메뉴 건너뛰기




Volumn 43, Issue 9, 2008, Pages 611-644

A refined difference field theory for symbolic summation

Author keywords

Difference fields; Nested depth; Symbolic summation

Indexed keywords


EID: 42649097573     PISSN: 07477171     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.jsc.2008.01.001     Document Type: Article
Times cited : (113)

References (41)
  • 1
    • 85031698900 scopus 로고
    • D'Alembertian solutions of linear differential and difference equations
    • von~zur Gathen J. (Ed), ACM Press
    • Abramov S., and Petkovšek M. D'Alembertian solutions of linear differential and difference equations. In: von~zur Gathen J. (Ed). Proc. ISSAC'94 (1994), ACM Press 169-174
    • (1994) Proc. ISSAC'94 , pp. 169-174
    • Abramov, S.1    Petkovšek, M.2
  • 3
    • 0042545136 scopus 로고    scopus 로고
    • Multibasic and mixed hypergeometric Gosper-type algorithms
    • Bauer A., and Petkovšek M. Multibasic and mixed hypergeometric Gosper-type algorithms. J.~Symbolic Comput. 28 4-5 (1999) 711-736
    • (1999) J.~Symbolic Comput. , vol.28 , Issue.4-5 , pp. 711-736
    • Bauer, A.1    Petkovšek, M.2
  • 4
    • 42649088797 scopus 로고    scopus 로고
    • Bierenbaum, I., Blümlein, J., Klein, S., Schneider, C., 2007. Difference equations in massive higher order calculations, in: Proc. ACAT 2007, volume PoS(ACAT)082
    • Bierenbaum, I., Blümlein, J., Klein, S., Schneider, C., 2007. Difference equations in massive higher order calculations, in: Proc. ACAT 2007, volume PoS(ACAT)082
  • 5
    • 5444238426 scopus 로고    scopus 로고
    • Algebraic relations between harmonic sums and associated quantities
    • Blümlein J. Algebraic relations between harmonic sums and associated quantities. Comput. Phys. Commun. 159 4 (2004) 19-54
    • (2004) Comput. Phys. Commun. , vol.159 , Issue.4 , pp. 19-54
    • Blümlein, J.1
  • 6
    • 17044378499 scopus 로고    scopus 로고
    • Harmonic sums and Mellin transforms up to two-loop order
    • Blümlein J., and Kurth S. Harmonic sums and Mellin transforms up to two-loop order. Phys. Rev. D60 (1999)
    • (1999) Phys. Rev. , vol.D60
    • Blümlein, J.1    Kurth, S.2
  • 7
    • 0041684683 scopus 로고    scopus 로고
    • On solutions of linear ordinary difference equations in their coefficient field
    • Bronstein M. On solutions of linear ordinary difference equations in their coefficient field. J.~Symbolic Comput. 29 6 (2000) 841-877
    • (2000) J.~Symbolic Comput. , vol.29 , Issue.6 , pp. 841-877
    • Bronstein, M.1
  • 8
    • 0345755649 scopus 로고    scopus 로고
    • An extension of Zeilberger's fast algorithm to general holonomic functions
    • Chyzak F. An extension of Zeilberger's fast algorithm to general holonomic functions. Discrete Math. 217 (2000) 115-134
    • (2000) Discrete Math. , vol.217 , pp. 115-134
    • Chyzak, F.1
  • 9
    • 33751501378 scopus 로고    scopus 로고
    • Padé approximations to the logarithm~III: Alternative methods and additional results
    • Driver K., Prodinger H., Schneider C., and Weideman J. Padé approximations to the logarithm~III: Alternative methods and additional results. Ramanujan~J. 12 3 (2006) 299-314
    • (2006) Ramanujan~J. , vol.12 , Issue.3 , pp. 299-314
    • Driver, K.1    Prodinger, H.2    Schneider, C.3    Weideman, J.4
  • 10
