Infinite products with strongly $B$-multiplicative exponents

Abstract

Let $N_{1,B}(n)$ denote the number of ones in the $B$-ary expansion of an integer $n$. Woods introduced the infinite product $P:={\prod }_{n\ge 0}(\frac{2n+1}{2n+2}{)}^{(-1{)}^{N_{1,2}(n)}}$ and Robbins proved that $P=1/\sqrt{2}$. Related products were studied by several authors. We show that a trick for proving that $P^2=1/2$ (knowing that $P$ converges) can be extended to evaluating new products with (generalized) strongly $B$-multiplicative exponents. A simple example is $$\prod _{n\ge 0}(\frac{Bn+1}{Bn+2}{)}^{(-1{)}^{N_{1,B}(n)}}=\frac{1}{\sqrt{B}}.$$

Updated [14] and journal reference

References in corpus (1)

• The generalized-Euler-constant function $\gamma (z)$ and a generalization of Somos's quadratic recurrence constant

Cited by in corpus (3)

• Transcendental Infinite Products Associated with the $\pm 1$ Thue-Morse Sequence
• Infinite products related to generalized Thue-Morse sequences
• Errata and Addenda to Mathematical Constants