Hypergeometry of the Parbelos

Abstract

In the December 2013 issue of this Monthly, Sondow introduced a parabolic analog of the classical arbelos figure called the parbelos. He found that the ratio of the perimeter of an arbitrary parbelos to that of the corresponding arbelos is a constant S that has an elegant symbolic evaluation involving $1/\pi$ and the universal parabolic constant. Using the On-Line Encyclopedia of Integer Sequences, in 2016 Campbell experimentally discovered a hypergeometric formula for S. Zudilin then proved the formula, using recent results of Borwein, Borwein, Glasser, and Wan on the moments of Ramanujan’s generalized elliptic integrals. In the present article, we offer a variety of new proofs of Campbell’s hypergeometric formula for the parbelos constant S, including a creative proof that makes use of a Fourier–Legendre expansion.

• MSC: Primary 33C20
• Secondary 33C05

Acknowledgments

The authors thank the two anonymous referees for providing useful feedback on our article.

Additional information

Notes on contributors

John M. Campbell

JOHN M. CAMPBELL received his M.Math. in pure mathematics from the University of Waterloo, and graduated first class with distinction with a Specialized Honours B.Sc. degree in mathematics from York University. He has been awarded the prestigious Carswell Scholarship and the Irvine R. Pounder Award, and has worked as a research assistant at York University and at the Fields Institute for Research in Mathematical Sciences.

Jacopo D’Aurizio

JACOPO D’AURIZIO studied at the University of Pisa and completed a master’s degree, and is a member in the development community for the Maxima computer algebra system. He has much experience in the Mathematics Olympiads, and is the webmaster of www.matemate.it and a moderator of math.stackexchange.com.

Jonathan Sondow

Jonathan Sondow graduated from Stuyvesant High School in New York City and the University of Wisconsin at Madison. In 1965 at the age of 22, he received a Ph.D. from Princeton University, where he wrote a thesis in differential topology under John Milnor. After two years as a postdoc in Paris, Sondow held numerous academic posts in the U.S. and worked at the Institute for Defense Analyses. Currently, he maintains a homepage at home.earthlink.net/∼jsondow/devoted to research in number theory. His Erdős number is two.