    • 0038840100 scopus 로고
    • Decision procedures for indefinite hypergeometric summation
    • Gosper R. Decision procedures for indefinite hypergeometric summation. Proc. Natl. Acad. Sci. USA 75 (1978) 40-42
    • (1978) Proc. Natl. Acad. Sci. USA , vol.75 , pp. 40-42
    • Gosper, R.1
  • 11
    • 0033092922 scopus 로고    scopus 로고
    • Solving difference equations in finite terms
    • Hendriks P., and Singer M. Solving difference equations in finite terms. J.~Symbolic Comput. 27 3 (1999) 239-259
    • (1999) J.~Symbolic Comput. , vol.27 , Issue.3 , pp. 239-259
    • Hendriks, P.1    Singer, M.2
  • 12
    • 84976849768 scopus 로고
    • Summation in finite terms
    • Karr M. Summation in finite terms. J.~ACM 28 (1981) 305-350
    • (1981) J.~ACM , vol.28 , pp. 305-350
    • Karr, M.1
  • 13
    • 84984634412 scopus 로고
    • Theory of summation in finite terms
    • Karr M. Theory of summation in finite terms. J.~Symbolic Comput. 1 (1985) 303-315
    • (1985) J.~Symbolic Comput. , vol.1 , pp. 303-315
    • Karr, M.1
  • 14
    • 33748984412 scopus 로고    scopus 로고
    • Application of unspecified sequences in symbolic summation
    • Dumas J. (Ed), ACM Press
    • Kauers M., and Schneider C. Application of unspecified sequences in symbolic summation. In: Dumas J. (Ed). Proc. ISSAC'06 (2006), ACM Press 177-183
    • (2006) Proc. ISSAC'06 , pp. 177-183
    • Kauers, M.1    Schneider, C.2
  • 15
    • 33746899050 scopus 로고    scopus 로고
    • Indefinite summation with unspecified summands
    • Kauers M., and Schneider C. Indefinite summation with unspecified summands. Discrete Math. 306 17 (2006) 2021-2140
    • (2006) Discrete Math. , vol.306 , Issue.17 , pp. 2021-2140
    • Kauers, M.1    Schneider, C.2
  • 16
    • 36349032286 scopus 로고    scopus 로고
    • Symbolic summation with radical expressions
    • Brown C. (Ed)
    • Kauers M., and Schneider C. Symbolic summation with radical expressions. In: Brown C. (Ed). Proc. ISSAC'07 (2007) 219-226
    • (2007) Proc. ISSAC'07 , pp. 219-226
    • Kauers, M.1    Schneider, C.2
  • 17
    • 42649139622 scopus 로고    scopus 로고
    • Kuba, M., Prodinger, H., Schneider, C., 2005. Generalized reciprocity laws for sums of harmonic numbers. SFB-Report 2005-17, J. Kepler University Linz
    • Kuba, M., Prodinger, H., Schneider, C., 2005. Generalized reciprocity laws for sums of harmonic numbers. SFB-Report 2005-17, J. Kepler University Linz
  • 18
    • 42649097299 scopus 로고    scopus 로고
    • Moch, S., Schneider, C., 2007. Feynman integrals and difference equations, in: Proc. ACAT 2007, volume PoS(ACAT)083
    • Moch, S., Schneider, C., 2007. Feynman integrals and difference equations, in: Proc. ACAT 2007, volume PoS(ACAT)083
  • 19
    • 42649103348 scopus 로고    scopus 로고
    • Osburn, R., Schneider, C., Gaussian hypergeometric series and extensions of supercongruences. Math. Comp., 2008 (in press)
    • Osburn, R., Schneider, C., Gaussian hypergeometric series and extensions of supercongruences. Math. Comp., 2008 (in press)
  • 20
    • 0001511724 scopus 로고
    • Greatest factorial factorization and symbolic summation
    • Paule P. Greatest factorial factorization and symbolic summation. J.~Symbolic Comput. 20 3 (1995) 235-268
    • (1995) J.~Symbolic Comput. , vol.20 , Issue.3 , pp. 235-268
