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## The method of differentiating under the integral sign (1990)

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Venue: | J. Symbolic Computation |

Citations: | 52 - 15 self |

### Citations

423 | Groebner-Bases: An Algorithmic Method in Polynomial Ideal Theory - Buchberger - 2003 |

211 | A holonomic systems approach to special functions identities
- Zeilberger
- 1990
(Show Context)
Citation Context ... are going to present a unified theory of differentiation under the integral sign for the important and wide class of so-called holonomic functions. We will recall the ”slow” elimination algorithm of =-=[Z1]-=-, and present it in a form that will hopefully lead to a fast version using Buchberger’s method of Grobner bases adapted to the Weyl algebra. For the subclass of ”hyperexponential” functions we will g... |

200 | The method of creative telescoping
- Zeilberger
- 1991
(Show Context)
Citation Context ...perexponential” functions we will give a fast algorithm, that given F (x, y), finds a linear differential equation satisfied by R(x). This ”fast” algorithm is a continuous analog of Zeilberger’s [Z2] =-=[Z3]-=- * Department of Mathematics, University of Lund, Box 118, 22100 Lund, Sweden. ** Department of Mathematics and Computer Science, Drexel University, Philadelphia, PA 19104. Supported in part by NSF gr... |

162 | A fast algorithm for proving terminating hypergeometric identities - Zeilberger - 1990 |

83 |
The problem of integration in finite terms,
- Risch
- 1969
(Show Context)
Citation Context ...f a MAPLE program that implements this algorithm, while Appendix 2 contains sample inputs and outputs. While the problem of indefinite integration in closed form is completely solved and implemented (=-=[Ri]-=-, [No], [Tr], [DST] ), there is currently no well-developed theory of definite integration ( see [No], 3.2). The present paper can be viewed as a first step toward developing a theory of definite inte... |

55 |
The diagonal of a D-finite power series is D-finite
- Lipshitz
- 1988
(Show Context)
Citation Context ...er bases algorithm in the Weyl algebra. A major step towards this goal was recently achieved by Takayama[Ta]. A related notion to holonomicity is that of D-finiteness, which was considered by Lipshitz=-=[Li]-=-. A formal power series f(x1, ..., xn) is D-finite if the vector space generated by all its partial derivatives is finite dimensional over the field of rational functions in (x1, ..., xn). Every holon... |

48 | Computer Algebra
- Davenport, Siret, et al.
(Show Context)
Citation Context ...that implements this algorithm, while Appendix 2 contains sample inputs and outputs. While the problem of indefinite integration in closed form is completely solved and implemented ([Ri], [No], [Tr], =-=[DST]-=- ), there is currently no well-developed theory of definite integration ( see [No], 3.2). The present paper can be viewed as a first step toward developing a theory of definite integration. The Theore... |

43 |
Integration of Algebraic Functions
- Trager
- 1984
(Show Context)
Citation Context ...ogram that implements this algorithm, while Appendix 2 contains sample inputs and outputs. While the problem of indefinite integration in closed form is completely solved and implemented ([Ri], [No], =-=[Tr]-=-, [DST] ), there is currently no well-developed theory of definite integration ( see [No], 3.2). The present paper can be viewed as a first step toward developing a theory of definite integration. The... |

41 | Some algorithmic questions on ideals of differential operators - Galligo - 1985 |

30 |
An approach to the zero recognition problem by Buchberger algorithm
- Takayama
- 1992
(Show Context)
Citation Context ...spect to P and Q and ”right” with respect to Dy, and so one has to develop an ambidextrous Grobner bases algorithm in the Weyl algebra. A major step towards this goal was recently achieved by Takayama=-=[Ta]-=-. A related notion to holonomicity is that of D-finiteness, which was considered by Lipshitz[Li]. A formal power series f(x1, ..., xn) is D-finite if the vector space generated by all its partial deri... |

25 |
Modules over a ring of differential operators. Study of fundamental solutions of equations with constant coefficients, Funct
- Bernstein
- 1981
(Show Context)
Citation Context ...jDi ,xiDj = Djxi (i �= j) Dixi − xiDi = 1 , Excellent expositions on the Weyl algebra and Bernstein’s theory can be found in [Bj] and [Eh]. It was proved in [Z1](section 6.4), using ideas of Bernstein=-=[Be]-=-, that if F (x, y) is holonomic and P (x, y, Dx, Dy) and Q(x, y, Dx, Dy) are operators that generate a ”holonomic ideal” then one can ”eliminate” y. In other words, it is possible to find operators A(... |

