S.A. Abramov (1995): Rational solutions of linear difference and q-difference equations with polynomial coefficients, these proceedings.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....to find polynomial as well as hypergeometric solutions of recurrences, all in the mixed hypergeometric case. We have also indicated how to extend the concept of GFF to this case. It remains to provide mixed hypergeometric generalizations of algorithms for finding rational solutions of recurrences [1, 11] and of algorithms for factorization of the corresponding operators [5] The more efficient algorithm of van Hoeij for finding hypergeometric solutions [12] should also admit of a generalization to the mixed hypergeometric case. Acknowledgements The authors wish to thank Bill Gosper for ....

S. A. Abramov, Rational solutions of linear difference and q-difference equations with polynomial coefficients, Programming and Comput. Software 21 (1995) 273--278. 19

....of greatest factorial factorization of polynomials [10] which seems to play a fundamental role in symbolic summation, from the basic [11] and bibasic [13] cases to the multibasic and mixed one. On the algorithmic plane, multibasic and mixed generalizations of algorithms for finding rational [1] and hypergeometric [12, 3] solutions of recurrences, and also of algorithms for factorization of recurrence operators [5] should be developed. 12 A Algorithm m m Poly INPUT: p0 ; p ae ; g; h1 ; hs 2 F [x; y] p0 ; p ae 6= 0 OUTPUT: general solution (f; 2 F [x; y] Theta F s ....

S. A. Abramov, Rational solutions of linear difference and q-difference equations with polynomial coefficients, Programming and Computer Software 21 (1995) 273-278.

....over k. Solutions of all completely factorable equations form the linear space Ak ae K of d Alembertian elements. The order of minimal operator over k which annihilates a 2 Ak is called the height of a. It is easy to see that Hk ae Ak and the height of any a 2 Hk is equal to 1. It is known ([12, 4]) that if L 2 k[ and f 2 Hk then all the hyperexponential solutions of the equation Ly = f ( have the form uf; u 2 k. Substitution y = uf , where u is a new unknown, gives us the equation Mu = g; M 2 k[ g 2 k: If all the solutions of ( in k are found, then all the hyperexponential ....

....side belong to k, is called the k problem. Fast algorithms to solve k problems are known for some concrete Ore polynomial rings (e.g. such algorithms for linear ordinary differential and (q )difference equations with rational functions coefficients and right hand sides have been given in [1, 2, 4]) They allow us to find hyperexponential solutions of the equation ( quickly. We consider in this paper the search for d Alembertian solutions of an equation of the form ( with f 2 Ak (the height of f is r 1) We show that in general case the Work reported herein was supported in part by ....

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Abramov S.A. (1995): Rational solutions of linear difference and q-difference equations with polynomial coefficients, in the Proceedings of ISSAC'95, Montreal, Canada, ACM Press, 285--289.

....c i (q k ) i for X(k) with as yet unknown coefficients c i in Equation (2.5) and solve the resulting linear system by comparing the coefficients of (q k ) i . This method involves solving a linear system in h Gamma l 1 variables which is inefficient if the difference h Gamma l is large. Abramov et al. 1995) introduced an alternative algorithm. It is based on the idea to convert the recurrence equation for X(k) into one for the coefficients c i and using this recurrence to calculate as many coefficients as possible. Abramov et al. 1995) suggest to use their algorithm if the difference h Gamma l is ....

....which is inefficient if the difference h Gamma l is large. Abramov et al. 1995) introduced an alternative algorithm. It is based on the idea to convert the recurrence equation for X(k) into one for the coefficients c i and using this recurrence to calculate as many coefficients as possible. Abramov et al. 1995) suggest to use their algorithm if the difference h Gamma l is greater or equal to the order of the recurrence equation that is to be solved. We want to illustrate this by calculating an antidifference for the function Fn (k) a; q) k (q; q) k Gamma a q n Delta Gammak ; 2.9) where a ....

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Abramov, S. A. (1995). Rational solutions of linear difference and q-difference equations. Programming and Comput. Software, 6:273--278. Transl. from Programmirovanie (1995) 6, 3-11.

