Creative Telescoping for Holonomic Functions

  • Christoph Koutschan
Part of the Texts & Monographs in Symbolic Computation book series (TEXTSMONOGR)


The aim of this article is twofold: on the one hand it is intended to serve as a gentle introduction to the topic of creative telescoping, from a practical point of view; for this purpose its application to several problems is exemplified. On the other hand, this chapter has the flavour of a survey article: the developments in this area during the last two decades are sketched and a selection of references is compiled in order to highlight the impact of creative telescoping in numerous contexts.


Left Ideal Computer Algebra System Closure Property Finite Function Holonomic Function 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


  1. 1.
    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(0), 110–115 (2010)CrossRefGoogle Scholar
  2. 2.
    Abramov, S.A.: When does Zeilberger’s algorithm succeed? Adv. Appl. Math. 30, 424–441 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Abramov, S.A., Le, H.Q.: A criterion for the applicability of Zeilberger’s algorithm to rational functions. Discrete Math. 259(1–3), 1–17 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Almkvist, G., Zeilberger, D.: The method of differentiating under the integral sign. J. Symb. Comput. 10(6), 571–591 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Amdeberhan, T., Zeilberger, D.: Hypergeometric series acceleration via the WZ method. Electron. J. Comb. 4(2), R3 (1997)MathSciNetGoogle Scholar
  6. 6.
    Amdeberhan, T., de Angelis, V., Lin, M., Moll, V.H., Sury, B.: A pretty binomial identity. Elem. Math. 67(1), 18–25 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Amdeberhan, T., Koutschan, C., Moll, V.H., Rowland, E.S.: The iterated integrals ofln(1 + x n). Int. J. Number Theory 8(1), 71–94 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Andrews, G.E., Paule, P.: Some questions concerning computer-generated proofs of a binomial double-sum identity. J. Symb. Comput. 16, 147–153 (1993)CrossRefzbMATHGoogle Scholar
  9. 9.
    Andrews, G.E., Paule, P., Schneider, C.: Plane partitions VI. Stembridge’s TSPP theorem. Adv. Appl. Math. 34, 709–739 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Apagodu, M., Zeilberger, D.: Multi-variable Zeilberger and Almkvist-Zeilberger algorithms and the sharpening of Wilf-Zeilberger theory. Adv. Appl. Math. 37(2), 139–152 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Bećirović, A., Paule, P., Pillwein, V., Riese, A., Schneider, C., Schöberl, J.: Hypergeometric summation algorithms for high order finite elements. Computing 78(3), 235–249 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Berkovich, A., Riese, A.: A computer proof of a polynomial identity implying a partition theorem of Göllnitz. Adv. Appl. Math. 28, 1–16 (2002)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Bernstein, J.N.: The analytic continuation of generalized functions with respect to a parameter. Funct. Anal. Appl. 6(4), 273–285 (1972)CrossRefGoogle Scholar
  14. 14.
    Beuchler, S., Pillwein, V.: Sparse shape functions for tetrahedral p-FEM using integrated Jacobi polynomials. Computing 80(4), 345–375 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Beuchler, S., Pillwein, V., Zaglmayr, S.: Sparsity optimized high order finite element functions for H(curl) on tetrahedra. Adv. Appl. Math. 50(5), 749–769 (2013)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Böing, H., Koepf, W.: Algorithms for q-hypergeometric summation in computer algebra. J. Symb. Comput. 28, 777–799 (1999)CrossRefzbMATHGoogle Scholar
  17. 17.
    Bostan, A., Chyzak, F., Lecerf, G., Salvy, B., Schost, É.: Differential equations for algebraic functions. In: Proceedings of the International Symposium on Symbolic and Algebraic Computation (ISSAC), Waterloo. ACM, New York (2007)Google Scholar
  18. 18.
