#### DMCA

## STRUCTURAL THEOREMS FOR SYMBOLIC SUMMATION

### Cached

### Download Links

Citations: | 9 - 7 self |

### Citations

211 | A holonomic systems approach to special functions identities
- Zeilberger
- 1990
(Show Context)
Citation Context ...as special case the summation of (q–)harmonic sums [10, 51, 29, 11] arising, e.g., in particle physics, and they can treat classes of multi-sums that are out of scope of, e.g., the holonomic approach =-=[53, 52, 15, 14]-=-. Karr’s algorithm can be considered as the discrete analogue of Risch’s algorithm [36, 37] for indefinite integration. Here the essential building blocks of exponentials and logarithms can be express... |

201 | The method of creative telescoping
- Zeilberger
- 1991
(Show Context)
Citation Context ...1) 2 + S1(k) ( S2(k)(k + 1) 3 + k + 1 ) + 1 S3(k)(k + 1) 3 + 1 − 1 ∈ F. This framework and extensions [42, 43, 44, 23, 48, 24, 45, 25] generalize, e.g., the (q–)hypergeometric algorithms presented in =-=[1, 18, 54, 34, 32, 35, 33, 5, 20, 3]-=-, they cover as special case the summation of (q–)harmonic sums [10, 51, 29, 11] arising, e.g., in particle physics, and they can treat classes of multi-sums that are out of scope of, e.g., the holono... |

169 |
Decision procedure for indefinite hypergeometric summation
- Gosper
- 1978
(Show Context)
Citation Context ...1) 2 + S1(k) ( S2(k)(k + 1) 3 + k + 1 ) + 1 S3(k)(k + 1) 3 + 1 − 1 ∈ F. This framework and extensions [42, 43, 44, 23, 48, 24, 45, 25] generalize, e.g., the (q–)hypergeometric algorithms presented in =-=[1, 18, 54, 34, 32, 35, 33, 5, 20, 3]-=-, they cover as special case the summation of (q–)harmonic sums [10, 51, 29, 11] arising, e.g., in particle physics, and they can treat classes of multi-sums that are out of scope of, e.g., the holono... |

100 | Non-commutative elimination in Ore algebras proves multivariate identities,
- Chyzak, Salvy
- 1998
(Show Context)
Citation Context ...as special case the summation of (q–)harmonic sums [10, 51, 29, 11] arising, e.g., in particle physics, and they can treat classes of multi-sums that are out of scope of, e.g., the holonomic approach =-=[53, 52, 15, 14]-=-. Karr’s algorithm can be considered as the discrete analogue of Risch’s algorithm [36, 37] for indefinite integration. Here the essential building blocks of exponentials and logarithms can be express... |

90 | 2000 An extension of Zeilberger’s fast algorithm to general holonomic functions Discrete Math
- Chyzak
(Show Context)
Citation Context ...as special case the summation of (q–)harmonic sums [10, 51, 29, 11] arising, e.g., in particle physics, and they can treat classes of multi-sums that are out of scope of, e.g., the holonomic approach =-=[53, 52, 15, 14]-=-. Karr’s algorithm can be considered as the discrete analogue of Risch’s algorithm [36, 37] for indefinite integration. Here the essential building blocks of exponentials and logarithms can be express... |

83 |
The problem of integration in finite terms,
- Risch
- 1969
(Show Context)
Citation Context ...ics, and they can treat classes of multi-sums that are out of scope of, e.g., the holonomic approach [53, 52, 15, 14]. Karr’s algorithm can be considered as the discrete analogue of Risch’s algorithm =-=[36, 37]-=- for indefinite integration. Here the essential building blocks of exponentials and logarithms can be expressed in terms of an elementary differential field F, and Risch’s algorithm can decide constru... |

81 |
Theory of summation in finite terms
- Karr
- 1985
(Show Context)
Citation Context ...lds. This leads to new and constructive structural theorems for symbolic summation. E.g., these results can be applied for harmonic sums which arise frequently in particle physics. 1. Introduction In =-=[21, 22]-=- M. Karr developed a summation algorithm in which indefinite nested sums and products can be simplified. More precisely, such expressions are rephrased in a ΠΣ-field F, a very general class of differe... |

