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How to Generate All Possible Rational Wilf-Zeilberger Pairs?

  • Shaoshi ChenEmail author
Conference paper
First Online:
Part of the Fields Institute Communications book series (FIC, volume 82)

Abstract

A Wilf–Zeilberger pair (F, G) in the discrete case satisfies the equation
$$\displaystyle F(n+1, k) - F(n, k) = G(n, k+1) - G(n, k). $$
We present a structural description of all possible rational Wilf–Zeilberger pairs and their continuous and mixed analogues.
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Notes

Acknowledgements

I would like to thank Prof. Victor J.W. Guo and Prof. Zhi-Wei Sun for many discussions on series for special constants, (super)-congruences and their q-analogues that can be proved using the WZ method. I am also very grateful to Ruyong Feng and Rong-Hua Wang for many constructive comments on the earlier version of this paper. I also thank the anonymous reviewers for their constructive and detailed comments.This work was supported by the NSFC grants 11501552, 11688101 and by the Frontier Key Project (QYZDJ-SSW-SYS022) and the Fund of the Youth Innovation Promotion Association, CAS.

References

  1. 1.
    Abramov SA (1975) The rational component of the solution of a first order linear recurrence relation with rational right hand side. Ž Vyčisl Mat i Mat Fiz 15(4):1035–1039, 1090Google Scholar
  2. 2.
    Abramov SA (1995) Indefinite sums of rational functions. In: ISSAC ’95: proceedings of the 1995 international symposium on symbolic and algebraic computation. ACM, New York, NY, pp 303–308Google Scholar
  3. 3.
    Abramov SA (2003) When does Zeilberger’s algorithm succeed? Adv Appl Math 30(3):424–441MathSciNetzbMATHGoogle Scholar
  4. 4.
    Abramov SA, Petkovšek M (2002) On the structure of multivariate hypergeometric terms. Adv Appl Math 29(3):386–411MathSciNetzbMATHGoogle Scholar
  5. 5.
    Amdeberhan T (1996) Faster and faster convergent series for ζ(3). Electron J Combin 3(1):Research Paper 13, approx. 2Google Scholar
  6. 6.
    Bailey DH, Borwein JM, Bradley DM (2006) Experimental determination of Apéry-like identities for ζ(2n + 2). Exp Math 15(3):281–289zbMATHGoogle Scholar
  7. 7.
    Baruah ND, Berndt BC, Chan HH (2009) Ramanujan’s series for 1∕π: a survey. Am Math Mon 116(7):567–587MathSciNetzbMATHGoogle Scholar
  8. 8.
    Bauer GC (1859) Von den Coefficienten der Reihen von Kugelfunctionen einer Variablen. J Reine Angew Math 56:101–121MathSciNetGoogle Scholar
  9. 9.
    Borwein J, Bailey D, Girgensohn R (2004) Experimentation in mathematics: computational paths to discovery. A K Peters/CRC Press, Natick, MA/Boca Raton, FLzbMATHGoogle Scholar
  10. 10.
    Bostan A, Kauers M (2010) The complete generating function for Gessel walks is algebraic. Proc Am Math Soc 138(9):3063–3078MathSciNetzbMATHGoogle Scholar
  11. 11.
    Bronstein M (2005) Symbolic integration I: transcendental functions, 2nd edn. Springer, BerlinzbMATHGoogle Scholar
  12. 12.
    Chen S (2011) Some applications of differential-difference algebra to creative telescoping. PhD Thesis, Ecole Polytechnique LIXGoogle Scholar
  13. 13.
    Chen S, Singer MF (2012) Residues and telescopers for bivariate rational functions. Adv Appl Math 49(2):111–133MathSciNetzbMATHGoogle Scholar
  14. 14.
    Chen S, Singer MF (2014) On the summability of bivariate rational functions. J Algebra 409:320–343MathSciNetzbMATHGoogle Scholar
  15. 15.
    Chen WYC, Hou Q-H, Mu Y-P (2005) Applicability of the q-analogue of Zeilberger’s algorithm. J Symb Comput 39(2):155–170MathSciNetzbMATHGoogle Scholar
  16. 16.
    Chen S, Feng R, Fu G, Kang J (2012) Multiplicative decompositions of multivariate q-hypergeometric terms. J Syst Sci Math Sci 32(8):1019–1032MathSciNetzbMATHGoogle Scholar
  17. 17.
    Chen S, Chyzak F, Feng R, Fu G, Li Z (2015) On the existence of telescopers for mixed hypergeometric terms. J Symb Comput 68(part 1):1–26MathSciNetzbMATHGoogle Scholar
  18. 18.
    Chen WYC, Hou Q-H, Zeilberger D (2016) Automated discovery and proof of congruence theorems for partial sums of combinatorial sequences. J Differ Equ Appl 22(6):780–788MathSciNetzbMATHGoogle Scholar
  19. 19.
