Skip to main content
SpringerLink
Log in
Book cover

Computer Algebra in Quantum Field Theory pp 325–360Cite as

Simplifying Multiple Sums in Difference Fields

  • Chapter
  • First Online:

Part of the book series: Texts & Monographs in Symbolic Computation ((TEXTSMONOGR))

Abstract

In this survey article we present difference field algorithms for symbolic summation. Special emphasize is put on new aspects in how the summation problems are rephrased in terms of difference fields, how the problems are solved there, and how the derived results in the given difference field can be reinterpreted as solutions of the input problem. The algorithms are illustrated with the Mathematica package Sigma by discovering and proving new harmonic number identities extending those from Paule and Schneider, 2003. In addition, the newly developed package EvaluateMultiSums is introduced that combines the presented tools. In this way, large scale summation problems for the evaluation of Feynman diagrams in QCD (Quantum ChromoDynamics) can be solved completely automatically.

This is a preview of subscription content, log in via an institution.

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   84.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD   109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Notes

  1. 1.

    For identities (1) and (2) we point also to [29]; for their indefinite versions see (10) and (11) below.

  2. 2.

    \(\mathbb{Z}\) are the integers, \(\mathbb{N} =\{ 0, 1, 2,\ldots \}\) are the non-negative integers, and all fields (resp. rings) contain the rational numbers \(\mathbb{Q}\) as a subfield (resp. subring). For a set A we define A ∗:= A∖{0}.

  3. 3.

    The solution set \(V =\{ (c_{0},\ldots,c_{d},g) \in {\mathbb{K}}^{d+1} \times \mathbb{F}\vert \alpha _{1}\sigma (g) + \alpha _{0}g =\sum _{ i=0}^{d}c_{i}f_{i}\}\) forms a \(\mathbb{K}\)-vector space of dimension ≤ d + 2 and the task is to get an explicit basis of V.

  4. 4.

    The solution set \(V =\{ (c_{0},\ldots,c_{d},g) \in {\mathbb{K}}^{d+1} \times \mathbb{F}\;\vert \; \text{ holds}\}\) forms a \(\mathbb{K}\)-vector space of dimension ≤ m + d + 1 and the task is to get an explicit basis of V.

  5. 5.

    \(\mathbb{K}(x)[p_{1},p_{1}^{-1},\ldots,p_{r},p_{r}^{-1}]\) stands for the polynomial Laurent ring in the variables \(p_{1},\ldots,p_{r}\), i.e., an element is of the form \(\sum _{(i_{1},\ldots,i_{r})\in S}f_{(i_{1},\ldots,i_{r})}p_{1}^{i_{1}}\ldots p_{r}^{i_{r}}\) where \(f_{(i_{1},\ldots,i_{r})} \in \mathbb{K}(x)\) and \(S \subseteq {\mathbb{Z}}^{r}\) is finite.

  6. 6.

    As observed in Remark 2 one might need in addition the alternating sign to represent all hypergeometric products. The underlying solution works analogously by adapted algorithms.

  7. 7.

    \(L(m_{1},\ldots,m_{u},n)\), u ≥ 0, stands for a linear combination of the m i and n with integer coefficients.

  8. 8.

    For a rigorous verification the proof certificate \((c_{0},\ldots,c_{d},G(k))\) of (41) with d = 2 is returned with the function call CreativeTelescoping[mySum,n].

References

  1. Ablinger, J.: Computer algebra algorithms for special functions in particle physics. Ph.D. thesis, J. Kepler University Linz (2012)

    Google Scholar 

  2. Ablinger, J., Blümlein, J.: Harmonic sums, polylogarithms, special numbers, and their generalizations. In: Blümlein, J., Schneider, C. (eds.) Computer Algebra in Quantum Field Theory: Integration, Summation and Special Functions. Springer, Vienna (2013)

    Google Scholar 

  3. Ablinger, J., Blümlein, J., Klein, S., Schneider, C.: Modern summation methods and the computation of 2- and 3-loop Feynman diagrams. Nucl. Phys. B (Proc. Suppl.) 205–206, 110–115 (2010). ArXiv::1006.4797 (math-ph)

    Google Scholar 

  4. Ablinger, J., Blümlein, J., Klein, S., Schneider, C., Wissbrock, F.: The O(α s 3) massive operator matrix elements of O(n f ) for the structure function F 2(x, Q 2) and transversity. Nucl. Phys. B 844, 26–54 (2011). ArXiv:1008.3347 (hep-ph)