20 |
Generalized hypergeometric series, Cambridge Math. Tracts 32
- Bailey
- 1935
(Show Context)
Citation Context ...s beta integral as well as four other, less trivial, recurrences. Applications To The Theory Of Special Functions It is perhaps surprising that while many deep hypergeometric sum identities are known(=-=[Ba]-=-), all of which can be handled by the algorithm of [Z2] and [Z3], relatively few hyperexponential integral identities of the form (1) are known, where both F (x, y) and R(x) are hyperexponential. In f... |

19 |
A combinatorial proof of the Mehler formula,
- Foata
- 1978
(Show Context)
Citation Context ... n y2 ) y (− 1 + 2 x y − y 2 ) n With the following input, the program proves the Mehler formula for Hermite polynomials (e.g. [Ho], p. 24, ex. 4). The Mehler formula came to the limelights when Foata=-=[Fo1]-=- gave a beautiful combinatorial proof that inaugurated the currently very active area of combinatorial special function theory. Unlike the computer-generated proof below, that is a verification, Foata... |

19 |
Companion to concrete mathematics,
- Melzak
- 1973
(Show Context)
Citation Context ...ating under the integral sign could hardly have been considered more than a trick, and the examples given in the few textbooks that treat it ([Ed]) are rather simple. One notable exception is Melzak’s=-=[Me]-=- delightful book, that devotes a whole section to this method. In this paper we are going to present a unified theory of differentiation under the integral sign for the important and wide class of so-... |

13 |
Towards computerized proofs of identities
- Wilf, Zeilberger
- 1990
(Show Context)
Citation Context ...Drexel University, Philadelphia, PA 19104. Supported in part by NSF grant DMS8800663. Address in 1999: Dept. of Mathematics, Temple Univ., Philadelphia, PA 19122, zeilberg@math.temple.edu 1s(see also =-=[W-Z]-=-) algorithm for hypergeometric summation, that uses Gosper’s[Go] algorithm as a subroutine. In Appendix 1 (which is published in [A-Z]) we give a listing of a MAPLE program that implements this algori... |

11 |
Twelve geometrical essays
- Coxeter
- 1968
(Show Context)
Citation Context ...METHOD OF DIFFERENTIATING UNDER THE INTEGRAL SIGN Gert Almkvist * and Doron Zeilberger** Appeared in J. Symbolic Computation 10 (1990), 571-591 ”I could never resist a definite integral” (G. H. Hardy =-=[Co]-=-) ”One thing I never did learn was contour integration. I had learned to do integrals by various methods shown in a book that my high-school physics teacher Mr. Bader had given me. ... The book also s... |

10 |
A combinatorial approach to the Mehler formulas for Hermite polynomials, Relations between combinatorics and other parts of mathematics
- Foata, Garsia
- 1978
(Show Context)
Citation Context ...part of the generating function arises naturally. Furthermore, Foata’s proof lead naturally to a proof of a multi-variate generalization of the Mehler formula that was accomplished by Foata and Garsia=-=[FG]-=-(see also [Fo2]). The theory of holonomic systems is incapable, even in principle, of proving multi-variate identities with an arbitrary number of variables. #Begin input #The Mehler formula for Hermi... |

9 |
Some Hermite polynomial identities and their combinatorics
- Foata
- 1981
(Show Context)
Citation Context ...nerating function arises naturally. Furthermore, Foata’s proof lead naturally to a proof of a multi-variate generalization of the Mehler formula that was accomplished by Foata and Garsia[FG](see also =-=[Fo2]-=-). The theory of holonomic systems is incapable, even in principle, of proving multi-variate identities with an arbitrary number of variables. #Begin input #The Mehler formula for Hermite polynomials ... |

