OFFSET
0,2
COMMENTS
The semiderivative is the fractional derivative of order 1/2. The Davison-Essex method is used.
REFERENCES
M. Davison and C. Essex, Fractional differential equations and initial value problems, The Mathematical Scientist, vol. 23, no. 2, pp. 108-116, 1998.
FORMULA
r(n) = Integral_{0..1}((d/dx)Bernoulli(n, x) / sqrt(1 - x)).
a(n) = numerator(r(n)).
EXAMPLE
r(n) = 0, 2, 2/3, 1/5, -8/105, -5/63, 4/77, 521/6435, -464/6435, -97/663, ...
a(n) = numerator(sdb_n(1) - sdb_n(0)), where
sdb_0(x) = 0;
sdb_1(x) = -2*sqrt(1-x);
sdb_2(x) = (-2 - 4*x)*sqrt(1-x) / 3;
sdb_3(x) = (-1 + 2*x - 6*x^2)*sqrt(1-x) / 5;
sdb_4(x) = (8 + 4*x + 108*x^2 - 120*x^3)*sqrt(1-x) / 105;
sdb_5(x) = (5 - 8*x - 6*x^2 + 100*x^3 - 70*x^4)*sqrt(1-x) / 63;
sdb_6(x) = (-12 - 6*x - 120*x^2 - 100*x^3 + 490*x^4 - 252*x^5)*sqrt(1-x) / 231.
MAPLE
r := n -> int(diff(bernoulli(n, t), t) / sqrt(1 - t), t = 0..1):
a := n -> numer(r(n)): seq(a(n), n = 0..9);
# Alternative:
fb := n -> sqrt(Pi)*fracdiff(bernoulli(n, x), x, 1/2):
seq(numer(simplify(subs(x=1, fb(n)))), n = 0..9);
CROSSREFS
KEYWORD
sign,frac
AUTHOR
Peter Luschny, Jul 31 2021
STATUS
approved