    • Paule, P.1
  • 21
    • 0001487910 scopus 로고    scopus 로고
    • A Mathematica q-analogue of Zeilberger's algorithm based on an algebraically motivated aproach to q-hypergeometric telescoping
    • Ismail M., and Rahman M. (Eds), Fields Institute Toronto, AMS
    • Paule P., and Riese A. A Mathematica q-analogue of Zeilberger's algorithm based on an algebraically motivated aproach to q-hypergeometric telescoping. In: Ismail M., and Rahman M. (Eds). Special Functions, q-Series and Related Topics vol. 14 (1997), Fields Institute Toronto, AMS 179-210
    • (1997) Special Functions, q-Series and Related Topics , vol.14 , pp. 179-210
    • Paule, P.1    Riese, A.2
  • 22
    • 0141838153 scopus 로고    scopus 로고
    • Computer proofs of a new family of harmonic number identities
    • Paule P., and Schneider C. Computer proofs of a new family of harmonic number identities. Adv. in Appl. Math. 31 2 (2003) 359-378
    • (2003) Adv. in Appl. Math. , vol.31 , Issue.2 , pp. 359-378
    • Paule, P.1    Schneider, C.2
  • 24
    • 34247505199 scopus 로고    scopus 로고
    • When is 0.999... equal to 1?
    • Pemantle R., and Schneider C. When is 0.999... equal to 1?. Amer. Math. Monthly 114 4 (2007) 344-350
    • (2007) Amer. Math. Monthly , vol.114 , Issue.4 , pp. 344-350
    • Pemantle, R.1    Schneider, C.2
  • 25
    • 84951900479 scopus 로고
    • The problem of integration in finite terms
    • Risch R. The problem of integration in finite terms. Trans. Amer. Math. Soc. 139 (1969) 167-189
    • (1969) Trans. Amer. Math. Soc. , vol.139 , pp. 167-189
    • Risch, R.1
  • 26
    • 84966223571 scopus 로고
    • The solution to the problem of integration in finite terms
    • Risch R. The solution to the problem of integration in finite terms. Bull. Amer. Math. Soc. 76 (1970) 605-608
    • (1970) Bull. Amer. Math. Soc. , vol.76 , pp. 605-608
    • Risch, R.1
  • 27
    • 42649089514 scopus 로고    scopus 로고
    • Schneider, C., 2001. Symbolic summation in difference fields. Technical Report 01-17, RISC-Linz, J. Kepler University, Ph.D. Thesis
    • Schneider, C., 2001. Symbolic summation in difference fields. Technical Report 01-17, RISC-Linz, J. Kepler University, Ph.D. Thesis
  • 28
    • 15844368051 scopus 로고    scopus 로고
    • The summation package Sigma: Underlying principles and a rhombus tiling application
    • Schneider C. The summation package Sigma: Underlying principles and a rhombus tiling application. Discrete Math. Theor. Comput. Sci. 6 2 (2004) 365-386
    • (2004) Discrete Math. Theor. Comput. Sci. , vol.6 , Issue.2 , pp. 365-386
    • Schneider, C.1
  • 29
    • 27644579000 scopus 로고    scopus 로고
    • Symbolic summation with single-nested sum extensions
    • Gutierrez J. (Ed), ACM Press
    • Schneider C. Symbolic summation with single-nested sum extensions. In: Gutierrez J. (Ed). Proc. ISSAC'04 (2004), ACM Press 282-289
    • (2004) Proc. ISSAC'04 , pp. 282-289
    • Schneider, C.1
  • 30
    • 33749560321 scopus 로고    scopus 로고
    • Finding telescopers with minimal depth for indefinite nested sum and product expressions
    • Kauers M. (Ed), ACM
    • Schneider C. Finding telescopers with minimal depth for indefinite nested sum and product expressions. In: Kauers M. (Ed). Proc. ISSAC'05 (2005), ACM 285-292