    Google Scholar 

  5. Ablinger, J., Blümlein, J., Schneider, C.: Harmonic sums and polylogarithms generated by cyclotomic polynomials. J. Math. Phys. 52(10), 1–52 (2011, to appear). arXiv:1007.0375 (hep-ph)

    Google Scholar 

  6. Ablinger, J., Blümlein, J., Freitas, A.D., Hasselhuhn, A., Klein, S., Raab, C., Round, M., Schneider, C., Wissbrock, F.: Three-loop contributions to the gluonic massive operator matrix elements at general values of N. In: Proceedings of the Loops and Legs in Quantum Field Theory 2012, PoS(LL2012)033, Wernigerode, pp. 1–12 (2012)

    Google Scholar 

  7. Ablinger, J., Blümlein, J., Hasselhuhn, A., Klein, S., Schneider, C., Wissbrock, F.: Massive 3-loop ladder diagrams for quarkonic local operator matrix elements. Nucl. Phys. B 864, 52–84 (2012). ArXiv:1206.2252v1 (hep-ph)

    Google Scholar 

  8. Ablinger, J., Blümlein, J., Round, M., Schneider, C.: Advanced computer algebra algorithms for the expansion of Feynman integrals. In: Loops and Legs in Quantum Field Theory 2012, PoS(2012), Wernigerode, pp. 1–14 (2012)

    Google Scholar 

  9. Ablinger, J., Blümlein, J., Schneider, C.: Analytic and algorithmic aspects of generalized harmonic sums and polylogarithms (2013). ArXiv:1302.0378 (math-ph)

    Google Scholar 

  10. Ablinger, J., Blümlein, J., Schneider, C.: Structural Relations of Harmonic Sums (2013, In preparation)

    Google Scholar 

  11. Abramov, S.A.: On the summation of rational functions. Zh. vychisl. mat. Fiz. 11, 1071–1074 (1971)

    MathSciNet  MATH  Google Scholar 

  12. Abramov, S., Petkovšek, M.: D’Alembertian solutions of linear differential and difference equations. In: von zur Gathen, J. (ed.) Proceedings of the ISSAC’94, Oxford, pp. 169–174. ACM (1994)

    Google Scholar 

  13. Abramov, S., Petkovšek, M.: Polynomial ring automorphisms, rational (w,σ)-canonical forms, and the assignment problem. J. Symb. Comput. 45(6), 684–708 (2010)

    Google Scholar 

  14. Abramov, S., Bronstein, M., Petkovšek, M., Schneider, C.: (2012, In preparation)

    Google Scholar 

  15. Apagodu, M., Zeilberger, D.: Multi-variable Zeilberger and Almkvist-Zeilberger algorithms and the sharpening of Wilf-Zeilberger theory. Adv. Appl. Math. 37, 139–152 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  16. Bauer, A., Petkovšek, M.: Multibasic and mixed hypergeometric Gosper-type algorithms. J. Symb. Comput. 28(4–5), 711–736 (1999)

    Article  MATH  Google Scholar 

  17. Bierenbaum, I., Blümlein, J., Klein, S., Schneider, C.: Two–loop massive operator matrix elements for unpolarized heavy flavor production to o(epsilon). Nucl.Phys. B 803(1–2), 1–41 (2008). arXiv:hep-ph/0803.0273

    Google Scholar 

  18. Blümlein, J.: Algebraic relations between harmonic sums and associated quantities. Comput. Phys. Commun. 159(1), 19–54 (2004). arXiv:hep-ph/0311046

    Google Scholar 

  19. Blümlein, J.: Structural relations of harmonic sums and Mellin transforms at weight w =6. In: Carey, A., Ellwood, D., Paycha, S., Rosenberg, S. (eds.) Motives, Quantum Field Theory, and Pseudodifferential Operators. Clay Mathematics Proceedings, vol. 12, pp. 167–186 (2010). DESY 08–206, SFB-CPP/09–002, arXiv:0901.0837 (math-ph)

    Google Scholar 

  20. Blümlein, J., Kurth, S.: Harmonic sums and Mellin transforms up to two-loop order. Phys. Rev. D60, 014018 (31p) (1999)

