M. Petkovsek & B. Salvy (1993): Finding all hypergeometric solutions of linear differential equations, in: (M. Bronstein, Ed.) Proceedings of ISSAC'93, ACM Press, New York 27--33.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....[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 solves P2. A short discussion of P1 in the case of hypergeometric coefficients is given in [7, Sec. 3.2] but a completely satisfactory solution has not been provided yet. In this paper, we show how to find all a and all solutions (2) of (1) for which there exists: ffl a polynomial p(x) ....

.... of (1) with rational coefficients wn , it suffices to consider the singularities of (1) as candidate expansion points a, and to use the algorithm of [1] at each of them to find rational solutions of the corresponding recurrence (6) Again, we need to compute in the splitting field of v r (x) In [7], a function is called d Alembertian if it can be written as f 1 (x) R f 2 (x) R Delta Delta Delta R f k (x)dx : dx dx where the f i have rational logarithmic derivatives. We want to show now that a power series with rational 3 coefficients is a d Alembertian function. Let f(x) ....

M. Petkovsek, B. Salvy (1993): Finding all hypergeometric solutions of linear differential equations, Proc. ISSAC '93, 27--33.

....3 2 ; Gamma 1 2 ; 1 2 ; z = 2 z 2 p 1 Gamma z 3 p z 2 sin Gamma1 Gamma p z Delta is a typical formula representation. Abilitiy to compute such representations is applicable to integration, differential equations, closed form summation, and difference equations [7] [10], 13] The Meijer G function, G( a; b; c; d; z) defined in the next section, is a generalization of the hypergeometric function F( a; b; z) Every hypergeometric function is a G function: F i a; b; z j = Gamma b a G i 1 Gamma a; 0; 1 Gamma b; log ( Gammaz) ....

Petkovsek, M. and Salvy, B. (1993), "Finding All hypergeometric Solutions of Linear Differential Equations", Proceedings of ISSAC '93 , ACM, New York.

.... c fl1996 ACM 0 89791 796 0 96 07: 3:50 s ; z) z 1 ( Gamma 1) 1) Theta F 1; Gamma 3 2 ; 3 2 ; Gamma z 2 4 hypergeometric functions are applicable to integration, differential equations, closed form summation, and difference equations [5] [6], 7] Some methods will create answers in terms of F. An algorithm like ours can often reexpress such answers in terms of better known functions. 2 2F1 Example Let pFq denote the restriction of F to C p Theta C q Theta C . Then 0F0 , 0F1 , and 1F0 are representable by F ( z) e z F ....

Petkovsek, M. and Salvy, B. (1993), "Finding All Hypergeometric Solutions of Linear Differential Equations ", Proceedings of ISSAC '93 , ACM, New York.

....of the (continuous or discrete) variable t. Those two types of equations are closely connected, the various algorithms for solving or otherwise manipulating them have some interesting similarities [1] and 1 on occasion, methods devised for one type of equation can be used on the other type [15]. A comparison of the algebraic properties of those equations points to the existence of some common mathematical abstraction behind them. This abstraction is provided by pseudo linear algebra, an area of mathematics with origins in the 1930 s, whose objects of study are skew polynomials [13] ....

.... of study are skew polynomials [13] which represent single equations (1,3) and pseudo linear operators [10] which represent systems (2,4) Algebraic algorithms originally developed for differential equations and systems [3, 5, 16, 17, 18] have been recently generalized to difference equations [14, 15] and arbitrary pseudo linear equations [6, 22] This enables us to present in this paper an algorithmic introduction to pseudo linear algebra. After introducing the basic objects of study in the first section, we describe their basic arithmetic operations, followed by a factorisation algorithm for ....

M. Petkovsek & B. Salvy (1993): Finding all hypergeometric solutions of linear differential equations, in: (M. Bronstein, Ed.) Proceedings of ISSAC'93, ACM Press, New York 27--33.

....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 expansion point a is presented in [11]. This solves S2. A short discussion of S1 in the case of hypergeometric coefficients is given in [11, Sec. 3.2] but a completely satisfactory solution has not been provided yet. Here we show how to find all a 2 K and all solutions (21) of Ly a = 0 for which there exists: 1. a polynomial p 2 ....

....the coefficients of w c which are thus hypergeometric. 2 Therefore the following algorithm will find all solutions (21) of Ly a = 0 with hypergeometric c n : 1. For each singular point a of L, find all solutions y = P 1 n=0 c n x n of L a y = 0 with hypergeometric c n , using the algorithm of [11]. Then the corresponding y a give all the hypergeometric series solutions at x = a. 2. Pick any ordinary point a of L. Find all solutions y = P 1 n=0 c n x n of L a y = 0 with hypergeometric c n , using either the algorithm of [11] or, since these solutions are d Alembertian, the algorithm ....

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M. Petkovsek, B. Salvy, Finding all hypergeometric solutions of linear differential equations, Proc. ISSAC '93, M. Bronstein, Ed., Kiev, Ukraine, July 6--8, 1993 (ACM Press, New York 1993) 27--33. 18

....(1990) for a survey of the algorithms) Petkovsek (1992) gave an algorithm to find hypergeometric solutions of linear recurrences with polynomial coefficients. This in turn gives an algorithm to find generalized hypergeometric solutions of linear differential equations with polynomial coefficients (Petkovsek Salvy 1993). Our prototype implementations should soon make their way into Maple s library. 4. Asymptotics The area of computer algebra needed to compute expansions of solutions of linear differential equations has undergone extensive research (Tournier 1987; Duval 1987; Thomann 1990) Completely solving ....

Petkovsek, M. and B. Salvy (1993, July). Finding all hypergeometric solutions of linear differential equations. In M. Bronstein (Ed.), ISSAC'93, pp. 27--33. ACM Press.

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M. Petkovsek, B. Salvy (1993): Finding All Hypergeometric Solutions of Linear Differential Equations. Proc. ISSAC'93, 27-33.

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