80 | D’Alembertian solutions of linear differential and difference equations
- Abramov, Petkovˇsek
- 1994
(Show Context)
Citation Context ...c summation. We stress that the suggested results and the underlying algorithms implemented in the summation package Sigma [46] play an important role in the simplification of d’Alembertian solutions =-=[30, 2, 39]-=-, a subclass of Liouvillian solutions [19] of a given recurrence relation. In this regard, special emphasize is put on the simplification of harmonic sum expressions that arise frequently in particle ... |

74 | Computer generated proofs of binomial multi-sum identities.
- Wegschaider
- 1997
(Show Context)
Citation Context |

67 | A Mathematica q-analogue of Zeilberger’s algorithm based on an algebraically motivated approach to q-hypergeometric telescoping
- Paule, Riese
- 1997
(Show Context)
Citation Context ...1) 2 + S1(k) ( S2(k)(k + 1) 3 + k + 1 ) + 1 S3(k)(k + 1) 3 + 1 − 1 ∈ F. This framework and extensions [42, 43, 44, 23, 48, 24, 45, 25] generalize, e.g., the (q–)hypergeometric algorithms presented in =-=[1, 18, 54, 34, 32, 35, 33, 5, 20, 3]-=-, they cover as special case the summation of (q–)harmonic sums [10, 51, 29, 11] arising, e.g., in particle physics, and they can treat classes of multi-sums that are out of scope of, e.g., the holono... |

66 | Greatest factorial factorization and symbolic summation.
- Paule
- 1995
(Show Context)
Citation Context |

63 | Harmonic sums, Mellin transforms and integrals
- Vermaseren
- 1999
(Show Context)
Citation Context ... [42, 43, 44, 23, 48, 24, 45, 25] generalize, e.g., the (q–)hypergeometric algorithms presented in [1, 18, 54, 34, 32, 35, 33, 5, 20, 3], they cover as special case the summation of (q–)harmonic sums =-=[10, 51, 29, 11]-=- arising, e.g., in particle physics, and they can treat classes of multi-sums that are out of scope of, e.g., the holonomic approach [53, 52, 15, 14]. Karr’s algorithm can be considered as the discret... |

62 | Symbolic summation in difference fields - Schneider - 2001 |

61 |
Nested sums, expansion of transcendental functions, and multiscale multiloop integrals.
- Moch, Uwer, et al.
- 2002
(Show Context)
Citation Context ... [42, 43, 44, 23, 48, 24, 45, 25] generalize, e.g., the (q–)hypergeometric algorithms presented in [1, 18, 54, 34, 32, 35, 33, 5, 20, 3], they cover as special case the summation of (q–)harmonic sums =-=[10, 51, 29, 11]-=- arising, e.g., in particle physics, and they can treat classes of multi-sums that are out of scope of, e.g., the holonomic approach [53, 52, 15, 14]. Karr’s algorithm can be considered as the discret... |

54 |
Symbolic Integration I – Transcendental Functions, 2nd ed
- Bronstein
- 2005
(Show Context)
Citation Context ...31]; a complete proof dealing also with algebraic extensions has been accomplished by Rosenlicht [38]. For an extensive list of literature and generalizations/refinements, like e.g. [50], we refer to =-=[12]-=-. To this end, Risch’s algorithm [36, 37] can be considered as a constructive breakthrough of Liouville’s structure theorem. For instance, let (F, D) be a differential field with K = constDF given by ... |

53 |
Hypergeometric solutions of linear recurrences with polynomial coefficients
- Petkovˇsek
- 1992
(Show Context)
Citation Context |

49 | Solving difference equations in finite terms
- Hendriks, Singer
- 1999
(Show Context)
Citation Context ... and the underlying algorithms implemented in the summation package Sigma [46] play an important role in the simplification of d’Alembertian solutions [30, 2, 39], a subclass of Liouvillian solutions =-=[19]-=- of a given recurrence relation. In this regard, special emphasize is put on the simplification of harmonic sum expressions that arise frequently in particle physics; we refer to [6, 7, 8] for typical... |

