On a formula of T. Rivoal

In an unpublished 2005 paper T. Rivoal proved a formula giving 4/pi as the infinite product of factors (1 + 1/(k+1)) to a power involving the integer part of the logarithm of k in base 2 and a 4-periodic sequence. We show how a lemma in a 1988 paper of J. Shallit and the author allows us to prove that formula, as well as a family of similar formulas involving occurrences of blocks of digits in the base-B expansion of the integer k, where B is an integer larger than 1.