    Bostan, A., Chen, S., Chyzak, F., Li, Z.: Complexity of creative telescoping for bivariate rational functions. In: Proceedings of the International Symposium on Symbolic and Algebraic Computation (ISSAC), Munich, pp. 203–210. ACM, New York (2010)Google Scholar
  19. 19.
    Bostan, A., Chyzak, F., van Hoeij, M., Pech, L.: Explicit formula for the generating series of diagonal 3D rook paths. Séminaire Lotharingien de Combinatoire 66, B66a (2011)Google Scholar
  20. 20.
    Bostan, A., Boukraa, S., Christol, G., Hassani, S., Maillard, J.-M.: Ising n-fold integrals as diagonals of rational functions and integrality of series expansions: integrality versus modularity. Technical report 1211.6031, arXiv (2012)Google Scholar
  21. 21.
    Bostan, A., Chen, S., Chyzak, F., Li, Z., Xin, G.: Hermite reduction and creative telescoping for hyperexponential functions. In: Proceedings of the International Symposium on Symbolic and Algebraic Computation (ISSAC), Boston, pp. 77–84. ACM, New York (2013)Google Scholar
  22. 22.
    Bostan, A., Lairez, P., Salvy, B.: Creative telescoping for rational functions using the Griffiths-Dwork method. In: Proceedings of the International Symposium on Symbolic and Algebraic Computation (ISSAC), Boston, pp. 93-100. ACM, New York (2013)Google Scholar
  23. 23.
    Buchberger, B.: Ein Algorithmus zum Auffinden der Basiselemente des Restklassenrings nach einem nulldimensionalen Polynomideal. PhD thesis, University of Innsbruck, Innsbruck (1965)Google Scholar
  24. 24.
    Cartier, P.: Démonstration “automatique” d’identités et fonctions hypergéometriques [d’après D. Zeilberger]. Astérisque 206, 41–91 (1991). Séminaire Bourbaki, 44ème année, 1991–1992, n746Google Scholar
  25. 25.
    Caruso, F.: A Macsyma implementation of Zeilberger’s fast algorithm. In: Strehl, V. (ed.) Séminaire Lotharingien de Combinatoire, S43c, pp. 1–8. Institut Girard Desargues, Université Claude Bernard Lyon-I, Villeurbanne (2000)Google Scholar
  26. 26.
    Chen, S., Kauers, M.: Order-degree curves for hypergeometric creative telescoping. In: Proceedings of the International Symposium on Symbolic and Algebraic Computation (ISSAC), Grenoble, pp. 122–129 (2012)Google Scholar
  27. 27.
    Chen, S., Kauers, M.: Trading order for degree in creative telescoping. J. Symb. Comput. 47(8), 968–995 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Chen, S., Singer, M.F.: Residues and telescopers for bivariate rational functions. Adv. Appl. Math. 49(2), 111–133 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Chen, W.Y.C., Hou, Q.-H., Mu, Y.-P.: A telescoping method for double summations. J. Comput. Appl. Math. 196(2), 553–566 (2006)MathSciNetADSCrossRefzbMATHGoogle Scholar
  30. 30.
    Chen, S., Chyzak, F., Feng, R., Fu, G., Li, Z.: On the existence of telescopers for mixed hypergeometric terms. Technical report 1211.2430, arXiv (2012)Google Scholar
  31. 31.
    Chen, S., Kauers, M., Singer, M.F.: Telescopers for rational and algebraic functions via residues. In: Proceedings of the International Symposium on Symbolic and Algebraic Computation (ISSAC), Grenoble, pp. 130–137. ACM, New York (2012)Google Scholar
  32. 32.
    Chyzak, F.: Fonctions holonomes en calcul formel. PhD thesis, École polytechnique (1998)Google Scholar
  33. 33.
    Chyzak, F.: Gröbner bases, symbolic summation and symbolic integration. In: Buchberger, B., Winkler, F. (eds.) Gröbner Bases and Applications. Volume 251 of London Mathematical Society Lecture Notes Series, pp. 32–60. Cambridge University Press, Cambridge (1998). Proceedings of the Conference 33 Years of Gröbner BasesGoogle Scholar
  34. 34.
