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Parameterized Telescoping Proves Algebraic Independence of Sums

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Abstract

Usually creative telescoping is used to derive recurrences for sums. In this article we show that the non-existence of a creative telescoping solution, and more generally, of a parameterized telescoping solution, proves algebraic independence of certain types of sums. Combining this fact with summation-theory shows transcendence of whole classes of sums. Moreover, this result throws new light on the question why, for example, Zeilberger’s algorithm fails to find a recurrence with minimal order.

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References

  1. Abramov S.A.: The summation of rational functions. Zh. Vychisl. Mat. Mat. Fiz. 11, 1071–1075 (1971)

    MathSciNet  MATH  Google Scholar 

  2. Abramov S.A.: When does Zeilberger’s algorithm succeed?. Adv. Appl. Math. 30(3), 424–441 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  3. Abramov S.A., Petkovšek M.: Rational normal forms and minimal decompositions of hypergeometric terms. J. Symbolic Comput. 33(5), 521–543 (2002)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  6. Gosper R.W.: Decision procedures for indefinite hypergeometric summation. Proc. Nat. Acad. Sci. USA 75(1), 40–42 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  7. Karr M.: Summation in finite terms. J. Assoc. Comput. Mach. 28(2), 305–350 (1981)

    MathSciNet  MATH  Google Scholar 

  8. Paule P., Nemes I.: A canonical form guide to symbolic summation. In: Miola, A., Temperini, M. (eds) Advances in the Design of Symbolic Computation Systems, pp. 84–110. Springer, Vienna (1997)

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  10. Paule, P.: Contiguous relations and creative telescoping. Preprint (2004)

  11. 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 Inst. Commun., 14, pp. 179–210. AMS, Providence, RI (1997)

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

    Article  MathSciNet  MATH  Google Scholar 

  13. 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 

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

  15. Schneider, C.: Symbolic Summation in Difference Fields. PhD Thesis, J. Kepler University, Linz (2001)

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

    MATH  Google Scholar 

  17. Schneider C.: The summation package Sigma: underlying principles and a rhombus tiling application. Discrete Math. Theor. Comput. Sci. 6(2), 365–386 (2004)

    MathSciNet  MATH  Google Scholar 

  18. Schneider C.: Degree bounds to find polynomial solutions of parameterized linear difference equations in ΠΣ-fields. Appl. Algebra Engrg. Comm. Comput. 16(1), 1–32 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  19. Schneider C.: Product representations in ΠΣ-fields. Ann. Combin. 9(1), 75–99 (2005)

    Article  MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

  21. Schneider C.: Simplifying Sums in ΠΣ*-Extensions. J. Algebra Appl. 6(3), 415–441 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  22. Schneider, C.: Symbolic summation assists combinatorics. Sém. Lothar. Combin. 56:#B56b. (2007)

    Google Scholar 

  23. Schneider C.: A refined difference field theory for symbolic summation. J. Symbolic Comput. 43(9), 611–644 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  24. van der Poorten, A.: A proof that Euler missed. . . Apéry’s proof of the irrationality of ζ(3). Math. Intelligencer 1(4), 195–203 (1979)

    Google Scholar 

  25. Zeilberger D.: The method of creative telescoping. J. Symbolic Comput. 11(3), 195–204 (1991)

    Article  MathSciNet  MATH  Google Scholar 

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  1. Research Institute for Symbolic Computation, J. Kepler University Linz, Altenbergerstraße 69, A-4040, Linz, Austria

    Carsten Schneider

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  1. Carsten Schneider
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Correspondence to Carsten Schneider.

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Supported by the SFB-grant F1305 and the grant P20347-N18 of the Austrian FWF.

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Schneider, C. Parameterized Telescoping Proves Algebraic Independence of Sums. Ann. Comb. 14, 533–552 (2010). https://doi.org/10.1007/s00026-011-0076-7

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  • DOI: https://doi.org/10.1007/s00026-011-0076-7

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