Proceedings of the Japan Academy, Series A, Mathematical Sciences
- Proc. Japan Acad. Ser. A Math. Sci.
- Volume 95, Number 6 (2019), 58-63.
Erdősian functions and an identity of Gauss
A famous identity of Gauss gives a closed form expression for the values of the digamma function $\psi(x)$ at rational arguments $x$ in terms of elementary functions. Linear combinations of such values are intimately connected with a conjecture of Erdős which asserts non vanishing of an infinite series associated to a certain class of periodic arithmetic functions. In this note we give a different proof for the identity of Gauss using an orthogonality like relation satisfied by these functions. As a by product we are able to give a new interpretation for $n$th Catalan number in terms of these functions.
Proc. Japan Acad. Ser. A Math. Sci., Volume 95, Number 6 (2019), 58-63.
First available in Project Euclid: 31 May 2019
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Chatterjee, Tapas; Khurana, Suraj Singh. Erdősian functions and an identity of Gauss. Proc. Japan Acad. Ser. A Math. Sci. 95 (2019), no. 6, 58--63. doi:10.3792/pjaa.95.58. https://projecteuclid.org/euclid.pja/1559268171