    Chyzak, F.: An extension of Zeilberger’s fast algorithm to general holonomic functions. Discrete Math. 217(1–3), 115–134 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Chyzak, F., Salvy, B.: Non-commutative elimination in Ore algebras proves multivariate identities. J. Symb. Comput. 26, 187–227 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Chyzak, F., Kauers, M., Salvy, B.: A non-holonomic systems approach to special function identities. In: Proceedings of the International Symposium on Symbolic and Algebraic Computation (ISSAC), Seoul, pp. 111–118. ACM, New York (2009)Google Scholar
  37. 37.
    Combot, T., Koutschan, C.: Third order integrability conditions for homogeneous potentials of degree − 1. J. Math. Phys. 53(8), 082704 (2012)MathSciNetADSCrossRefGoogle Scholar
  38. 38.
    Coutinho, S.C.: A Primer of Algebraic D-Modules. Volume 33 of London Mathematical Society Student Texts. Cambridge University Press, Cambridge/New York (1995)Google Scholar
  39. 39.
    Ekhad, S.B., Zeilberger, D.: A WZ proof of Ramanujan’s formula for π. In: Rassias, J.M. (ed.) Geometry, Analysis, and Mechanics, pp. 107–108. World Scientific, Singapore (1994)Google Scholar
  40. 40.
    Feynman, R.P., Leighton, R. (eds.): Surely You’re Joking, Mr. Feynman!: Adventures of a Curious Character. W. W. Norton, New York (1985)Google Scholar
  41. 41.
    Garoufalidis, S., Koutschan, C.: Irreducibility of q-difference operators and the knot 74. Algebr. Geom. Topol. (2013, To appear). Preprint on arXiv:1211.6020Google Scholar
  42. 42.
    Garoufalidis, S., Lê, T.T.Q.: The colored Jones function is q-holonomic. Geom. Topol. 9, 1253–1293 (2005). (electronic)Google Scholar
  43. 43.
    Garoufalidis, S., Sun, X.: The non-commutative A-polynomial of twist knots. J. Knot Theory Ramif. 19(12), 1571–1595 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  44. 44.
    Gradshteyn, I.S., Ryzhik, J.M., Jeffrey, A., Zwillinger, D. (eds.): Table of Integrals, Series, and Products, 7th edn. Academic/Elsevier, Amsterdam (2007)zbMATHGoogle Scholar
  45. 45.
    Graham, R.L., Knuth, D.E., Patashnik, O: Concrete Mathematics, 2nd edn. Addison-Wesley, Reading (1994)Google Scholar
  46. 46.
    Guillera, J.: Generators of some Ramanujan formulas. Ramanujan J. 11, 41–48 (2006)MathSciNetADSCrossRefzbMATHGoogle Scholar
  47. 47.
    Guo, Q.H., Hou, Q.-H., Sun, L.H.: Proving hypergeometric identities by numerical verifications. J. Symb. Comput. 43(12), 895–907 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  48. 48.
    Ishikawa, M., Koutschan, C.: Zeilberger’s holonomic ansatz for Pfaffians. In: Proceedings of the International Symposium on Symbolic and Algebraic Computation (ISSAC), Grenoble, pp. 227–233. ACM, (2012)Google Scholar
  49. 49.
    Kandri-Rody, A., Weispfenning, V.: Non-commutative Gröbner bases in algebras of solvable type. J. Symb. Comput. 9(1), 1–26 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  50. 50.
    Kauers, M.: Summation algorithms for Stirling number identities. J. Symb. Comput. 42(10), 948–970 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  51. 51.
    Kauers, M.: The holonomic toolkit. In: Blümlein, J., Schneider, C. (eds.) Computer Algebra in Quantum Field Theory: Integration, Summation and Special Functions. Springer, Wien (2013)Google Scholar
  52. 52.
