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Journal of the London Mathematical Society 1989 s2-39(2):193-204; doi:10.1112/jlms/s2-39.2.193
© 1989 by London Mathematical Society
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© Oxford University Press

Infinite Products Associated with Counting Blocks in Binary Strings

J.-P. Allouche and J. O. Shallit

U. A. 226, Mathématiques et Informatique, Université de Bordeaux 351 cours de la Libération, 33405 Talence Cedex, France
Department of Mathematics and Computer Science, Dartmouth College Hanover, New Hampshire 03755, USA

Let w be a string of zeros and ones, and let aw(n) be the function which counts the number of (possibly overlapping) occurrences of w in the binary expansion of n. We show that there exists an effectively computable rational function bw) such that


Formula

By setting X = – 1 and exponentiating, we recover previous results and also obtain some new ones; for example,


Formula

Our work is a generalization of previous results of D. Woods, D. Robbins, H. Cohen, M. Mendes France, and the authors.


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