    Google Scholar 

  21. Blümlein, J., Kauers, M., Klein, S., Schneider, C.: Determining the closed forms of the O(a s 3) anomalous dimensions and Wilson coefficients from Mellin moments by means of computer algebra. Comput. Phys. Commun. 180, 2143–2165 (2009)

    Article  ADS  MATH  Google Scholar 

  22. Blümlein, J., Broadhurst, D.J., Vermaseren, J.A.M.: The multiple zeta value data mine. Comput. Phys. Commun. 181, 582–625 (2010). arXiv:0907.2557 (math-ph)

    Google Scholar 

  23. Blümlein, J., Hasselhuhn, A., Schneider, C.: Evaluation of multi-sums for large scale problems. In: Proceedings of RADCOR 2011 PoS(RADCOR2011), Durham, vol. 32, pp. 1–9 (2012)

    Google Scholar 

  24. Blümlein, J., Klein, S., Schneider, C., Stan, F.: A symbolic summation approach to Feynman integral calculus. J. Symb. Comput. 47, 1267–1289 (2012)

    Article  MATH  Google Scholar 

  25. Blümlein, J., Hasselhuhn, A., Klein, S., Schneider, C.: The \(O(\alpha _{s}^{3}n_{f}T_{F}^{2}C_{A,F})\) contributions to the gluonic massive operator matrix elements. Nucl. Phys. B 866, 196–211 (2013)

    Google Scholar 

  26. Bronstein, M.: On solutions of linear ordinary difference equations in their coefficient field. J. Symb. Comput. 29(6), 841–877 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  27. Brown, F.: The massless higher-loop two-point function. Math. Phys. 287, 925–958 (2009)

    Article  ADS  MATH  Google Scholar 

  28. Chen, S., Kauers, M.: Order-degree curves for hypergeometric creative telescoping. In: van der Hoeven, J., van Hoeij, M. (eds.) Proceedings of ISSAC 2012, Grenoble, pp. 122–129 (2012)

    Google Scholar 

  29. Chu, W., Donno, L.D.: Hypergeometric series and harmonic number identities. Adv. Appl. Math. 34(1), 123–137 (2005)

    Article  MATH  Google Scholar 

  30. Chyzak, F.: An extension of Zeilberger’s fast algorithm to general holonomic functions. Discret. Math. 217, 115–134 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  31. Cohn, R.: Difference Algebra. Interscience/Wiley, New York (1965)

    MATH  Google Scholar 

  32. Driver, K., Prodinger, H., Schneider, C., Weideman, J.: Padé approximations to the logarithm III: alternative methods and additional results. Ramanujan J. 12(3), 299–314 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  33. Eröcal, B.: Algebraic extensions for summation in finite terms. Ph.D. thesis, RISC, Johannes Kepler University, Linz (2011)

    Google Scholar 

  34. Gosper, R.: Decision procedures for indefinite hypergeometric summation. Proc. Natl. Acad. Sci. U.S.A. 75, 40–42 (1978)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  35. Hardouin, C., Singer, M.: Differential Galois theory of linear difference equations. Math. Ann. 342(2), 333–377 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  36. Hendriks, P., Singer, M.: Solving difference equations in finite terms. J. Symb. Comput. 27(3), 239–259 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  37. Hoeij, M.: Finite singularities and hypergeometric solutions of linear recurrence equations. J. Pure Appl. Algebra 139(1–3), 109–131 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  38. Horn, P., Koepf, W., Sprenger, T.: m-fold hypergeometric solutions of linear recurrence equations revisited. Math. Comput. Sci. 6(1), 61–77 (2012)

    Google Scholar 

  39. Karr, M.: Summation in finite terms. J. ACM 28, 305–350 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  40. Karr, M.: Theory of summation in finite terms. J. Symb. Comput. 1, 303–315 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  41. Kauers, M.: The holonomic toolkit. In: Blümlein, J., Schneider, C. (eds.) Computer Algebra in Quantum Field Theory: Integration, Summation and Special Functions. Springer, Vienna (2013)

    Google Scholar 

  42. Kauers, M., Zimmermann, B.: Computing the algebraic relations of C-finite sequences and multisequences. J. Symb. Comput. 43(11), 787–803 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  43. Koornwinder, T.: On Zeilberger’s algorithm and its q-analogue. J. Comput. Appl. Math. 48, 91–111 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  44. Koutschan, C.: A fast approach to creative telescoping. Math. Comput. Sci. 4(2–3), 259–266 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  45. Krattenthaler, C., Rivoal, T.: Hypergéométrie et fonction zêta de Riemann. Mem. Am. Math. Soc. 186(875) (2007)