    Kauers, M., Paule, P.: The Concrete Tetrahedron. Text and Monographs in Symbolic Computation, 1st edn. Springer, Wien (2011)Google Scholar
  53. 53.
    Kauers, M., Schneider, C.: Automated proofs for some Stirling number identities. Electron. J. Comb. 15(1), 1–7 (2008). R2Google Scholar
  54. 54.
    Klein, S.: Heavy flavor coefficient functions in deep-inelastic scattering at O(a s 2) and large virtualities. Diplomarbeit, Universität Potsdam (2006)Google Scholar
  55. 55.
    Koepf, W.: Algorithms for m-fold hypergeometric summation. J. Symb. Comput. 20, 399–417 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  56. 56.
    Koepf, W.: REDUCE package for the indefinite and definite summation. SIGSAM Bull. 29(1), 14–30 (1995)CrossRefzbMATHGoogle Scholar
  57. 57.
    Koepf, W.: Hypergeometric Summation. An Algorithmic Approach to Summation and Special Function Identities. Advanced Lectures in Mathematics. Vieweg Verlag, Braunschweig/Wiesbaden (1998)CrossRefzbMATHGoogle Scholar
  58. 58.
    Koepf, W., Schmersau, D.: Representations of orthogonal polynomials. J. Comput. Appl. Math. 90, 57–94 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  59. 59.
    Koornwinder, T.H.: On Zeilberger’s algorithm and its q-analogue. J. Comput. Appl. Math. 48(1–2), 91–111 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  60. 60.
    Koornwinder, T.H.: Identities of nonterminating series by Zeilberger’s algorithm. J. Comput. Appl. Math. 99(1–2), 449–461 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  61. 61.
    Koutschan, C.: Advanced applications of the holonomic systems approach. PhD thesis, Research Institute for Symbolic Computation (RISC), Johannes Kepler University, Linz (2009)Google Scholar
  62. 62.
    Koutschan, C.: Eliminating human insight: an algorithmic proof of Stembridge’s TSPP theorem. In: Amdeberhan, T., Medina, L.A., Moll, V.H. (eds.) Gems in Experimental Mathematics. Volume 517 of Contemporary Mathematics, pp. 219–230. American Mathematical Society, Providence (2010)CrossRefGoogle Scholar
  63. 63.
    Koutschan, C.: A fast approach to creative telescoping. Math. Comput. Sci. 4(2–3), 259–266 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  64. 64.
    Koutschan, C.: HolonomicFunctions (user’s guide). Technical report 10-01, RISC Report Series, Johannes Kepler University, Linz (2010).
  65. 65.
    Koutschan, C.: Lattice Green’s functions of the higher-dimensional face-centered cubic lattices. J. Phys. A Math. Theor. 46(12), 125005 (2013)MathSciNetADSCrossRefGoogle Scholar
  66. 66.
    Koutschan, C., Moll, V.H.: The integrals in Gradshteyn and Ryzhik. Part 18: some automatic proofs. SCIENTIA Ser. A Math. Sci. 20, 93–111 (2011)Google Scholar
  67. 67.
    Koutschan, C., Thanatipanonda, T.: Advanced computer algebra for determinants. Ann. Comb. (2013, To appear). Preprint on arXiv:1112.0647Google Scholar
  68. 68.
    Koutschan, C., Kauers, M., Zeilberger, D.: Proof of George Andrews’s and David Robbins’s q-TSPP conjecture. Proc. Natl. Acad. Sci. 108(6), 2196–2199 (2011)MathSciNetADSCrossRefzbMATHGoogle Scholar
  69. 69.
    Koutschan, C., Lehrenfeld, C., Schöberl, J.: Computer algebra meets finite elements: an efficient implementation for Maxwell’s equations. In: Langer, U., Paule, P. (eds.) Numerical and Symbolic Scientific Computing: Progress and Prospects. Volume 1 of Texts and Monographs in Symbolic Computation, pp. 105–121. Springer, Wien (2012)CrossRefGoogle Scholar
  70. 70.
