Integer Powers of Arcsin - CiteSeerX

where G 0 (k) = 1 and
G N (k) :=
∑k−1
n 1 =0
n
1 ∑ 1 −1
(2n 1 + 1) 2
n 2 =0
n N−1 −1
1
(2n 2 + 1) 2 · · · ∑
n N =0
1
(2n N + 1) 2 .
The convention is that the sum is zero if the starting index exceeds the
finishing index.
Nested sums are not new. The last decade saw many interesting results
concerning Euler sums or multizeta values, wonderful generalizations of the
classical ζ-function, whose discovery can be traced to a letter from Goldbach
to Euler [2, pp.99–100] and [3, Chapter 3]—a letter that played a seminal
role in the discovery of the ζ-function.
When Gauss was criticized for the lack of motivation in his writings, he
remarked that the architects of great cathedrals do not obscure the beauty
of their work by leaving the scaffolding in place after the construction has
been completed. While we find (1) and (2) worthy of undistracted attention,
in truth their discovery was greatly facilitated by the use of experimental
mathematics—the relatively new approach to doing mathematical research
with the intelligent use of computers. This perspective is elucidated throughout
this paper. It is also illustrative of the changing speed of mathematical
communication that the special cases (3), (4), (5), and (6) given below are
already online at [11].
2 Experiments and Proofs
The first identity below is very well known:
(
arcsin 2 x
)
= 1 ∞∑
2 2
k=1
x 2k
( 2 k
k
)
k 2 . (3)
It is explored at some length in [4, pp.384–386]. While it is seen in various
calculus books (see [6, pp.88–90], where the series for arcsin 3 (x) is also
proven), it dates back at least two centuries and was given by Ramanujan
among many others; see [10, pp.262–63]. As often in Mathematics, history is
complicated. Equation (3) has been rediscovered repeatedly. For example,
an equivalent form is elegantly solved as a 1962 MAA Monthly problem (“A
Well-Known Constant”, Problem E 1509, p.232). We quote in extenso, the
editors’ attempts to trace the history of the formula:
The series was located in the Smithsonian Mathematical Formulae and
Tables of Elliptic Functions, 6.42 No. 5, p. 122; Chrystal, Algebra, vol.
2