44 | Euler sums and contour integral representations
- Flajolet
(Show Context)
Citation Context ...1,...,mr (k) = k∑ i1=1 1 i m1 1 i1∑ i2=1 1 i m2 2 ir−1 ∑ · · · e.g., the shift S1,3(k + 1) = S1,3(k) + S3(k+1) k+1 is reflected by σ(s1,3) = s1,3 + σ(s3) k . In this way, also the truncated Euler sum =-=[17]-=- ∑k S2(i)S3(i) i=1 i is rephrased by e. Similarly, q–analogues of harmonic sums [4, 16, 11] can be formulated in ΠΣ ∗ -fields. 2.3. Karr’s Structural theorem. In [21, 22] Karr arrives at the following... |

42 | Solving parameterized linear difference equations in ΠΣ-fields. SFB-Report 02-19
- Schneider
- 2002
(Show Context)
Citation Context ...n summing the telescoping equation (1) over k leads to the simplification k∑ i=1 f(i) = S2(k)(k + 1) 2 + S1(k) ( S2(k)(k + 1) 3 + k + 1 ) + 1 S3(k)(k + 1) 3 + 1 − 1 ∈ F. This framework and extensions =-=[42, 43, 44, 23, 48, 24, 45, 25]-=- generalize, e.g., the (q–)hypergeometric algorithms presented in [1, 18, 54, 34, 32, 35, 33, 5, 20, 3], they cover as special case the summation of (q–)harmonic sums [10, 51, 29, 11] arising, e.g., i... |

41 |
On solutions of linear ordinary difference equations in their coefficient field
- Bronstein
- 2000
(Show Context)
Citation Context ...e γ = 0.5772... denotes Euler’s constant. Similarities between elementary unimonomial extensions and ΠΣ∗-extensions in the algebraic setting of difference/differential fields are worked out, e.g., in =-=[13]-=-. As it turns out, the discrete version of Liouville’s structural theorem in the context of ΠΣ ∗ -extensions can be stated in the following surprisingly simple form: a sum of f ∈ F is either expressib... |

37 | Harmonic sums and Mellin transforms up to two-loop order.
- Blumlein, Kurth
- 1999
(Show Context)
Citation Context ... [42, 43, 44, 23, 48, 24, 45, 25] generalize, e.g., the (q–)hypergeometric algorithms presented in [1, 18, 54, 34, 32, 35, 33, 5, 20, 3], they cover as special case the summation of (q–)harmonic sums =-=[10, 51, 29, 11]-=- arising, e.g., in particle physics, and they can treat classes of multi-sums that are out of scope of, e.g., the holonomic approach [53, 52, 15, 14]. Karr’s algorithm can be considered as the discret... |

28 |
On the summation of rational functions,
- Abramov
- 1971
(Show Context)
Citation Context |

28 |
Rational normal forms and minimal decompositions of hypergeometric terms,
- Abramov, Petkovšek
- 2002
(Show Context)
Citation Context |

26 | Symbolic summation with single-nested sum extensions (extended version). SFB-Report 2004-7
- Schneider
- 2004
(Show Context)
Citation Context ... [21], but covering all sums and products treated explicitly by Karr’s work. For such ΠΣ ∗ -extensions we shall be able to make Karr’s structural theorem constructive: based on the algorithm given in =-=[40]-=- we show that any ΠΣ ∗ -field can be transformed to a reduced ΠΣ ∗ -field in which Karr’s structural theorem can be applied. In addition, we complement Karr’s structural results by taking into account... |

25 | Product representations in ΠΣ-fields
- Schneider
- 2003
(Show Context)
Citation Context ...n summing the telescoping equation (1) over k leads to the simplification k∑ i=1 f(i) = S2(k)(k + 1) 2 + S1(k) ( S2(k)(k + 1) 3 + k + 1 ) + 1 S3(k)(k + 1) 3 + 1 − 1 ∈ F. This framework and extensions =-=[42, 43, 44, 23, 48, 24, 45, 25]-=- generalize, e.g., the (q–)hypergeometric algorithms presented in [1, 18, 54, 34, 32, 35, 33, 5, 20, 3], they cover as special case the summation of (q–)harmonic sums [10, 51, 29, 11] arising, e.g., i... |

