Automatic Conjecturing and Proving of Exact Values of Some Infinite Families of Infinite Continued Fractions

By Robert Dougherty-Bliss and Doron Zeilberger

.pdf    .tex

Appeared in the Ramanujan Journal v. 61, no. 1, (2023), 31-47.

In fond memory of Richard A. Askey (1933-2019), that taught us that Special Functions are Useful Functions

First Version: March 31, 2020 . This version: May 20, 2020.

We are sure that Richard Askey would have appreciated this paper, since it is yet another example how Special Functions, combined with symbolic computation, could lead to interesting new results.

Acknowledgment: The authors would like to thank the OEIS for its help in identifying promising sequences of integers that arose in the evaluation of these continued fractions, as well as Alex Kontorovich who forwarded us some very useful feedback on an earlier version from a very distinguished and esteemed number theorist (not to be confused with his equally brilliant brother), who out of natural modesty, asked not to be named, and we reluctantly agreed to keep him anonymous.

# Maple package

• GCF.txt, a Maple package to conjecture and PROVE exact expressions for generalized continued fractions

# Sample Input and Output for GCF.txt

• If you want to see the PROVED exact values for the infinite continued fraction a(1)/b(1)/(a(2)+b(2)/(a(3)+... where a(n)=n+k, and b(n)=n as well as a(n)=n+k, and b(n)=-n for k from 1 to 40 then
The input file yields the output file
• If you want to see computer-generated articles with full self-contained proofs for these continued fractions where a(n)=n+k, and b(n)=n as well as a(n)=n+k, and b(n)=-n for k from 1 to 10 then
The input file yields the output file

Articles of Doron Zeilberger