from fractions import Fraction as Q
def derivative_rayleigh(nu, n):
nu = Q(nu)
if nu < 0 or not isinstance(n, int) or n < 1:
raise ValueError("Require rational nu >= 0 and integer n >= 1")
order = Q(1) if nu == 0 else nu
c, base = [Q(1)], Q(1)
for m in range(1, n + 1):
base = -base / (4*m*(order+m))
c.append(base if nu == 0 else base*(nu+2*m)/nu)
s = [Q(0)]
for m in range(1, n + 1):
s.append(-m*c[m] - sum(c[m-k]*s[k] for k in range(1, m)))
return s[n]
for nu in range(4):
for n in range(1, 13):
print(nu, n, derivative_rayleigh(nu, n))
All 48 values were computed in exact Sage rational arithmetic from the normalized Bessel series [5] and derivative product [4]. No floating-point division or truncated root sum enters the stored values.
An independent Python Fraction implementation using the Riccati recurrence obtained from [1], equations (19)–(21), agrees on every entry. The first three closed forms agree at each positive selected order. The order-zero values agree with the order-one Bessel Rayleigh sums. Positivity, decrease and strict log convexity were checked throughout the finite selection. Supplementary sums over 32 strictly positive derivative roots per order were computed at 70 and 100 decimal digits and compared with an explicit bound for the infinite tail. Those root checks test signs, scale and indexing conventions; they do not certify root indexing or 100 digits of an infinite sum. Known nonzero controls and deliberately perturbed values test the checkers. The attached checks are repeated on the stored entries.