    Google Scholar 

  46. Levin, A.: Difference Algebra. Algebra and Applications, vol. 8. Springer, New York (2008)

    Google Scholar 

  47. Moch, S., Uwer, P., Weinzierl, S.: Nested sums, expansion of transcendental functions, and multiscale multiloop integrals. J. Math. Phys. 6, 3363–3386 (2002)

    Article  MathSciNet  ADS  Google Scholar 

  48. Moch, S., Vermaseren, J., Vogt, A.: The three-loop splitting functions in QCD: the non-singlet case. Nucl. Phys. B 688, 101 (2004). ArXiv:hep-ph/0403192v1

    Google Scholar 

  49. Nemes, I., Paule, P.: A canonical form guide to symbolic summation. In: Miola, A., Temperini, M. (eds.) Advances in the Design of Symbolic Computation Systems. Texts and Monographs in Symbolic Computation, pp. 84–110. Springer, Wien/New York (1997)

    Google Scholar 

  50. Nørlund, N.E.: Vorlesungen über Differenzenrechnung. Springer, Berlin (1924)

    Book  Google Scholar 

  51. Osburn, R., Schneider, C.: Gaussian hypergeometric series and extensions of supercongruences. Math. Comput. 78(265), 275–292 (2009)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  52. Paule, P.: Greatest factorial factorization and symbolic summation. J. Symb. Comput. 20(3), 235–268 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  53. Paule, P., Riese, A.: A Mathematica q-analogue of Zeilberger’s algorithm based on an algebraically motivated aproach to q-hypergeometric telescoping. In: Ismail, M., Rahman, M. (eds.) Special Functions, q-Series and Related Topics. Fields Institute Communications, vol. 14, pp. 179–210 AMS, Providence (1997)

    Google Scholar 

  54. Paule, P., Schneider, C.: Computer proofs of a new family of harmonic number identities. Adv. Appl. Math. 31(2), 359–378 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  55. Paule, P., Schorn, M.: A mathematica version of Zeilberger’s algorithm for proving binomial coefficient identities. J. Symb. Comput. 20(5–6), 673–698 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  56. Pemantle, R., Schneider, C.: When is 0.999…equal to 1? Am. Math. Mon. 114(4), 344–350 (2007)

    Google Scholar 

  57. Petkovšek, M.: Hypergeometric solutions of linear recurrences with polynomial coefficients. J. Symb. Comput. 14(2–3), 243–264 (1992)

    Article  MATH  Google Scholar 

  58. Petkovšek, M., Zakrajšek, H.: Solving linear recurrence equations with polynomial coefficients. In: Blümlein, J., Schneider, C. (eds.) Computer Algebra in Quantum Field Theory: Integration, Summation and Special Functions. Springer (2013, to appear)

    Google Scholar 

  59. Petkovšek, M., Wilf, H.S., Zeilberger, D.: A = B. A. K. Peters, Wellesley (1996)

    Google Scholar 

  60. Riese, A.: qMultisum – a package for proving q-hypergeometric multiple summation identities. J. Symb. Comput. 35, 349–377 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  61. Risch, R.: The problem of integration in finite terms. Trans. Am. Math. Soc. 139, 167–189 (1969)

    Article  MathSciNet  MATH  Google Scholar 

  62. Schneider, C.: Symbolic summation in difference fields, Technical Report 01–17. Ph.D. thesis, RISC-Linz, J. Kepler University (2001)

    Google Scholar 

  63. Schneider, C.: A collection of denominator bounds to solve parameterized linear difference equations in Π Σ-extensions. An. Univ. Timişoara Ser. Mat.-Inform. 42(2), 163–179 (2004)

    MATH  Google Scholar 

  64. Schneider, C.: The summation package Sigma: underlying principles and a rhombus tiling application. Discret. Math. Theor. Comput. Sci. 6, 365–386 (2004)

    MATH  Google Scholar 

  65. Schneider, C.: Symbolic Summation with Single-Nested Sum Extensions. In: Gutierrez, J. (ed.) Proceedings of the ISSAC’04, Santander, pp. 282–289. ACM (2004)

    Google Scholar 

  66. Schneider, C.: Degree bounds to find polynomial solutions of parameterized linear difference equations in Π Σ-fields. Appl. Algebra Eng. Commun. Comput. 16(1), 1–32 (2005)