....(y) Delta Delta Delta a 1 (y) a 0 y = 0 where a i 2 C(x) with an 6= 0; a 0 6= 0: 2) Inhomogeneous equations can be reduced to homogeneous equations (cf. section 2.2) so treating the homogeneous case is sufficient. Computing a bound for the denominator is a key step in the algorithm (cf. [1, 2]) for computing rational solutions, i.e. solutions Y 2 C(x) n or y 2 C(x) The purpose of this paper is to give a sharper bound for the denominator of rational solutions of equation (1) A consequence of having a smaller denominator is that the numerator one needs to compute is also smaller, and ....

.... that this lower bound is non zero for only finitely many p 2 C) To compute a bound for the numerator of y one needs to find a lower bound for the valuation at infinity and use equation (3) To find such lower bound for v 1 (y) for solutions y of equation (2) we can use the same approach as in [1]. Bounds at infinity for equation (1) can be obtained from chapter 7 in [3] see also [5, 4] In the rest of this paper we will only consider the problem of bounding the denominator, in other words finding lower bounds for the valuations at finite points. If f 1 ; fm 2 C [x] then v p ....

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S. A. Abramov. Rational solutions of linear difference and q-difference equations with polynomial coefficients, Programming and Comput. Software 21, No 6, p. 273 -- 278. Transl. from Programmirovanie, No 6, p. 3 -- 11, (1995).

....this approach can be modified for solving also this more general problem. We present the result of this modification in form of a proposition. Its proof is entirely analogous to what we did above and is left to the reader. For alternative methods and the general n th order case see, for instance, Abramov (1995). Proposition 5.2. Given nonzero a; b; c 2 K[x] the problem of solving a Delta E Gamma b Delta = c (5.16) for 2 K(x) can be decided constructively as follows: i) Compute a G form hV; q; ri for a=b with V; q; r 2 K[x] such that bj(r Delta V ) ii) If b Delta q Delta EU Gamma b ....

Abramov, S.A. (1995). Rational solutions of linear difference and q-difference equations with polynomial coefficients. In: (Levelt, T., ed) Proc. ISSAC '95 , pp. 285--289, New York: ACM Press.

....L be a linear ordinary difference operator with coefficients in k. A classical problem in the theory of difference equations is to compute all the solutions in k of the equation L(y) b. If C denotes a constant field and if k = C(n) and oen = n 1 or oen = qn, there are known algorithms (see [2] for example) Manuel Bronstein presents here a generalization to monomial extensions of C(n) see [6] for details and generalization) 1. Historical Context The rational solutions of linear differential equations (equations of the form P n i=0 a i y (i) have been first studied a long time ....

....difference equations (equations of the form P n i=0 a i y(x i) The link between the linear differential equations and the linear difference equations is now clear, and in [1] an algorithm to compute the rational solutions of this two types of equations with coefficients in C(x) is given. In [2], the author extends the previous algorithm to q linear difference equations (equations of the form P n i=0 a i y(q i x) Algorithms to compute the rational solutions of linear differential, difference and q difference equations with coefficients in C(x) are now available, and extensions of ....

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Abramov (S. A.). -- Rational solutions of linear difference and q-difference equations with polynomial coefficients. In Levelt (A. H. M.) (editor), Symbolic and Algebraic Computation. pp. 285--289. -- ACM Press, New York, 1995. Proceedings of ISSAC'95, Montreal, Canada.

....n (y) Delta Delta Delta a1 (y) a0y = 0 (2) where a i 2 C(x) with an 6= 0, a0 6= 0. Inhomogeneous equations can be reduced to homogeneous equations (cf. section 2.2) so treating the homogeneous case is sufficient. Computing a bound for the denominator is a key step in the algorithm (cf. [1, 3, 2]) for computing rational solutions, i.e. solutions Y 2 C(x) n or y 2 C(x) The purpose of this paper is to give a sharper bound for the denominator of rational solutions of equation (1) A consequence of having a smaller denominator is that the numerator one needs to compute is also smaller, and ....

.... that this lower bound is nonzero for only finitely many p 2 C) To compute a bound for the numerator of y one needs to find a lower bound for the valuation at infinity and use equation (3) To find such lower bound for v1 (y) for solutions y of equation (2) we can use the same approach as in [1, 2]. Bounds at infinity for equation (1) can be obtained from [2] similar to the differential case [5] In the rest of this paper we will only consider the problem of bounding the denominator, in other words finding lower bounds for the valuations at finite points. If f1 ; fm 2 C [x] then ....

