First Written: Dec. 8, 2004.

This version: Aug. 15, 2005.

Last Update of this page: June 10, 2024 : inspired by a question of Frank Calegari

Previous 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)

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

  • Added June 10, 2024 (inspired by a question of Frank Calegari)

    To get the third-order recurrence satisfied by the Beukers-style integral

    Int(Int(Int((x*(1-x)*y*(1-y)*z*(1-z))^n/(1-x*y*z)^(n+1),x=0..1),y=0..1),z=0..1)

    the input yields the output

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  Doron Zeilberger's List of Papers

  Doron Zeilberger's Home Page