    Article  MATH  Google Scholar 

  67. Schneider, C.: Finding telescopers with minimal depth for indefinite nested sum and product expressions. In: Kauers, M. (ed.) Proceedings of the ISSAC’05, Beijing, pp. 285–292. ACM (2005)

    Google Scholar 

  68. Schneider, C.: A new Sigma approach to multi-summation. Adv. Appl. Math. 34(4), 740–767 (2005)

    Article  MATH  Google Scholar 

  69. Schneider, C.: Product representations in Π Σ-fields. Ann. Comb. 9(1), 75–99 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  70. Schneider, C.: Solving parameterized linear difference equations in terms of indefinite nested sums and products. J. Differ. Equ. Appl. 11(9), 799–821 (2005)

    Article  MATH  Google Scholar 

  71. Schneider, C.: Apéry’s double sum is plain sailing indeed. Electron. J. Comb. 14 (2007)

    Google Scholar 

  72. Schneider, C.: Simplifying sums in Π Σ-extensions. J. Algebra Appl. 6(3), 415–441 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  73. Schneider, C.: Symbolic summation assists combinatorics. Sém. Lothar. Combin. 56, 1–36 (2007). Article B56b

    Google Scholar 

  74. Schneider, C.: A refined difference field theory for symbolic summation. J. Symb. Comput. 43(9), 611–644 (2008). arXiv:0808.2543v1

    Google Scholar 

  75. Schneider, C.: A symbolic summation approach to find optimal nested sum representations. In: Carey, A., Ellwood, D., Paycha, S., Rosenberg, S. (eds.) Motives, Quantum Field Theory, and Pseudodifferential Operators. Clay Mathematics Proceedings, vol. 12, pp. 285–308. American Mathematical Society (2010). ArXiv:0808.2543

    Google Scholar 

  76. Schneider, C.: Parameterized telescoping proves algebraic independence of sums. Ann. Comb. 14(4), 533–552 (2010). arXiv:0808.2596

    Google Scholar 

  77. Schneider, C.: Structural theorems for symbolic summation. Appl. Algebra Eng. Commun. Comput. 21(1), 1–32 (2010)

    Article  MATH  Google Scholar 

  78. Vermaseren, J.: Harmonic sums, Mellin transforms and integrals. Int. J. Mod. Phys. A14, 2037–2976 (1999)

    Article  MathSciNet  ADS  Google Scholar 

  79. Wegschaider, K.: Computer generated proofs of binomial multi-sum identities. Master’s thesis, RISC, J. Kepler University (1997)

    Google Scholar 

  80. Wilf, H., Zeilberger, D.: An algorithmic proof theory for hypergeometric (ordinary and “q”) multisum/integral identities. Invent. Math. 108, 575–633 (1992)

    Article  MathSciNet  ADS  Google Scholar 

  81. Zeilberger, D.: A holonomic systems approach to special functions identities. J. Comput. Appl. Math. 32, 321–368 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  82. Zeilberger, D.: The method of creative telescoping. J. Symb. Comput. 11, 195–204 (1991)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

Supported by the Austrian Science Fund (FWF) grants P20347-N18 and SFB F50 (F5009-N15) and by the EU Network LHCPhenoNet PITN-GA-2010-264564.

Author information

Authors and Affiliations

  1. Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz, Linz, Austria

    Carsten Schneider

Authors
  1. Carsten Schneider
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Carsten Schneider .

Editor information

Editors and Affiliations

  1. RISC, Johannes Kepler University, Linz, Austria

    Carsten Schneider

  2. DESY, Zeuthen, Germany

    Johannes Blümlein

Rights and permissions

Reprints and permissions

Copyright information

© 2013 Springer-Verlag Wien

About this chapter

Cite this chapter

Schneider, C. (2013). Simplifying Multiple Sums in Difference Fields. In: Schneider, C., Blümlein, J. (eds) Computer Algebra in Quantum Field Theory. Texts & Monographs in Symbolic Computation. Springer, Vienna. https://doi.org/10.1007/978-3-7091-1616-6_14

Download citation

  • DOI: https://doi.org/10.1007/978-3-7091-1616-6_14

  • Published:

  • Publisher Name: Springer, Vienna

  • Print ISBN: 978-3-7091-1615-9

  • Online ISBN: 978-3-7091-1616-6

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation