OFFSET
0,2
COMMENTS
Integral representation as n-th moment of a positive function W(x) expressed in terms of Meijer's G-function on the positive axis, in Maple notation: a(n)= int(x^n*W(x),x=0..infinity)= int(x^n*(1/2)*MeijerG([[], []], [[1, 1/2, 0], []], (1/16)*x) /(sqrt(x)*Pi),x=0..infinity), n=0,1... .
Explicit form of the function W(x) is
W(x)=1/2*(-1/2*Pi*x^(1/2)*hypergeom([],[1/2, 3/2],-1/16*x)+Pi^(1/2)-1/16*Pi^(1/2)*x*sum((-1)^(2*j)* (-Psi(3/2+j)+Pi*tan(Pi*j)-Psi(2+j)-Psi(1+j)-4*log(2)+log(x))*2^(-4*j)* (x^j)*sec(Pi*j)*2^(2*j)/(1/2+j)/GAMMA(1+2*j)/GAMMA(2+j), j = 0..infinity))/(x^(1/2)*Pi);
This is the solution of the Stieltjes moment problem with the moments a(n).
This solution may not be unique.
FORMULA
Hypergeometric generating function: sum(a(n)*x^n/(n!)^4, n=0..infinity)= -2*EllipticE(4*sqrt(x))/((16*x-1)*Pi).
PROG
(Magma) [Factorial(2*n)*Factorial(2*n+1)/Factorial(n): n in [0..20]]; // Vincenzo Librandi, Jul 06 2015
CROSSREFS
KEYWORD
nonn
AUTHOR
Karol A. Penson, Jun 03 2009
STATUS
approved