    Christopher C (1999) Liouvillian first integrals of second order polynomial differential equations. Electron J Differ Equ 49:1–7MathSciNetzbMATHGoogle Scholar
  20. 20.
    Dreyfus T, Hardouin C, Roques J, Singer MF (2018) On the nature of the generating series of walks in the quarter plane. Invent Math 213(1):139–203. https://doi.org/10.1007/s00222-018-0787-z MathSciNetzbMATHGoogle Scholar
  21. 21.
    Ekhad SB, Zeilberger D (1994) A WZ proof of Ramanujan’s formula for π. In: Geometry, Analysis and Mechanics. World Scientific Publishing, River Edge, NJ, pp 107–108zbMATHGoogle Scholar
  22. 22.
    Gessel IM (1995) Finding identities with the WZ method. J. Symb. Comput. 20(5–6):537–566MathSciNetzbMATHGoogle Scholar
  23. 23.
    Guillera J (2002) Some binomial series obtained by the WZ-method. Adv Appl Math 29(4):599–603MathSciNetzbMATHGoogle Scholar
  24. 24.
    Guillera J (2006) Generators of some Ramanujan formulas. Ramanujan J 11(1):41–48MathSciNetzbMATHGoogle Scholar
  25. 25.
    Guillera J (2008) Hypergeometric identities for 10 extended Ramanujan-type series. Ramanujan J 15(2):219–234MathSciNetzbMATHGoogle Scholar
  26. 26.
    Guillera J (2010) On WZ-pairs which prove Ramanujan series. Ramanujan J 22(3):249–259MathSciNetzbMATHGoogle Scholar
  27. 27.
    Guillera J (2013) WZ-proofs of “divergent” Ramanujan-type series. In: Advances in combinatorics. Springer, Heidelberg, pp 187–195Google Scholar
  28. 28.
    Guo VJW (2017) Some generalizations of a supercongruence of van Hamme. Integr Transforms Spec Funct (1):1–12MathSciNetzbMATHGoogle Scholar
  29. 29.
    Guo VJW (2018) A q-analogue of a Ramanujan-type supercongruence involving central binomial coefficients. J Math Anal Appl 458(1):590–600MathSciNetzbMATHGoogle Scholar
  30. 30.
    Guo VJW (2018) A q-analogues of the (J.2) supercongruence of van Hamme. J Math Anal Appl 466(1):776–788MathSciNetzbMATHGoogle Scholar
  31. 31.
    Guo VJW, Liu J-C (2018) q-analogues of two Ramanujan-type formulas for 1∕π. J Differ Equ Appl 24(8):1368–1373MathSciNetzbMATHGoogle Scholar
  32. 32.
    Guo VJW, Zudilin W (2018) Ramanujan-type formulae for 1∕π: q-analogues. Integral Transforms Spec Funct 29(7):505–513. https://doi.org/10.1080/10652469.2018.1454448 MathSciNetzbMATHGoogle Scholar
  33. 33.
    Hermite C (1872) Sur l’intégration des fractions rationnelles. Ann Sci École Norm Sup (2) 1:215–218MathSciNetzbMATHGoogle Scholar
  34. 34.
    Hou Q-H, Wang R-H (2015) An algorithm for deciding the summability of bivariate rational functions. Adv Appl Math 64:31–49MathSciNetzbMATHGoogle Scholar
  35. 35.
    Hou Q-H, Krattenthaler C, Sun Z-W (2018) On q-analogues of some series for π and π 2. Proceedings of the American Mathematical Society. https://doi.org/10.1090/proc/14374 Google Scholar
  36. 36.
    Kauers M, Koutschan C, Zeilberger D (2009) Proof of Ira Gessel’s lattice path conjecture. Proc Natl Acad Sci U S A 106(28):11502–11505MathSciNetzbMATHGoogle Scholar
  37. 37.
    Kondratieva M, Sadov S (2005) Markov’s transformation of series and the WZ method. Adv Appl Math 34(2):393–407MathSciNetzbMATHGoogle Scholar
  38. 38.
    Koutschan C, Kauers M, Zeilberger D (2011) Proof of George Andrews’s and David Robbins’s q-TSPP conjecture. Proc Natl Acad Sci U S A 108(6):2196–2199MathSciNetzbMATHGoogle Scholar
  39. 39.
    Liu Z-G (2012) Gauss summation and Ramanujan-type series for 1∕π. Int J Number Theory 08(02):289–297MathSciNetzbMATHGoogle Scholar
  40. 40.
    Liu Z-G (2015) A q-extension of a partial differential equation and the Hahn polynomials. Ramanujan J 38(3):481–501MathSciNetzbMATHGoogle Scholar
  41. 41.
    Long L (2011) Hypergeometric evaluation identities and supercongruences. Pac J Math 249(2):405–418MathSciNetzbMATHGoogle Scholar
  42. 42.
    Pilehrood KH, Pilehrood TH (2008a) Simultaneous generation for zeta values by the Markov-WZ method. Discrete Math Theor Comput Sci 10(3):115–123MathSciNetzbMATHGoogle Scholar
  43. 43.
    Pilehrood KH, Pilehrood TH (2008b) Generating function identities for ζ(2n + 2), ζ(2n + 3) via the WZ method. Electron J Combin 15(1):Research Paper 35, 9Google Scholar
  44. 44.