24 | Symbolic summation assists combinatorics.
- Schneider
- 2007
(Show Context)
Citation Context ...and we derive new structural theorems that contribute in the field of symbolic summation. We stress that the suggested results and the underlying algorithms implemented in the summation package Sigma =-=[46]-=- play an important role in the simplification of d’Alembertian solutions [30, 2, 39], a subclass of Liouvillian solutions [19] of a given recurrence relation. In this regard, special emphasize is put ... |

23 |
Some q-series identities related to divisor functions
- Dilcher
- 1995
(Show Context)
Citation Context ...k + 1) = S1,3(k) + S3(k+1) k+1 is reflected by σ(s1,3) = s1,3 + σ(s3) k . In this way, also the truncated Euler sum [17] ∑k S2(i)S3(i) i=1 i is rephrased by e. Similarly, q–analogues of harmonic sums =-=[4, 16, 11]-=- can be formulated in ΠΣ ∗ -fields. 2.3. Karr’s Structural theorem. In [21, 22] Karr arrives at the following conclusion: one can predict the structure of a solution g for (5) in a refined version of ... |

23 | Finding telescopers with minimal depth for indefinite nested sum and product expressions
- Schneider
- 2005
(Show Context)
Citation Context ... can be used to simplify the nested depth of a given sum expression. Finally, we relate these results with the difference field theory of depth-optimal ΠΣ ∗ -fields that have been introduced recently =-=[41, 47, 49]-=-. Comparing Karr’s approach and depth-optimal ΠΣ ∗ -extensions we obtain additional insight in ΠΣ-difference field theory and we derive new structural theorems that contribute in the field of symbolic... |

22 | A new Sigma approach to multi-summation.
- Schneider
- 2005
(Show Context)
Citation Context ...n summing the telescoping equation (1) over k leads to the simplification k∑ i=1 f(i) = S2(k)(k + 1) 2 + S1(k) ( S2(k)(k + 1) 3 + k + 1 ) + 1 S3(k)(k + 1) 3 + 1 − 1 ∈ F. This framework and extensions =-=[42, 43, 44, 23, 48, 24, 45, 25]-=- generalize, e.g., the (q–)hypergeometric algorithms presented in [1, 18, 54, 34, 32, 35, 33, 5, 20, 3], they cover as special case the summation of (q–)harmonic sums [10, 51, 29, 11] arising, e.g., i... |

22 | A refined difference field theory for symbolic summation.
- Schneider
- 2008
(Show Context)
Citation Context ... can be used to simplify the nested depth of a given sum expression. Finally, we relate these results with the difference field theory of depth-optimal ΠΣ ∗ -fields that have been introduced recently =-=[41, 47, 49]-=-. Comparing Karr’s approach and depth-optimal ΠΣ ∗ -extensions we obtain additional insight in ΠΣ-difference field theory and we derive new structural theorems that contribute in the field of symbolic... |

21 |
A computer algebra toolbox for harmonic sums related to particle physics.
- Ablinger
- 2009
(Show Context)
Citation Context ...c3 ∈ Q. In previous work [42–44,23,48,24,45,25] we incorporated and generalized, e.g., the (q–)hypergeometric algorithms presented in [2,18,54,34,32,35,33,6, 20,4], the summation of (q–)harmonic sums =-=[10,51,29,11,1]-=- arising, e.g, in particle physics, and parts of the holonomic approach [53,52,15,14] in Karr’s unified framework of ΠΣ-difference fields [21]. Here we restricted ourself to ΠΣ ∗ -extensions and ΠΣ ∗ ... |

19 | Parameterized Telescoping Proves Algebraic Independence of Sums
- Schneider
(Show Context)
Citation Context |

18 | A symbolic summation approach to find optimal nested sum representations. In:
- Schneider
- 2011
(Show Context)
Citation Context ... can be used to simplify the nested depth of a given sum expression. Finally, we relate these results with the difference field theory of depth-optimal ΠΣ ∗ -fields that have been introduced recently =-=[41, 47, 49]-=-. Comparing Karr’s approach and depth-optimal ΠΣ ∗ -extensions we obtain additional insight in ΠΣ-difference field theory and we derive new structural theorems that contribute in the field of symbolic... |

