Abstract
We use George Andrews’ “reverse-engineering” method to re-prove, using experimental mathematics, a conjecture of D.H. Lehmer.
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Notes
Typing qzeil((-1)**a*X**a*q**(a*(a-1))*qbin(n-a,a),S,a,n,[]); in qEKHAD gives \(Xq^n-S+S^2, Xq^n\) that is the recurrence operator annihilating the sum (S is the forward shift operator in n) followed by the ‘certificate’ (i.e., the proof, see [4])).
You start out with a generic polynomial of degree 0, and keep raising the degree until success (or failure).
References
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Dedicated to George Andrews on his 80th birthday.
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Ekhad, S.B., Zeilberger, D. D.H. Lehmer’s Tridiagonal Determinant: An Étude in (Andrews-Inspired) Experimental Mathematics. Ann. Comb. 23, 717–724 (2019). https://doi.org/10.1007/s00026-019-00441-y
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DOI: https://doi.org/10.1007/s00026-019-00441-y