    Lyons, R., Paule, P., Riese, A.: A computer proof of a series evaluation in terms of harmonic numbers. Appl. Algebra Eng. Commun. Comput. 13, 327–333 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  71. 71.
    Majewicz, J.E.: WZ-style certification and Sister Celine’s technique for Abel-type sums. J. Differ. Equ. Appl. 2(1) 55–65 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  72. 72.
    Mohammed, M. Zeilberger, D.: Sharp upper bounds for the orders of the recurrences output by the Zeilberger and q-Zeilberger algorithms. J. Symb. Comput. 39(2), 201–207 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  73. 73.
    Ore, Ø.: Theory of non-commutative polynomials. Ann. Math. 34(3), 480–508 (1933)MathSciNetCrossRefzbMATHGoogle Scholar
  74. 74.
    Paule, P.: Short and easy computer proofs of the Rogers-Ramanujan identities and of identities of similar type. Electron. J. Comb. 1, 1–9 (1994)MathSciNetADSGoogle Scholar
  75. 75.
    Paule, P., Riese, A.: A Mathematica q-analogue of Zeilberger’s algorithm based on an algebraically motivated approach to q-hypergeometric telescoping. In: Ismail, M.E.H., Masson, D.R., Rahman, M. (eds.) Special Functions, q-Series and Related Topics. Volume 14 of Fields Institute Communications, pp. 179–210. American Mathematical Society, Providence (1997)Google Scholar
  76. 76.
    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).
  77. 77.
    Paule, P., Strehl, V.: Symbolic summation—some recent developments. In: Fleischer, J. et al. (eds.) Computer Algebra in Science and Engineering—Algorithms, Systems, and Applications, pp. 138–162. World Scientific, Singapore (1995)Google Scholar
  78. 78.
    Paule, P., Suslov, S.: Relativistic Coulomb integrals and Zeilbergers holonomic systems approach I. In: Blümlein, J., Schneider, C. (eds.) Computer Algebra in Quantum Field Theory: Integration, Summation and Special Functions. Springer, Wien (2013)Google Scholar
  79. 79.
    Paule, P., Pillwein, V., Schneider, C., Schöberl, J.: Hypergeometric summation techniques for high order finite elements. In: PAMM, Weinheim, vol. 6, pp. 689–690. Wiley InterScience (2006)Google Scholar
  80. 80.
    Petkovšek, M., Wilf, H.S., Zeilberger, D.: A = B. A. K. Peters, Ltd., Wellesley (1996)Google Scholar
  81. 81.
    Prodinger, H.: Descendants in heap ordered trees or a triumph of computer algebra. Electron. J. Comb. 3(1), R29 (1996)MathSciNetGoogle Scholar
  82. 82.
    Raab, C.G.: Definite integration in differential fields. PhD thesis, Johannes Kepler Universität Linz (2012)Google Scholar
  83. 83.
    Riese, A.: A Mathematica q-analogue of Zeilberger’s algorithm for proving q-hypergeometric identities. Master’s thesis, RISC, Johannes Kepler University Linz (1995)Google Scholar
  84. 84.
    Riese, A.: Fine-tuning Zeilberger’s algorithm—the methods of automatic filtering and creative substituting. In: Garvan, F.G., Ismail, M.E.H. (eds.) Symbolic Computation, Number Theory, Special Functions, Physics and Combinatorics. Volume 4 of Developments in Mathematics, pp. 243–254. Kluwer, Dordrecht (2001)CrossRefGoogle Scholar
  85. 85.
    Riese, A.: qMultiSum—A package for proving q-hypergeometric multiple summation identities. J. Symb. Comput. 35, 349–376 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  86. 86.
