OFFSET
0,3
COMMENTS
REFERENCES
Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 3.1, Shapiro-Drinfeld constant, p. 210.
Hillel Gauchman, Solution to Problem 10528(b), unpublished note, 1998.
LINKS
Vasile Cârtoaje, Jeremy Dawson and Hans Volkmer, Solution to Problem 10528(a,b), American Mathematical Monthly, 105 (1998), 473-474. [A comment was made about Hillel Gauchman's solution to part (b) of the problem that involves this constant, but no solution was published.]
Petros Hadjicostas, Plot of the curves y = (1 - exp(x/2))/(exp(x) + exp(x/2)) and y = (exp(-x) - 1)/2 and their common tangent, 2020.
FORMULA
We solve the following system of equations:
exp(-c) = (exp(b/2) + 2*exp(b) - exp(3*b/2))/(exp(b) + exp(b/2))^2 and
2*(1 - exp(b/2)) = (exp(b) + exp(b/2))*(exp(-c)*(1 + c - b) - 1).
Then the constant equals (1 - exp(b/2))/(exp(b) + exp(b/2)).
It turns out that b = -A335810 = -0.387552... and c = A335809 = 0.330604... even though A335810 and A335809 are also involved in the calculation of the Shapiro cyclic sum constant mu (A086278).
As a result, the constant also equals A335825 - 1.
EXAMPLE
0.11723849199621197165722389607046320202248089118...
PROG
(PARI) default("realprecision", 200)
c(b) = -log((exp(b/2) + 2*exp(b) - exp(3*b/2))/(exp(b) + exp(b/2))^2);
a = solve(b=-2, 0, (exp(b) + exp(b/2))*(-1 + exp(-c(b))*(1 + c(b) - b)) - 2*(1 - exp(b/2)));
(1 - exp(a/2))/(exp(a) + exp(a/2))
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
Petros Hadjicostas, Jul 05 2020
STATUS
approved