The HOLONOMIC ANSATZ II: Automatic DISCOVERY(!) and PROOF(!!) of Holonomic Determinant Evaluations

By Doron Zeilberger

[Appeared in Annals of Combinatorics 11(2007), 241-247.]

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Written: Feb. 17, 2006.
There are lots of ways to evaluate (symbolic) determinants, and there are quite a few humans (Tewodros Amdeberhan, George Andrews, Mihai Ciucu, Christian Krattenthaler and Herb Wilf, for example), who are very good at it.

In my Lieber Opa article, I taught Shalosh how to evaluate determinants using Charles Dodgson's Condensation method, but its scope was rather limited, since it was restricted to the case of what I called the hyperhypergeometric framework. Moving up to the holnomic ansatz can do many more determinant evaluations. Alas, not all of them!

# Sample Input and Output for DET

• Input and Output for Rproof
• To (symbolically!) evaluate and (prove!) the determinant of the famous Hilbert matrix,
the input yields the output.
• To (symbolically!) evaluate and (prove!) a special case [the general case is below] of the determinant in Theorem 33 of Christian Krattenthaler's "Advanced Determinant Claculus: a Complement"
the input yields the output.
• To (symbolically!) evaluate and (prove!) the determinant (1/(i+j+1)!),
the input yields the output.

• Input and Output for RproofP
• To (symbolically!) evaluate and (prove!) the determinant of an arbitrary minor of the Hilbert matrix,
the input yields the output.
• To (symbolically!) evaluate and (prove!) (essentially) the determinant (1.2) in Christian Krattenthaler's "Advanced Determinant Calculus"
the input yields the output.
• To (symbolically!) evaluate and (prove!) a determinant conjectured by Greg Kuperberg and Jim Propp and first proved, in a cute paper by Shalosh B. Ekhad and Tewodros Amdberhan,
the input yields the output.
• To (symbolically!) evaluate and (prove!) the determinant in Theorem 33 of Christian Krattenthaler's "Advanced Determinant Claculus: a Complement"
the input yields the output.
• To (symbolically!) evaluate and (prove!), as an explicit expression in n and p, the determinant of the (n+1) by (n+1) matrix whose (i,j) entry is binomial(2p+2i+2j,p+i+j)
the input yields the output.
• To (symbolically!) evaluate and (prove!), as an explicit expression in n and p, the determinant of the (n+1) by (n+1) matrix whose (i,j) entry is (i+j+p)!,
the input yields the output.
• And, finally, TaTa!, to (symbolically!) evaluate and (prove!), as an explicit expression in n and p, the determinant of the (n+1) by (n+1) matrix whose (i,j) entry is binomial(p+i+j,2*i-j+1), which is essentially the Mills-Robbins-Rumsey determinant, (MRR) mentioned in the body of the article,
the input yields the output.

• Input and Output for SRproof
• To (symbolically!) evaluate and provide a semi-rigorous proof of seven determinant evaluations of Ira Gessel and Guoce Xin, reproduced in Theorem 31 of Christian Krattenthaler's "Advanced Determinant Claculus: a Complement"
the input yields the output.

• Input and Output for SRproofI
• To (symbolically!) evaluate and provide a semi-rigorous proof of the determinant of the (n+1) by (n+1) matrix whose (i,j) entry is delta(i,j)+binomial(i+j,j), where delta(i,j) is 1 if i=j and 0 otherwise (the celeberated determinant, first evaluated in George Andrews's article "Plane Partitions (III): The Weak Macdoland conjecture", published in the "prestigious" journal Invent. Math. v. 53 (1979), 193-225, and used by him to enumerate cyclically symmetric plane partitions [note that the q-analog of this, that would be doable with the forthcoming q-analog, of DET, qDET, is the determinant that made William Mills, Dave Robbins and Howard Rumsey famous (proving the "strong" form of Ian Macdonald's conjecture about the generating function of Cyclically Symmetric Plane Partitions, that Richard Stanely, at the time, called "the most interesting open problem in enumeration")
the input yields the output.

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