17 | Two-loop massive operator matrix elements for unpolarized heavy flavor production to O(ǫ).
- Bierenbaum, Blumlein, et al.
- 2008
(Show Context)
Citation Context ...illian solutions [19] of a given recurrence relation. In this regard, special emphasize is put on the simplification of harmonic sum expressions that arise frequently in particle physics; we refer to =-=[6, 7, 8]-=- for typical examples in the frame of difference fields. The general structure of this article is as follows. In Section 2 we state Liouville’s structural theorem, and we relate it to Karr’s results i... |

15 | Multibasic and mixed hypergeometric Gosper-type algorithms
- Bauer, Petkovˇsek
- 1999
(Show Context)
Citation Context |

12 | Indefinite summation with unspecified summands.
- Kauers, Schneider
- 2006
(Show Context)
Citation Context |

12 | Simplifying Sums in ΠΣ-Extensions
- Schneider
- 2007
(Show Context)
Citation Context |

10 | Duality for finite multiple harmonic q-series
- Bradley
(Show Context)
Citation Context |

9 | A note on alternating sums
- Kirschenhofer
- 1996
(Show Context)
Citation Context ...), σ) with (k+1) 6 , we get g = s1,3s6 and f ′ = − σ(s3)(σ(s6)(k+1) 6 −1) (k+1) 7 σ(y) = y + σ(s3)(σ(s6)(k+1)6−1) (k+1) 7 ) such that In particular, we get the Q-isomorphism ∆(s1,3s6 − y) = σ(s1,3) . =-=(26)-=- (k + 1) 6 µ: Q(k)(s1)(s2)(s3)(s6)(s1,3)(x)(s6,1,3) → Q(k)(s1)(s2)(s3)(s6)(s1,3)(x)(y) by keeping all variables fixed except µ(s6,1,3) = s1,3s6 − y. (27) To sum up, we managed to transform the ΠΣ ∗ -f... |

8 |
Determining the closed forms of the O(a 3 s) anomalous dimensions and Wilson coefficients from Mellin moments by means of computer algebra, Comp. Phys. Commun. (in print
- Blümlein, Kauers, et al.
- 2009
(Show Context)
Citation Context ...illian solutions [19] of a given recurrence relation. In this regard, special emphasize is put on the simplification of harmonic sum expressions that arise frequently in particle physics; we refer to =-=[6, 7, 8]-=- for typical examples in the frame of difference fields. The general structure of this article is as follows. In Section 2 we state Liouville’s structural theorem, and we relate it to Karr’s results i... |

8 | An Extension of Liouville’s Theorem on Integration in Finite Terms
- Singer, Saunders, et al.
- 1985
(Show Context)
Citation Context ...been provided by [31]; a complete proof dealing also with algebraic extensions has been accomplished by Rosenlicht [38]. For an extensive list of literature and generalizations/refinements, like e.g. =-=[50]-=-, we refer to [12]. To this end, Risch’s algorithm [36, 37] can be considered as a constructive breakthrough of Liouville’s structure theorem. For instance, let (F, D) be a differential field with K =... |

7 |
Identities in combinatorics IV: differentiation and harmonic numbers
- Andrews, Uchimura
- 1995
(Show Context)
Citation Context ...ote also that one can verify by the same mechanism that the base field (F, σ) constructed in Ex. 3 forms a ΠΣ ∗ -field over Q. We remark that Karr’s framework covers also q–analogues of harmonic sums =-=[5, 16,11]-=- or generalized harmonic sums [29]; for a package which combines the ideas of [10,51,29] with the difference field approach see [1]. 7 2.3 Karr’s structural theorem In [21,22] Karr arrives at the foll... |

6 |
Liouville’s Theorem on Functions with Elementary Integrals
- Rosenlicht
- 1968
(Show Context)
Citation Context ...ds contain the rational numbers Q as subfield. 12 CARSTEN SCHNEIDER here D denotes the differential operator acting on the elements of F. In this regard, Liouville’s theorem of integration, see e.g. =-=[28, 31, 38]-=-, plays an important role. In a nutshell, it states that for integration with elementary functions it suffices to restrict to logarithmic extensions, i.e., one can neglect exponential and algebraic fu... |