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S. A. Abramov. Rational solutions of linear difference and q-difference equations with polynomial coefficients, Programming and Comput. Software 21, No 6, p. 273 -- 278. Transl. from Programmirovanie, No 6, p. 3 -- 11, (1995).

....parts: P1 Find all candidate values of a for which (1) may have solutions of the form (2) with wn expressible in closed form. P2 Find closed form solutions of the corresponding recurrence (6) Once P1 has been solved and the candidate expansion points a have been found, the algorithms of [2] [1], and [6] resp. can be used for each a to find all polynomial, rational, resp. hypergeometric solutions of the corresponding recurrence (6) In particular, a detailed description of an algorithm to find all hypergeometric series solutions of (1) given the expansion point a is presented in [7] ....

....has to be a rational function r(x) such that wn = r(n) for all n 0. In particular, this means that r(x) can have no nonnegative integer poles. For a nonconstant irreducible polynomial f(n) and a nonzero polynomial g(n) denote deg(f(n) g(n) maxfk 0; f(n) k j g(n)g: Proposition 1 (cf. [1]) Let wn = p(n) q(n) be a rational function of n with p and q relatively prime, and f(n) any nonconstant, irreducible polynomial. If there are polynomials p 0 (n) p 1 (n) p s (n) such that s X j=0 p j (n)wn j = 0; p 0 ; p s 6= 0; 10) then deg(f(n) q(n) 1 X i=0 deg(f(n ....

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S.A. Abramov (1995): Rational solutions of linear difference and q-difference equations with polynomial coefficients, Proc. ISSAC '95, 285--289.

....4.2, we can find the general solution (f; of equation (4.1) as follows: First compute the set R as given in (4.14) Then distinguish three cases: 1. ff maxR fi Set = maxR and look for f in the form f = t(x)y f 1 (4. 15) where f 1 = o(y ) To find t(x) apply the algorithm of (Abramov et al. 1995) to T = 0 (an ordinary homogeneous recurrence relation) Then remove maxR from R and find f 1 recursively by solving Lf 1 = rhs( Gamma L (t(x)y ) 4.16) 2. ff maxR fi (a) ff fi Set = fi Gamma ff and look for f in the form (4.15) To find t(x) apply the algorithm of (Abramov ....

....al. 1995) to T = 0 (an ordinary homogeneous recurrence relation) Then remove maxR from R and find f 1 recursively by solving Lf 1 = rhs( Gamma L (t(x)y ) 4.16) 2. ff maxR fi (a) ff fi Set = fi Gamma ff and look for f in the form (4. 15) To find t(x) apply the algorithm of (Abramov et al. 1995) to T = r fi (an ordinary parametric nonhomogeneous recurrence relation) Then remove maxR from R (only in case that ff maxR = fi) and find f 1 recursively by solving (4.16) Mixed Gosper Type Algorithms 11 (b) ff 6 fi Let = 0 be the solution of the system of linear algebraic equations ....

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Abramov, S. A. (1995). Rational solutions of linear difference and q-difference equations with polynomial coefficients. Programming and Comput. Software 21, 273--278.

....F[q n ] 4.4) with given q hypergeometric term G(n) then L(n) F(n) G(n) is in F(q n ) To solve Equation (4.4) we divide it by G(n) and search for all rational solutions L(n) of J X j=0 oe j (n) j Gamma1 Y i=0 fl(n i) L(n) 1; fl(n) G(n 1) G(n) 2 F(q n ) 4. 5) Abramov (1995) showed how to determine a multiple of the denominator of L(n) therefore leaving the easier problem of finding polynomial solutions of inhomogeneous recurrence equations. 19 As an example, we search for all q hypergeometric solutions F(k) of the inhomogeneous recurrence equation q k (q 1) ....

Abramov, S. A. (1995). Rational solutions of linear difference and q-difference equations. Programming and Comput. Software, 6:273--278. Transl. from Programmirovanie (1995) 6, 3-11.

....c fl1997 ACM 0 89791 875 4 97 0007 3. 50 famous ones as Dixon s theorem [10] and the first RogersRamanujan identity [11] The related problem of solving difference and differential equations with rational coefficients is investigated by Risch [23] Rothstein [24] Davenport [9] Abramov [2, 3, 4], Bronstein [6, 7] Petkovsek [21] and Abramov et al. 5] We report on a particular class of solutions of the key equation in Gosper s algorithm, where polynomials r; s; v 2 F [x] over a field F of characteristic zero are given and a polynomial u 2 F [x] satisfying r Delta Eu Gamma s Delta ....

Abramov, S. A. Rational solutions of linear difference and q-difference equations with polynomial coefficients. In Proc. 1995 Internat. Symp. Symb. Alg. Comput. (1995), ACM Press, pp. 285--289.

....operator of (ordinary or q ) difference, because both operator algebras K [ffi; oe; ffi] and K [oe; oe; 0] are equal when ffi = oe Gamma 1. After the uncoupling step described above, we are led to linear equations in the shift or q shift operator. In each case, an algorithm of Abramov s applies [2, 1]. The case of (ordinary) differential equations. In this case, oe is the identity, so that the change of Ore operators in the uncoupling step above is trivial ( 0 = We next solve each uncoupled differential equation by Abramov s algorithm [1] Finally, note that the value 1 in the ....

....close to questions related to the factorization of operators. The crucial step of Algorithm 2 for definite summation and integration is the resolution of the linear system (4) which we perform by first uncoupling the system using an algorithm in [4] before appealing to specialized algorithms [1, 2] to solve equations in a single unknown function. Other uncoupling algorithms are available [6, 10] but we emphasize the desire for an algorithm that works directly at the level of systems of Ore operators. Indeed, from our first experiments, the uncoupling step is the computational bottleneck of ....

Abramov, S. A. Rational solutions of linear difference and q-difference equations with polynomial coefficients. In Symbolic and algebraic computation (New York, 1995), A. Levelt, Ed., ACM Press, pp. 285--289. Proceedings ISSAC'95, Montreal, Canada.

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S.A. Abramov (1995): Rational solutions of linear difference and q-difference equations with polynomial coefficients, these proceedings.

....P1. Expanding a function as a power series and subsequently investigating the expansion. An annihilator lets one construct the recurrence for the series coefficients and manipulate them ( 14, 17] P2. Solving linear inhomogeneous equations. Some methods use annihilators of the right hand side ([4, 8]) P3. Integrating. If the minimal annihilator L; ord L = n, of f is given, then one can check whether there exists a primitive of f with an n th order minimal annihilator. If yes, then it is possible to express the primitive explicitly via f ( 9] P4. Recognizing the equivalence of two given ....

S. A. Abramov (1995): Rational solutions of linear difference and q-difference equations with polynomial coefficients, Programming and Comput. Software, No 6.

....infinite product. For integer m let [m] 1 Gamma q m ) 1 Gamma q) Note that Gammaq m [ Gammam] m] 1 q : q m Gamma1 for m 1. Thus [m] turns into m as q 1. The same applies to [m] d defined as [m] with q replaced by q d , d a positive integer. Let [m] [1][2] Delta : Delta [m] for m 1 and [0] 1. For integer n and nonnegative integer k, the Gaussian polynomial or the q binomial coefficient is defined as Theta n k = n] n Gamma 1] Delta : Delta [n Gamma k 1] k] These definitions are introduced to make the analogy with the ....

....= qx, and the corresponding right divisor L 1 = Gamma qx of L. To find other first order right divisors, the remaining combinations for a(x) and b(x) could be tried. Using our Mathematica implementation to carry this out, it turns out that there are in fact no other such divisors: In[1]: qHyper[x y[q3 x] q3 x2 y[q2 x] x2 q) y[q x] q x (x2 q) y[x] 0, y[x] Solutions All, Quadratics True] Out[1] q x 2 Example 4 Consider the operator L = 2 Gamma (1 q) q(1 Gamma qx 2 )I. As shown by qHyper, 2 available at ....

