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 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.
Published Version
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