    Schneider, C.: Symbolic summation in difference fields. PhD thesis, RISC, Johannes Kepler University, Nov 2001Google Scholar
  87. 87.
    Schneider, C.: Simplifying multiple sums in difference fields. In: Blümlein, J., Schneider, C. (eds.) Computer Algebra in Quantum Field Theory: Integration, Summation and Special Functions. Springer, Wien (2013)CrossRefGoogle Scholar
  88. 88.
    Slater, P.B.: A concise formula for generalized two-qubit Hilbert-Schmidt separability probabilities. Technical report 1301.6617, arXiv (2013)Google Scholar
  89. 89.
    Strehl, V.: Binomial identities—combinatorial and algorithmic aspects. Discrete Math. 136 (1–3), 309–346 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  90. 90.
    Takayama, N.: An algorithm of constructing the integral of a module—an infinite dimensional analog of Gröbner basis. In: Proceedings of the International Symposium on Symbolic and Algebraic Computation (ISSAC), Tokyo, pp. 206–211. ACM, New York (1990)Google Scholar
  91. 91.
    Tefera, A.: MultInt, a MAPLE package for multiple integration by the WZ method. J. Symb. Comput. 34(5), 329–353 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  92. 92.
    van der Poorten, A.: A proof that Euler missed—Apéry’s proof of the irrationality of ζ(3). An informal report. Math. Intell. 1, 195–203 (1979)CrossRefzbMATHGoogle Scholar
  93. 93.
    Wegschaider, K.: Computer generated proofs of binomial multi-sum identities. Master’s thesis, RISC, Johannes Kepler University Linz, May 1997.
  94. 94.
    Wilf, H.S., Zeilberger, D.: An algorithmic proof theory for hypergeometric (ordinary and “q”) multisum/integral identities. Invent. Math. 108(1), 575–633 (1992)MathSciNetADSCrossRefGoogle Scholar
  95. 95.
    Yen, L.: A two-line algorithm for proving terminating hypergeometric identities. J. Math. Anal. Appl. 198(3), 856–878 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  96. 96.
    Yen, L.: A two-line algorithm for proving q-hypergeometric identities. J. Math. Anal. Appl. 213(1), 1–14 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  97. 97.
    Zeiberger, D.: A fast algorithm for proving terminating hypergeometric identities. Discrete Math. 80(2), 207–211 (1990)MathSciNetCrossRefGoogle Scholar
  98. 98.
    Zeilberger, D.: A holonomic systems approach to special functions identities. J. Comput. Appl. Math. 32(3), 321–368 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  99. 99.
    Zeilberger, D.: The method of creative telescoping. J. Symb. Comput. 11, 195–204 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  100. 100.
    Zeilberger, D.: Three recitations on holonomic systems and hypergeometric series. J. Symb. Comput. 20(5–6), 699–724 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  101. 101.
    Zeilberger, D.: The holonomic ansatz II. Automatic discovery(!) and proof(!!) of holonomic determinant evaluations. Ann. Comb. 11(2), 241–247 (2007)Google Scholar
  102. 102.
    Zhang, X.-K., Wan, J., Lu, J.-J., Xu, X.-P.: Recurrence and Pólya number of general one-dimensional random walks. Commun. Theor. Phys. 56(2), 293 (2011)ADSCrossRefzbMATHGoogle Scholar
  103. 103.
    Zudilin, W.: An Apéry-like difference equation for Catalan’s constant. Electron. J. Comb. 10(1), #R14 (2003)Google Scholar
  104. 104.
    Zudilin, W.: Apéry’s theorem. Thirty years after [an elementary proof of Apéry’s theorem]. Int. J. Math. Comput. Sci. 4(1), 9–19 (2009)Google Scholar

Copyright information

© Springer-Verlag Wien 2013

Authors and Affiliations

  1. 1.Johann Radon Institute for Computational and Applied Mathematics (RICAM)Austrian Academy of Sciences (ÖAW)LinzAustria

Personalised recommendations