First Written: Dec. 8, 2004.

This version: Aug. 15, 2005.

Last Update of this page: March 20, 2012 (featuring extension of the Maple package SMAZ, and sample input and output, inspired by questions of Miklos Bona)

Previous Update of this page: Feb. 26, 2012 (featuring extension of the Maple package MultiAlmkvistZeilberger, and sample input and output, inspired by questions of Miklos Bona)

This is a much improved, and much sharper redoing of multi-WZ theory! It is written in a very terse style, and some of the proofs are only sketched. It would be a very instructive excercise to fill-in the details.

• MultiZeilberger [Revised and extended Feb. 26, 2012] that computes recurrences and certificates for hypergeometric multiple sums.
• MultiZeilbergerDen that also computes recurrences and certificates for hypergeometric multiple sums, but the user has the freedom to assign denominators to the certificates, thereby having a chance to reduce the order of the recurrence.
• qMultiZeilberger that computes recurrences and certificates for q-hypergeometric multiple sums. It does what Axel Riese's Mathematica package does.
• MultiAlmkvistZeilberger that computes recurrences and certificates for hypergeometric (in fact hyper-exponential) multiple integrals. [Revised and extended Feb. 26, 2012, including new procedures MAZpaper and DIAGpaper] [For the record, here is the Old MultiAlmkvistZeilberger

• SMAZ that computes recurrences and certificates for hypergeometric (in fact hyper-exponential) multiple integrals that are SYMMETRIC in their variables.

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  SAMPLE Input and Output Files
  • For the Maple package MultiZeilberger
    • The input     yields the output.
    • The input     yields the output.
    • [Added Feb. 12, 2013: inspired by a conjecture of Juan Sebastian Pererya and Federico Echenique, that we can "almost" solve (and could if we had more time)] the input     yields the output.
  • For the Maple package MultiZeilbergerDen
    • The input     yields the output    
  • For the Maple package qMultiZeilberger
    • The input     yields the output
  • For the Maple package MultiAlmkvistZeilberger [revised Feb. 26, 2012]
    • The input yields the output
    • The input yields the output
    • The input yields the output
    • The input yields the output
    • The input yields the output
    • The input yields the output

    • Sample input and output files for procedure DIAGpaper (written Feb. 26, 2012)
      • Miklos Bona asked us to prove a recurrence for the diagonal of the formal power series 1/((1-x)²+(1-y)²-1)^1/2. To get it
        the input yields the output

      • More generally, for the diagonal of the formal power series 1/((1-x)²+(1-y)²-1)^a. To get it
        the input yields the output

      • For the diagonal of the formal power series 1/(1-x-y-z+4xyz), to get it
        the input yields the output

      • More generally, for the diagonal of the formal power series 1/(1-x-y-z+4xyz)^a, to get it
        the input yields the output

  For the Maple package SMAZ
  • The input (for the diagonal coefficients of the rational function 1/(1-x-y-z+4xyz))     yields the output    
  • The input (for the 3D Mehta integral)     yields the output    
  • [added March 20, 2012] Miklos Bona asked us to prove a recurrence for the sequence of diagonal of the formal power series ((1-x)²+(1-y)²+(1-z)²-2)^-1/2. To get it
    the input yields the output
  • [added March 21, 2012] Miklos Bona asked us to prove a recurrence for the sequence of diagonal coefficients of the formal power series ((1-x)²+(1-y)²+(1-z)²-2)^1/2. To get it
    the input yields the output
  • [added March 21, 2012] More generally, to get a rigorously-proved linear recurrence for the sequence of diagonal coefficients of the formal power series ((1-x)²+(1-y)²+(1-z)²-2)^-a, for any (i.e. symbolic) a,
    
    the input yields the output

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  Doron Zeilberger's List of Papers

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