D.H. Lehmer's Tridiagonal determinant: An Étude in (Andrews-Inspired) Experimental Mathematics

By Shalosh B. Ekhad and Doron Zeilberger

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First Written: Aug. 20, 2018

[Appeared in Annals of Combinatorics, 23(3) (2019) , 717-724; special issue in honor of George Andrews' 80th birthday]

Dedicated to George Andrews on his forthcoming 80th birthday
In the last chapter of George Andrews' classic booklet

"q-Series: Their development and applications in Analysis, Number Theory, Combinatorics, Physics, and Computer Algebra" CBMS series No. 66, AMS, 1986, based on 10 amazing lectures, given at Arizona State, May 1985 (organized by Mourad Ismail and Ed Ihrig), that one of us (DZ) was lucky to attend, George Andrews described how he used the computer algebra system SCRATCHPAD to prove a deep conjecture by "fancy" mathematicians: Lusztig, Macdonald and Wall.

In this short note, dedicated with admiration to George Andrews, and submitted to the special issue in honor of his 80th birthday, we use his methodology to discover an explicit expression for the determinant of a tridiagonal matrix discussed by Derrick Henry Lehmer in 1974, that inspired OEIS sequence A039924. Lehmer only did the infinite case, and here we do the finite case, that immediately implies (faster than the original) the infinite case, by taking the limit as n goes to infinity. Like in George Andrews' case, the hard part is the discovery. Once discovered, the proof can be done via the q-Zeilberger algorithm (using procedure qzeil in the Maple package qEKHAD), but in fact, is fairly simple for humans, even those who are not George Andrews.

Maple package

Sample Input and Output for LehmerDet.txt

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