (In [11] we can ¯nd the construction of complete systems of solutions of factorized linear di®erential equations of arbitrary order). Observation 1. If we try to solve (3) by replacing A = f ¡ B into the second equation, we obtain 0 2 that B is a solution of the Ricatti equation B = B ¡ fB + g. But precisely B and y1 are related 0 by means of the Ricatti transformation B = ¡y =y. Therefore, y1 is just the solution of L[y] = 0 obtained by means of the standard Ricatti transformation B = ¡y0=y. It is then appropriate to give the name Ricatti system to the system (3). It is straightforward to see that y2 is precisely a second solution obtained from y1 by the method of variation of parameters. We see that, in principle, the factorization L = LALB gives us the same couple of independent solutions y1 and y2 obtained by the method of the Ricatti transformation plus variation of parameters. It is not obvious that the resolution of the Ricatti equation is equivalent to the resolution of system (3). Observation 2. Observe also when the operator L has constant coe±cients, we can choose A and B constants, and the di®erential system (3) becomes the algebraic system: ( f = A + B; g = AB; 4 the same system (1) that we have for the algebraic equation p2(x) = 0. In this case the above solutions y1 and y2 of L[y] = 0 are just the ones obtained by the standard method applied to equations of constant coe±cients, ( ¡Bx ¡Ax y1(x) = e ; y2(x) = e ; if A 6= B; ¡Bx ¡Bx y1(x) = e ; y2(x) = xe ; if A = B: Then, somehow, the factorization of a linear di®erential operator of the second order may be considered as a generalization of the factorization of a second degree polynomial.