6 |
Surely you are joking, Mr Feynman
- Feynman, Leighton
- 1985
(Show Context)
Citation Context ...a great reputation for doing integrals, only because my box of tools was different from everybody else’s, and they had tried all their tools on it before giving the problem to me.” (Richard P. Feynman=-=[F]-=-) Introduction The method of differentiating under the integral sign can be described as follows. Given a function F (x, y) of x and y, one is interested in evaluating � ∞ R(x) := F (x, y)dy . (1) −∞ ... |

3 |
A Treatise on
- EDWARDS
- 1955
(Show Context)
Citation Context ...ferential equation. Although termed a ”method”, differentiating under the integral sign could hardly have been considered more than a trick, and the examples given in the few textbooks that treat it (=-=[Ed]-=-) are rather simple. One notable exception is Melzak’s[Me] delightful book, that devotes a whole section to this method. In this paper we are going to present a unified theory of differentiation under... |

3 |
Decision procedure of indefinite summation
- Gosper
(Show Context)
Citation Context ...SF grant DMS8800663. Address in 1999: Dept. of Mathematics, Temple Univ., Philadelphia, PA 19122, zeilberg@math.temple.edu 1s(see also [W-Z]) algorithm for hypergeometric summation, that uses Gosper’s=-=[Go]-=- algorithm as a subroutine. In Appendix 1 (which is published in [A-Z]) we give a listing of a MAPLE program that implements this algorithm, while Appendix 2 contains sample inputs and outputs. While ... |

2 |
Summation in finite terms. In Computer Algebra: symbolic and algebraic computation, 2nd Ed
- Lafon
- 1983
(Show Context)
Citation Context ..., that is, in some sense, a special case of Risch’s algorithm, but that we will be able to extend to find both ¯ S and G of (6). This algorithm, that is a continuous analog of Gosper’s ([Go],see also =-=[La]-=-, [GKP]) decision procedure for indefinite hypergeometric summation, will now be presented. or: Gosper’s Algorithm Translated To The Continuous” This section follows closely Gosper’s[Go] classical pap... |

2 |
Integration in Finite Terms, Computer Algebra: Symbolic and Algebraic Computation
- Norman
- 1983
(Show Context)
Citation Context ...PLE program that implements this algorithm, while Appendix 2 contains sample inputs and outputs. While the problem of indefinite integration in closed form is completely solved and implemented ([Ri], =-=[No]-=-, [Tr], [DST] ), there is currently no well-developed theory of definite integration ( see [No], 3.2). The present paper can be viewed as a first step toward developing a theory of definite integratio... |

1 |
A MAPLE program that finds, and proves, recurrences and differential equations satisfied by hyperexponential definite integrals
- Almkvist, Zeilberger
(Show Context)
Citation Context ...iv., Philadelphia, PA 19122, zeilberg@math.temple.edu 1s(see also [W-Z]) algorithm for hypergeometric summation, that uses Gosper’s[Go] algorithm as a subroutine. In Appendix 1 (which is published in =-=[A-Z]-=-) we give a listing of a MAPLE program that implements this algorithm, while Appendix 2 contains sample inputs and outputs. While the problem of indefinite integration in closed form is completely sol... |

1 |
The Weyl algebra, in: ”Algebraic D-modules”, by A. Borel et
- Ehlers
- 1987
(Show Context)
Citation Context ...principle” where 1 is the identity operator. xixj = xjxi ,DiDj = DjDi ,xiDj = Djxi (i �= j) Dixi − xiDi = 1 , Excellent expositions on the Weyl algebra and Bernstein’s theory can be found in [Bj] and =-=[Eh]-=-. It was proved in [Z1](section 6.4), using ideas of Bernstein[Be], that if F (x, y) is holonomic and P (x, y, Dx, Dy) and Q(x, y, Dx, Dy) are operators that generate a ”holonomic ideal” then one can ... |

1 |
Lehrbuch der hoeren
- Herr
- 1857
(Show Context)
Citation Context ... the existence of (3) could have also been established by Lipshitz’s[Li] method. 2. The Secret Behind Many Definite Integrals The secret behind many definite integrals is a lemma that can be found in =-=[He]-=-. It is probably not the earliest reference. It is likely that the trick was known to Euler or at least to Cauchy (Herr quotes Cauchy, among others , in the preface ). Lemma: Proof: t = x − a x so, We... |