    • (2005) Proc. ISSAC'05 , pp. 285-292
    • Schneider, C.1
  • 32
    • 15844383473 scopus 로고    scopus 로고
    • Product representations in Π Σ-fields
    • Schneider C. Product representations in Π Σ-fields. Ann. Comb. 9 1 (2005) 75-99
    • (2005) Ann. Comb. , vol.9 , Issue.1 , pp. 75-99
    • Schneider, C.1
  • 33
    • 24344486864 scopus 로고    scopus 로고
    • Solving parameterized linear difference equations in terms of indefinite nested sums and products
    • Schneider C. Solving parameterized linear difference equations in terms of indefinite nested sums and products. J. Difference Equ. Appl. 11 9 (2005) 799-821
    • (2005) J. Difference Equ. Appl. , vol.11 , Issue.9 , pp. 799-821
    • Schneider, C.1
  • 34
    • 42649098957 scopus 로고    scopus 로고
    • Simplifying Sums in Π Σ-Extensions
    • Schneider C. Simplifying Sums in Π Σ-Extensions. J. Algebra Appl. 6 3 (2007) 415-441
    • (2007) J. Algebra Appl. , vol.6 , Issue.3 , pp. 415-441
    • Schneider, C.1
  • 35
    • 33846672625 scopus 로고    scopus 로고
    • Symbolic summation assists combinatorics
    • Article B56b
    • Schneider C. Symbolic summation assists combinatorics. Sém.~Lothar. Combin. 56 (2007) 1-36 Article B56b
    • (2007) Sém.~Lothar. Combin. , vol.56 , pp. 1-36
    • Schneider, C.1
  • 36
    • 42649108098 scopus 로고    scopus 로고
    • Schneider, C., 2007c. Symbolic summation finds optimal nested sum representations. SFB-Report 2007-26, SFB F013, J. Kepler University Linz
    • Schneider, C., 2007c. Symbolic summation finds optimal nested sum representations. SFB-Report 2007-26, SFB F013, J. Kepler University Linz
  • 37
    • 42649139966 scopus 로고    scopus 로고
    • Schneider, C., 2008. Parameterized telescoping proves algebraic independence of sums. Ann. Comb. (in press)
    • Schneider, C., 2008. Parameterized telescoping proves algebraic independence of sums. Ann. Comb. (in press)
  • 38
    • 3242701424 scopus 로고
    • An extension of Liouville's theorem on integration in finite terms
    • Singer M., Saunders B., and Caviness B. An extension of Liouville's theorem on integration in finite terms. SIAM J. Comput. 14 4 (1985) 966-990
    • (1985) SIAM J. Comput. , vol.14 , Issue.4 , pp. 966-990
    • Singer, M.1    Saunders, B.2    Caviness, B.3
  • 39
    • 0001764998 scopus 로고    scopus 로고
    • Harmonic sums, Mellin transforms and integrals
    • Vermaseren J. Harmonic sums, Mellin transforms and integrals. Internat. J. Modern Phys. A14 (1999) 2037-2976
    • (1999) Internat. J. Modern Phys. , vol.A14 , pp. 2037-2976
    • Vermaseren, J.1
  • 40
    • 84891291550 scopus 로고    scopus 로고
    • The third-order qcd corrections to deep-inelastic scattering by photon exchange
    • Vermaseren J., Vogt A., and Moch S. The third-order qcd corrections to deep-inelastic scattering by photon exchange. Nuclear Phys. B724 (2005) 3-182
    • (2005) Nuclear Phys. , vol.B724 , pp. 3-182
    • Vermaseren, J.1    Vogt, A.2    Moch, S.3
  • 41
    • 73949145301 scopus 로고
    • The method of creative telescoping
    • Zeilberger D. The method of creative telescoping. J.~Symbolic Comput. 11 (1991) 195-204
    • (1991) J.~Symbolic Comput. , vol.11 , pp. 195-204
    • Zeilberger, D.1


* 이 정보는 Elsevier사의 SCOPUS DB에서 KISTI가 분석하여 추출한 것입니다.