Scaling factor for the Fourier coefficients of Legendre polynomials

ann.awk

function abs(n)
{
  return (n>0)? n : -n
}

function f1(n,     i)
{
  return sqrt(1 - 1 / (4 * n * n))
}

function f2(n,     i)
{
  return sqrt((2*n-1)/(2*n) * (2*n+1)/(2*n))
}

function f3(n,     i)
{
  return sqrt(((2*n-1)*(2*n+1))/(4*n*n))
}

BEGIN {
  n = 32
  for (i = 5; i <= 20; i++) {
    x1 = 1
    x2 = 1
    x3 = 1
    for (j = 1; j <=n; j++) {
      x1 *= f1(j)
      x2 *= f2(j)
      x3 *= f3(j)
    }
#    printf("%7d %20.17e %20.17e %20.17e %20.17e %20.17e\n",  n, x1, x2, x3, abs(x1-x2), abs(x2-x3))
    printf("%7d %20.17e %20.17e %20.17e\n",  n, x1, x2, abs(x1-x2))
    n *= 2
  }
}