    Pilehrood KH, Pilehrood TH (2011) A q-analogue of the Bailey-Borwein-Bradley identity. J Symb Comput 46(6):699–711MathSciNetzbMATHGoogle Scholar
  45. 45.
    Mohammed M (2005) The q-Markov-WZ method. Ann Comb 9(2): 205–221MathSciNetzbMATHGoogle Scholar
  46. 46.
    Mohammed M, Zeilberger D (2004) The Markov-WZ method. Electron J Combin 11(1): 205–221MathSciNetzbMATHGoogle Scholar
  47. 47.
    Mortenson E (2008) A p-adic supercongruence conjecture of van Hamme. Proc Am Math Soc 136(12):4321–4328MathSciNetzbMATHGoogle Scholar
  48. 48.
    Ore O (1930) Sur la forme des fonctions hypergéométriques de plusieurs variables. J Math Pures Appl (9) 9(4):311–326zbMATHGoogle Scholar
  49. 49.
    Ostrogradskiı̆ MV (1845) De l’intégration des fractions rationnelles. Bull de la classe physico-mathématique de l’Acad Impériale des Sciences de Saint-Pétersbourg 4:145–167, 286–300Google Scholar
  50. 50.
    Petkovšek M, Wilf HS, Zeilberger D (1996) A = B. A K Peters Ltd., Wellesley, MA. With a foreword by Donald E. KnuthGoogle Scholar
  51. 51.
    Riordan J (1968) Combinatorial identities. Wiley, Hoboken, NJzbMATHGoogle Scholar
  52. 52.
    Sato M (1990) Theory of prehomogeneous vector spaces (algebraic part)—the English translation of Sato’s lecture from Shintani’s note. Nagoya Math J 120:1–34. Notes by Takuro Shintani. Translated from the Japanese by Masakazu MuroMathSciNetzbMATHGoogle Scholar
  53. 53.
    Sun Z-W (2011) Super congruences and Euler numbers. Sci China Math 54(12):2509–2535MathSciNetzbMATHGoogle Scholar
  54. 54.
    Sun X (2012) Some discussions on three kinds of WZ-equations. Master thesis, Soochow University, April 2012. Supervised by Xinrong MaGoogle Scholar
  55. 55.
    Sun Z-W (2012) A refinement of a congruence result by van Hamme and Mortenson. Ill J Math 56(3):967–979MathSciNetzbMATHGoogle Scholar
  56. 56.
    Sun Z-W (2013) Conjectures involving arithmetical sequences. In: Kanemitsu S, Li H, Liu J (eds) Number theory: arithmetic in Shangri-La, Proceedings of the 6th China-Japan Seminar (Shanghai, August 15–17, 2011). World Scientific Publishing, Singapore, pp 244–258Google Scholar
  57. 57.
    Sun Z-W (2013) Products and sums divisible by central binomial coefficients. Electron J Combin 20(1):91–109(19)MathSciNetGoogle Scholar
  58. 58.
    Sun Z-W (2018) Two q-analogues of Euler’s formula ζ(2) = π 2∕6. Preprint. arXiv:arXiv:1802.01473Google Scholar
  59. 59.
    Tefera A (2010) What is … a Wilf-Zeilberger pair? Not Am Math Soc 57(4):508–509MathSciNetzbMATHGoogle Scholar
  60. 60.
    van Hamme L (1997) Some conjectures concerning partial sums of generalized hypergeometric series. In: p-adic functional analysis (Nijmegen, 1996). Lecture notes in pure and applied mathematics, vol 192. Dekker, New York, pp 223–236Google Scholar
  61. 61.
    Wilf HS, Zeilberger D (1992a) An algorithmic proof theory for hypergeometric (ordinary and “q”) multisum/integral identities. Invent Math 108(3):575–633MathSciNetzbMATHGoogle Scholar
  62. 62.
    Wilf HS, Zeilberger D (1992b) Rational function certification of multisum/integral/“q” identities. Bull Am Math Soc (N.S.) 27(1):148–153Google Scholar
  63. 63.
    Zeilberger D (1993) Closed form (pun intended!). In: A tribute to Emil Grosswald: number theory and related analysis. Contemporary mathematics, vol 143. American Mathematical Society, Providence, RI, pp 579–607Google Scholar
  64. 64.
    Zoladek H (1998) The extended monodromy group and Liouvillian first integrals. J Dynam Control Syst 4(1):1–28MathSciNetzbMATHGoogle Scholar
  65. 65.
    Zudilin W (2007) More Ramanujan-type formulas for 1∕π 2. Russ Math Surv 62(3):634–636MathSciNetzbMATHGoogle Scholar
  66. 66.
    Zudilin W (2009) Ramanujan-type supercongruences. J Number Theory 129(8):1848–1857MathSciNetzbMATHGoogle Scholar
  67. 67.
    Zudilin W (2011) Arithmetic hypergeometric series. Russ Math Surv 66(2):369–420MathSciNetzbMATHGoogle Scholar

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Authors and Affiliations

  1. 1.KLMM, Academy of Mathematics and Systems Science, Chinese Academy of SciencesBeijingChina
  2. 2.School of Mathematical SciencesUniversity of Chinese Academy of SciencesBeijingChina

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