M. Petkovsek (1992): Hypergeometric solutions of linear recurrences with polynomial coefficients, J. Symb. Comp. 14, 243--264.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....of [2, p. 56 58] There the problem is to calculate the minimal polynomial satisfied by a product # 1 # k of k distinct branches of an algebraic function defined by P (z, #) 0. Gosper Petkovsek normal forms for rational functions. In many algorithms for symbolic summation (e.g. 15] [21] and [19] one has to solve a linear first order difference equation, the key equation. For example, Gosper s algorithm for hypergeometric indefinite summation [15] reduces the search for a hypergeometric solution f of a di#erence equation f(x 1) f(x) g(x) with hypergeometric right hand ....

M. Petkovsek. Hypergeometric solutions of linear recurrences with polynomial coe#cients. J. Symbolic Comput., 14(2-3):243--264, 1992.

....any t 0, applying that equality to s = tm j 1 and remarking that a s im = a (t i)m j 1 = a t i , we obtain a t N i=0 b i (tm j 1)a t i = 0 for all t 0. We therefore need to look for all the hypergeometric solutions of L j for m. We can use the algorithm Hyper [20, 21], which, given a linear ordinary di#erence operator R with coe#cients in C(x) returns # 1 , # t C(x) # and finitely many polynomials p rs C[x] such that the hypergeometric solutions of Ry = 0 are exactly all the sequences satisfying #y = # r s c rs #p rs s c rs p rs (5) ....

M. Petkovsek. Hypergeometric solutions of linear recurrences with polynomial coefficients. J. Symbolic Computation, 14(2 and 3):243--264, August&September 1992.

....classical that the Taylor coe#cients satisfy a linear recurrence that can be derived from the di#erential equation. This provides a fast method for Taylor expansion. This is implemented in the gfun package [8] Closed form for Taylor coe#cients. From the recurrence above, Petkovsek s algorithm [6] decides whether the Taylor coe#cients u n are hypergeometric (i.e. u n 1 u n is a rational function of n) This is implemented in the LREtools package of Maple. Success then leads to a closed form for the coe#cients in terms of the Gamma function. Note that failure of this algorithm is a proof ....

Marko Petkovsek. Hypergeometric solutions of linear recurrences with polynomial coe#cients. Journal of Symbolic Computation, 14(2-3):243--264, 1992.

....that the Taylor coefficients satisfy a linear recurrence that can be derived from the differential equation. This provides a fast method for Taylor expansion. This is implemented in the gfun package [8] Closed form for Taylor coefficients. From the recurrence above, Petkovsek s algorithm [6] decides whether the Taylor coefficients un are hypergeometric (i.e. un 1=un is a rational function of n) This is implemented in the LREtools package of Maple. Success then leads to a closed form for the coefficients in terms of the Gamma function. Note that failure of this algorithm is a proof ....

Marko Petkovsek. Hypergeometric solutions of linear recurrences with polynomial coefficients. Journal of Symbolic Computation, 14(2-3):243--264, 1992.

....series representations with point of development x 0 = 0 for the Gegenbauer polynomials which are specific Jacobi polynomials, but failed in the Jacobi case, though. One might ask whether such a representation exists. This question can be completely answered by an algorithm of Petkovsek [18]. Petkovsek s algorithm finds all hypergeometric term solutions of holonomic recurrence equations, i.e. homogeneous linear recurrence equations with polynomial coefficients. Using the recurrence equation (29) an application of Petkovsek s algorithm proves that the Jacobi polynomials do not ....

Petkovsek, M.: Hypergeometric solutions of linear recurrences with polynomial coefficients. J. Symbolic Comp. 14, 1992, 243--264.

....main ingredient in Zeilberger s algorithm is Gosper s algorithm for indefinite hy A Mathematica Version of Zeilberger s Algorithm 3 pergeometric summation. One peculiarity of our implementation is that it computes Petkovsek s canonical form of Gosper s representation of a rational function, see Petkovsek (1992). We also want to point out that due to the special structure of the input, see section 3, which delivers information essentially in already factored form, this form is computed without any resultant computation. In more general contexts this procedure is unavoidable, see also section 5. Besides ....

.... Zb[2( 4m) Binomial[4m,2m] 4 ( 1)j Binomial[m 1,j] Binomial[2m 1,2j] Binomial[4m 1,4j] 1) 2m 4j 1) 4m 4j 1) j,0,m 1, m, 2] Out[13] 1 m) SUM[m] SUM[1 m] 3 m) SUM[2 m] 0 This recursion finds no hypergeometric solution T (m) which can be seen, for instance, by applying Petkovsek s (1992) algorithm. Consequently, the term 2 4m Gamma 4m 2m Delta Gamma1 we extracted constitutes the hypergeometric part of EO(m) Nevertheless, the remainder T (m) is almost hypergeometric which is made more precise as follows. Denoting TE(n) T (2n) for n 0, one gets In[14] Zb[2( 8n) ....

Petkovsek, M. (1992). Hypergeometric solutions of linear recurrences with polynomial coefficients. J. Symb. Comp. 14, 243--264.

....GFF is more efficient. As the referee points out, the heuristic explanation is that it is often better to have more operations on smaller polynomials than to have fewer operations on larger ones. ii) Another alternative to compute the GFF form can be derived from the fact that the algorithm of Petkovsek (1992) for computing the Gosper Petkovsek representation ( GP form ) for rational functions, a normalized version of the G form representation also described in Section 5, contains the GFF form computation as a special case; see Lemma 5.2. This also was briefly described in Paule Strehl (1995) 2 3. ....

....polynomials p; q; r 2 K[x] is called a G form for the rational function a=b 2 K(x) if a b = Ep p Delta q r and gcd(q; E k r) 1 for all k 0: In the previous section we used Algorithm VMULT to compute multiples P i of the GFF constituents p i of the rational solution denominator v. Petkovsek (1992) used exactly the same algorithm in order to compute a canonical G form representation. That canonical form, called GP form, serves as a key ingredient for his algorithm Hyper ; it is defined as the unique G form where additionally p and r are supposed to be monic, and gcd(p; q) gcd(Ep; r) ....

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Petkovsek, M. (1992). Hypergeometric solutions of linear recurrences with polynomial coefficients. J. Symbolic Computation 14, 243--264.

....with the computational help of Shalosh B. Ekhad, provided us with relations satisfied by the sums. Then Marko Petkovsek reduced the order of one of the recursions and applied his algorithm for determining all hypergeometric solutions to linear difference equations with polynomial coefficients [25]. Since these sums were not among them, they have no closed form. For completeness, we state the recurrence relations here; the reader should consult Petkovsek s article for details of the proof. In what follows, E denotes the shift operator with respect to m, i.e. E(f(m) f(m 1) 12 ....

M. Petkovsek, Hypergeometric solutions of linear recurrences with polynomial coefficients, J. Symbolic Computation, to appear.

....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] This ....

M. Petkovsek (1992): Hypergeometric solutions of linear recurrences with polynomial coefficients, J. Symb. Comput. 14, 243--264.

.... 1 2 Gamma(x Gamma 1) x Gamma 2) u(0) Gamma 16 x (x Gamma 2) u(1) 70 x (x Gamma 1) u(2) 3 x Gamma 1) 3 x 1) Gamma(x 1) 3 All solutions of this difference operator L have the same type; each solution u(x) is a rational function times 1= Gamma(x 1) However, the algorithm in [15] will still distinguish a number of different cases. The number of cases in this example can be reduced to 1 by computing the g p (L) defined in section 4. In fact, whenever all solutions of a difference operator L are hypergeometric and of the same type, the number of cases in the algorithm in ....

....cases. The number of cases in this example can be reduced to 1 by computing the g p (L) defined in section 4. In fact, whenever all solutions of a difference operator L are hypergeometric and of the same type, the number of cases in the algorithm in section 5 is always 1, whereas in the method in [15] the number of cases to check can still be exponentially high. Both methods are fast on this example though, because of the fact that all roots of the leading and trailing coefficients (the coefficients of the highest and of the lowest power of in L) are rational numbers, which makes this example ....

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M. Petkovsek. Hypergeometric solutions of linear recurrences with polynomial coefficients. J. Symbolic Comput., 14(2-3):243--264, 1992.

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M. Petkovsek (1992): Hypergeometric solutions of linear recurrences with polynomial coefficients, J. Symb. Comp. 14, 243--264.

....In this section E denotes the shift operator corresponding to x, so that Ef(x) f(x 1) for every f 2 K(x) Theorem 4 For every rational function F 2 K(x) there are polynomials a; b; c 2 K[x] such that (i) F = ii) a E b for all k 2 N , iii) a c and b Ec. For a proof, see [13] or [14] The original version of this theorem (without (iii) is due to Gosper [6] Definition 7 (PNF) If a, b, c, F satisfy (i) and (ii) of Theorem 4, then (a; b; c) is a polynomial normal form or PNF of F . A PNF which satisfies (iii) of Theorem 4 is strict. Lemma 1 If (a; b; c) is a strict ....

....0, it follows that b 1 E a 1 for k 0 as well, proving (ii) 2 Definition 9 (RNF) If u, v, F are as in Theorem 5, u; v) is a rational normal form, or RNF, of F . We denote the set of all RNF s of F by RNF x (F ) Note that together with an algorithm for computing strict PNF (to be found in [13] or [14] the proof of Theorem 5 provides an algorithm for computing an element of RNF x (F ) Theorem 6 Let (u; v) and (u 1 ; v 1 ) be two RNF s of F 2 K(x)nf0g. Write u = zp=q and u 1 = z 1 p 1 =q 1 where z; z 1 2 K, p; q; p 1 ; q 1 2 K[x] are monic, p q, and p 1 q 1 . Then z = z 1 , deg p ....

M. Petkovsek, Hypergeometric solutions of linear recurrences with polynomial coefficients, J. Symb. Comp. 14 (1992) 243--264.

....7 8] it is not immediately clear how to extend that to the multibasic case. In Section 7 we provide algorithm MixedHyper which finds all mixed hypergeometric solutions of a homogeneous polynomial coefficient recurrence of any order. This is a common generalization of the algorithms presented in [19] and [3] In Section 8 we extend the concept of greatest factorial factorization [16] to an arbitrary automorphism oe of the multivariate polynomial ring. Notation. The set of integers is denoted by Z, the set of nonnegative integers by N 0 , and the field of rational numbers by Q . If n; m 2 N ....

M. Petkovsek, Hypergeometric solutions of linear recurrences with polynomial coefficients, J. Symbolic Comp. 14 (1992) 243--264.

....E denotes the shift operator corresponding to x, so that Ef(x) f(x 1) for every f 2 K(x) Theorem 4 For every rational function F 2 K(x) there are polynomials a; b; c 2 K[x] such that (i) F = a b Delta Ec c , ii) a E k b for all k 2 N , iii) a c and b Ec. For a proof, see [6] or [7] The original version of this theorem (without (iii) is due to Gosper [3] Definition 7 If a, b, c, F satisfy (i) and (ii) of Theorem 4, then (a; b; c) is a polynomial normal form or PNF of F . A PNF which satisfies (iii) of Theorem 4 is strict. Lemma 1 If (a; b; c) is a strict PNF of ....

M. Petkovsek, Hypergeometric solutions of linear recurrences with polynomial coefficients, J. Symb. Comp. 14 (1992) 243--264.

....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 of Lf = g P s j=1 j h ....

M. Petkovsek, Hypergeometric solutions of linear recurrences with polynomial coefficients, J. Symb. Comp. 14 (1992) 243--264.

.... Summation and Gosper Petkovsek Representation 3 ffl to show (from local considerations) how this information can be obtained from the Gosper Petkovsek representation of rational functions, which was originally invented in the context of indefinite hypergeometric summation, see Gosper (1978) Petkovsek (1992), and for which we give a purely combinatorial equivalent, Section 5] ffl to propose a new summation method for rational functions, based on (known) algorithms for producing the Gosper Petkovsek representation of rational functions, Section 6] In Section 5.1 we give a detailed example, ....

.... of rational functions is at the basis of Gosper s classical decision method for indefinite hypergeometric summation: For any rational function ff 2 k(X) there are polynomials p; q; r 2 k[X] such that ff = E p p Delta q E r with gcd(q; E i r) 1 for all i 1 Petkovsek showed in Petkovsek (1992) that a presentation of ff 2 k(X) as ff = c Delta E p p Delta q E r with gcd(q; E i r) 1 for all i 1 ; gcd(p; r) 1 = gcd(p; q) with monic polynomials p; q; r 2 k[X] and c 2 k is unique. Note that the usual algorithms for computing p; q; r (and c) as outlined in Gosper (1978) and ....

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Petkovsek, M. (1992), Hypergeometric solutions of linear recurrences with polynomial coefficients. Journal of Symbolic Computation, 14:243--264.

....an operator L 2 k[X] as input and returns a hyperexponential solution of the equation Ly = 0 if it exists. In the case when k is the field of rational functions F (x) over some field F of characteristic 0 such algorithms are given, e.g. in [Sin91] or [Bro92b] for differential operators, and in [Pet92] for difference operators. We show that algorithm H, used recursively in combination with reduction of order, will construct a basis for the space of all d Alembertian solutions of Ly = 0. We call this algorithm A. Algorithm A INPUT: A nonzero linear operator L 2 k[X] OUTPUT: A basis for the ....

M. Petkovsek (1992): Hypergeometric solutions of linear recurrences with polynomial coefficients, J. Symb. Comp. 14, 243--264.

....The overview of the paper is as follows. An algorithm qHyper for finding first order right divisors of linear q difference operators with rational coefficients is presented in Section 4. It is a q analogue of algorithm Hyper for finding hypergeometric solutions of difference equations described in [13]. Note that by clearing denominators in (3) we can restrict attention to operators L with polynomial coefficients p i 2 IF[x] In preparation, we show how to find polynomial solutions of (3) in Section 2, and give a suitable normal form for rational functions in Section 3. In Section 5, we discuss ....

....the Zeilberger recurrence for the sum expression on the left hand side of n X k=0 ( Gamma1) k n k d k n = Gammad) n (29) for a fixed positive integer d is of order d Gamma 1 instead of order 1 according to its hypergeometric evaluation. Here one applies algorithm Hyper of [13] to the recurrence in order to find its hypergeometric solutions. In this section a brief discussion of applications of qHyper in connection with definite q hypergeometric summation is given. 7.1 A new q summation identity Let d and n be positive integers, then n X k=0 ( Gamma1) k q d( ....

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M. Petkovsek, Hypergeometric solutions of linear recurrences with polynomial coefficients, J. Symb. Comp. 14 (1992) 243--264.

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M. Petkovsek (1992): Hypergeometric solutions of linear recurrences with polynomial coefficients, J. Symb. Comp. 14, 243--264.

....for finding first order right divisors of linear q difference operators with rational coefficients is presented in Section 1 All these concepts are relative to the field IF. 2 4. It is a q analogue of algorithm Hyper for finding hypergeometric solutions of difference equations described in [13]. Note that by clearing denominators in (3) we can restrict attention to operators L with polynomial coefficients p i 2 IF[x] In preparation, we show how to find polynomial solutions of (3) in Section 2, and give a suitable normal form for rational functions in Section 3. In Section 5, we discuss ....

....the Zeilberger recurrence for the sum expression on the left hand side of n X k=0 ( Gamma1) k n k d k n = Gammad) n (29) for a fixed positive integer d is of order d Gamma 1 instead of order 1 according to its hypergeometric evaluation. Here one applies algorithm Hyper of [13] to the recurrence in order to find its hypergeometric solutions. In this section a brief discussion of applications of qHyper in connection with definite q hypergeometric summation is given. 14 7.1 A new q summation identity Let d and n be positive integers, then n X k=0 ( Gamma1) k q ....

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M. Petkovsek, Hypergeometric solutions of linear recurrences with polynomial coefficients, J. Symb. Comp. 14 (1992) 243--264.

....inverse transformation: R Gamma1 C : En 7 E Gamma 1; n 7 x(1 Gamma E Gamma1 ) n i E Gammai n 7 x i E Gammai : 7 Example 5 Let L = x 4)E 4 Gamma(7x 24)E 3 Gamma(x 2 Gamma8x Gamma28)E 2 (6x 2 10x Gamma1)E Gamma5(x 1) 2 . Algorithm Hyper of [10] shows that L has no right or left first order factors in K(x) E] where K is any field of characteristic 0, so the full factorization algorithm of [8] needs to be used to check for existence of second order factors. Instead, we compute here the induced recurrence operator R C L = n 4)E 4 n ....

....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 given the ....

M. Petkovsek, Hypergeometric solutions of linear recurrences with polynomial coefficients, J. Symb. Comput. 14 (1992) 243--264.

.... methods The methods that have achieved the complete solution of this class of problems are the following: Sister Celine s method [1] Gosper s algorithm [3] Zeilberger s algorithm ct ( creative telescoping ) 11] Wilf and Zeilberger s WZ method [9] Petkovsek s algorithm Hyper [6] Here is a brief description of the scope of each of these algorithms (full descriptions are in [8] Computer programs, in Maple or Mathematica versions, that carry out each of these algorithms are available free at http: www.cis.upenn.edu #wilf AeqB.html. Sister Celine s algorithm has been ....

M. Petkovsek, Hypergeometric solutions of linear recurrences with polynomial coefficients, J. Symb. Comput. 14 (1992) 243--264.

.... The methods that have achieved the complete solution of this class of problems are the following: ffl Sister Celine s method [1] ffl Gosper s algorithm [3] ffl Zeilberger s algorithm ct ( creative telescoping ) 11] ffl Wilf and Zeilberger s WZ method [9] ffl Petkovsek s algorithm Hyper [6] Here is a brief description of the scope of each of these algorithms (full descriptions are in [8] Computer programs, in Maple or Mathematica versions, that carry out each of these algorithms are available free at http: www.cis.upenn.edu wilf AeqB.html. Sister Celine s algorithm has been ....

M. Petkovsek, Hypergeometric solutions of linear recurrences with polynomial coefficients, J. Symb. Comput. 14 (1992) 243--264.

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M. Petkovsek. Hypergeometric solutions of linear recurrences with polynomial coefficients. J. Symbolic Comput., 14(2-3):243--264, 1992.

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M. Petkovsek. Hypergeometric solutions of linear recurrences with polynomial coefficients. J. Symbolic Comput. , 14(2-3):243--264, 1992.

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