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J. Ramanujan Math. Soc. 38, No.3 (2023) 265–273

Hölder and Kurokawa meet Borwein–Dykshoorn and Adamchik

J.-P. Allouche

CNRS, IMJ-PRG, Sorbonne, 4 Place Jussieu, F-75252 Paris Cedex 05, France

e-mail: jean-paul.allouche@imj-prg.fr

Communicated by: Prof. Sanoli Gun

Received: May 19, 2022

Abstract. Following our discovery of a nice identity in a recent preprint of Hu and Kim, we show a link between

the Kurokawa multiple trigonometric functions and two functions introduced respectively by Borwein-Dykshoorn

and by Adamchik. In particular several identities involving ζ (3), π and the Catalan constant G that are proved in

these three papers are related.

2010 Mathematics Subject Classification: 11M06; 33B15; 11M35; 33E20.

1. Introduction

There is a wealth of special functions arising from geometry and from transcendence theory. The purpose of this

paper is to provide identities relating some of them. The beginning of the story here is a recent preprint of Hu and

Kim [11] that gives the nice identity

ζ (3) = 4π

2

21

log

e

4G

π C3

1

4

16

√

2

!

·

Here G = ∑n≥0

(−1)

n

(2n+1)

2

is the Catalan constant, and C3 is the Kurokawa-Koyama triple cosine function (see the

beginning of Section 4):

C3(x) = ∏n≥1





e

x

2

1−

4x

2

(2n−1)

2

(2n−1)

2

4



.

The (slightly hidden) occurrences of 7ζ (3)/4π

2

and of e

G/2π

reminded us two (out of four) identities in a paper

of Kachi and Tzermias [12], namely:

lim

n→∞

2n+1

∏

k=1

e

−1/4

1−

1

k +1

k(k+1)

2

(−1)

k

= exp

7ζ (3)

4π

2

+

1

4

and

lim

n→∞

2n+1

∏

k=1

1−

2

2k +1

k(−1)

k

= exp

2G

π

+

1

2

·

In that paper Kachi and Tzermias proved four identities, and they indicated that they were not able to deduce them

directly from the values of a function introduced in 1993 by Borwein and Dykshoorn [4]. We provided in [2] a proof

of their identities, using the paper of Borwein and Dykshoorn and their function D for two of the identities, and a

paper of Adamchik [1] and his function E for the remaining two. It was thus tempting to relate these functions D and

E to the Kurokawa-Koyama triple cosine.

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