5 | Difference equations in massive higher order calculations. In:
- Bierenbaum, Blumlein, et al.
- 2007
(Show Context)
Citation Context ...illian solutions [19] of a given recurrence relation. In this regard, special emphasize is put on the simplification of harmonic sum expressions that arise frequently in particle physics; we refer to =-=[6, 7, 8]-=- for typical examples in the frame of difference fields. The general structure of this article is as follows. In Section 2 we state Liouville’s structural theorem, and we relate it to Karr’s results i... |

5 |
Théorie analytique des probabilités. Vol
- Laplace
- 1995
(Show Context)
Citation Context ...et the Q-isomorphism ∆(s1,3s6 − y) = σ(s1,3) . (26) (k + 1) 6 µ: Q(k)(s1)(s2)(s3)(s6)(s1,3)(x)(s6,1,3) → Q(k)(s1)(s2)(s3)(s6)(s1,3)(x)(y) by keeping all variables fixed except µ(s6,1,3) = s1,3s6 − y. =-=(27)-=- To sum up, we managed to transform the ΠΣ ∗ -field (G, σ) to the Q-ordered and completely reduced ΠΣ ∗ -extension (Q(k)(s1)(s2)(s3)(s6)(s1,3)(x)(y), σ) of (Q, σ) with σ(k) = k + 1, σ(s1) = s1 + 1 k+1... |

5 |
Mémoire sur l’intégration d’une classe de fonctions transcendantes, J. reine und angewandte Mathematik 13
- Liouville
(Show Context)
Citation Context ...en f ∈ F there exists an antiderivative g ∈ F, i.e., D(g) = f; (2) here D denotes the differential operator acting on the elements of F. In this regard, Liouville’s theorem of integration, see, e.g., =-=[28,31,38]-=-, plays an important role. In a nutshell, it states that for integration with elementary functions it suffices to restrict to logarithmic extensions, i.e., one can neglect exponential and algebraic fu... |

4 |
From moments to functions in quantum chromodynamics
- Blümlein, Kauers, et al.
- 2009
(Show Context)
Citation Context ... results lead to fine-tuned telescoping algorithms that enables one to handle efficiently a tower of up to 100 Σ δ -extensions in the summation package Sigma; for an example from particle physics see =-=[9]-=-. Besides this, we emphasize Theorem 53. [[47, Result 6]] Let (E, σ) be a ΠΣ δ -ext. of (F, σ); let f ∈ E. If there is a ΠΣ ∗ -extension (H, σ) of (F, σ) with g ∈ H s.t. (5), then there is a Σ δ -exte... |

4 | Symbolic summation with radical expressions
- Kauers, Schneider
- 2007
(Show Context)
Citation Context |

4 | Automated proofs for some Stirling number identities
- Kauers, Schneider
(Show Context)
Citation Context |

3 |
Finite singularities and hypergeometric solutions of linear recurrence equations
- Hoeij
- 1999
(Show Context)
Citation Context |

1 |
Mémoire sur l’int´gration d’une classe de fonctions transcendantes
- Liouville
(Show Context)
Citation Context ...ds contain the rational numbers Q as subfield. 12 CARSTEN SCHNEIDER here D denotes the differential operator acting on the elements of F. In this regard, Liouville’s theorem of integration, see e.g. =-=[28, 31, 38]-=-, plays an important role. In a nutshell, it states that for integration with elementary functions it suffices to restrict to logarithmic extensions, i.e., one can neglect exponential and algebraic fu... |

1 |
Sur l’intégrabilité élémentaire de quelques classes d’expressions
- Ostrowski
- 1946
(Show Context)
Citation Context ...ds contain the rational numbers Q as subfield. 12 CARSTEN SCHNEIDER here D denotes the differential operator acting on the elements of F. In this regard, Liouville’s theorem of integration, see e.g. =-=[28, 31, 38]-=-, plays an important role. In a nutshell, it states that for integration with elementary functions it suffices to restrict to logarithmic extensions, i.e., one can neglect exponential and algebraic fu... |