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On infinite products associated with sums of digits. (English) Zbl 0573.10005

This paper is concerned with the evaluation of some unusual infinite products. They are derived from the following theorem, whose proof occupies the major part of the paper, and two generalizations. The theorem states that if \(k\geq 2\) is an integer and \(1\leq j\leq k-1\) then
\[ \prod^{\infty}_{i=0} \frac{1+c_i}{1+d_i} = k^{-1/k}.\]
The \(c_i\) and \(d_i\) are such that \(ki\le c_i\), \(d_i<k(i+1)\), \(s_k(c_i)\equiv j-1\pmod k\) and \(s_k(d_i)\equiv j \pmod k\). Here, \(s_k(n)\) denotes the sum of the digits of the number obtained if \(n\) is written in base \(k\). For example, if \(k=2\) we obtain the following infinite product:
\[ 2^{-1/2}=(1/2)(4/3)(6/5)(7/8)\cdots \]
This result was first established by D. Robbins and D. R. Woods in a problem (E 2692) which was proposed in the American Mathematical Monthly in 1978 and solved in 1979.

MSC:

11A63 Radix representation; digital problems
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References:

[1] Henrici, P, (), 4
[2] Robbins, D, Solution to problem E 2692, Amer. math. monthly, 86, 394-395, (1979)
[3] Woods, D.R, Elementary problem proposal E 2692, Amer. math. monthly, 85, 48, (1978)
[4] Wright, E.M, Prouhet’s 1851 solution of the tarry-escott problem of 1910, Amer. math. monthly, 66, 199-201, (1959) · Zbl 0087.03805
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