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S. A. Abramov, Rational solutions of linear difference and q-difference equations with polynomial coefficients, Programming and Comput. Software (1995) 6, 273--278. Transl. from Programmirovanie (1995) 6, 3--11. 20

....in differential and difference algebra contain these tasks as subproblems which, however conceptually simple, often account for a fair share of the overall computing time. For instance, the algorithms for ffl finding all the rational solutions of differential and (q )difference equations [Abr89b, Abr95], ffl finding Liouvillian solutions of differential equations [Sin91] ffl finding (q )hypergeometric solutions of (q )difference equations [Pet92, Abr Pet95] ffl factoring linear differential and difference operators with rational coefficients [Bro Pet94] ffl indefinite hypergeometric ....

.... 6 x 10 q 6 1 Gamma q 6 q 12 1 Gamma q 12 x 13 Delta Delta Delta = C 1 X k=0 q 3k(k 1) q 6 ; q 6 )k x 3k 7 where (a; q)k = 1 Gamma a) 1 Gamma aq) Delta Delta Delta (1 Gamma aq k Gamma1 ) In the differential and (q )difference cases, we can compute [Abr89b, Abr95] a universal denominator d(x) 2 K[x] such that any solution of Ly = f of the form z(x) p(x) 18) where z(x) 2 K[ Pn) 1 n=0 ] and p 2 K[x] can be written as z(x) d(x) where z(x) 2 K[ Pn) 1 n=0 ] So we can apply our algorithm after doing the change of variable z(x) d(x)y(x) ....

S.A. Abramov (1995): Rational solutions of linear difference and q-difference equations with polynomial coefficients, these proceedings.

....infinite product. For integer m let [m] 1 Gamma q m ) 1 Gamma q) Note that Gammaq m [ Gammam] m] 1 q : q m Gamma1 for m 1. Thus [m] turns into m as q 1. The same applies to [m] d defined as [m] with q replaced by q d , d a positive integer. Let [m] [1][2] Delta : Delta [m] for m 1 and [0] 1. For integer n and nonnegative integer k, the Gaussian polynomial or the q binomial coefficient is defined as h n k i = n] n Gamma 1] Delta : Delta [n Gamma k 1] k] These definitions are introduced to make the analogy with the ....

....= qx, and the corresponding right divisor L 1 = Gamma qx of L. To find other first order right divisors, the remaining combinations for a(x) and b(x) could be tried. Using our Mathematica implementation to carry this out, it turns out that there are in fact no other such divisors: In[1]: qHyper[x y[q3 x] q3 x2 y[q2 x] x2 q) y[q x] q x (x2 q) y[x] 0, y[x] Solutions All, Quadratics True] Out[1] q x 2 Example 4 Consider the operator L = 2 Gamma (1 q) q(1 Gamma qx 2 )I. As shown by qHyper, In[2] qHyper[y[q2 x] 1 q) y[q x] q (1 q x2) ....

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S. A. Abramov, Rational solutions of linear difference and q-difference equations with polynomial coefficients, in: T. Levelt, ed., Proc. ISSAC '95 (ACM Press, New York, 1995) 285--289.

....of a for which Ly a = 0 may have solutions of the form (21) with nice c n . S2 For each candidate value of a, find nice solutions c = hc n i 1 n=0 of the corresponding recurrence (26) Once S1 has been solved and the candidate expansion points a have been found, the algorithms of [2] [1], and [10] resp. can be used at each a (assuming there are finitely many of them) to find all polynomial, rational, resp. hypergeometric solutions of the corresponding recurrence (26) In particular, a detailed description of an algorithm to find all hypergeometric series solutions of Ly a = 0 ....

....has a solution with rational logarithmic derivative [3, Theorem 4] and so does Ly = 0. 2 Therefore to find solutions (21) of Ly a = 0 with non polynomial rational coefficients c n , it suffices to consider the singular points of (17) as candidate expansion points a, and to use the algorithm of [1] at each of them to find rational solutions of the corresponding recurrence (26) Example 8 The equation 2x(x Gamma 1)y 00 (x) 7x Gamma 3)y 0 (x) 2y(x) 0 (33) is singular at x = 0 and x = 1. Let us find power series solutions at x = 0. Recurrence (26) in this case is (n 1) 2n 3)c ....

S. A. Abramov, Rational solutions of linear difference and q-difference equations with polynomial coefficients, Progr. and Comp. Software 21 (1995) 273--278.

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