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# Computation and checks for T693
Write x = pi^2. If exp(-z lambda(u)) = sum P_k(z)u^k and
lambda(u) = sum_{r>=1} B_{r+1}u^r/(r+1)!, differentiation gives
k P_k(z) = -z sum_{r=1}^k r B_{r+1} P_{k-r}(z)/(r+1)!.
With Q_k(x) = z^k P_k(z) at z^2=-4x, the odd r alone survive:
Q_0 = 1,
k Q_k = -sum_{r odd, 1<=r<=k}
r B_{r+1} (-4x)^((r+1)/2) Q_{k-r}/(r+1)!.
The generator forms C_ell(x) = sum_{k=0}^ell H(k+1,ell-k) Q_k(x)/4^(ell-k)
over QQ[x]. H(nu,j) is the product of (4 nu^2-(2r-1)^2)/(4r), r=1,...,j.
It equals 2^j times the polynomial a_j(nu) stored by T569. All divisions
in the generator are exact Sage rational divisions.
A second construction in checks.py inverts (exp(u)-1)/u by Cauchy products,
obtaining the Bernoulli coefficients without the generator's bernoulli routine.
It computes the exponential by its powers divided by m!, and uses the
Pochhammer product for H. Exact agreement is required for all 61 full
polynomials. This is an independent implementation of the same finite
coefficient formula, not by itself an external mathematical validation.
An external structural check comes from Kummer's equation
z y''-z y'-n y=0 for y=U(n,0,z), DLMF 13.2.1. Put z=t^2 and
y = (constant in n) exp(z/2) exp(-2 sqrt(n) t) t^(1/2) F(t).
Direct differentiation yields
F''-4 sqrt(n) F'-(t^2+3/(4t^2)) F=0.
For F=sum a_ell(t) n^(-ell/2), the coefficient identity is
4 a_(ell+1)' = a_ell''-(t^2+3/(4t^2))a_ell,
a_ell(t) = t^(-ell) C_ell(-t^4/4).
All 60 adjacent identities are checked exactly as Laurent polynomials. The
checker deliberately changes C_1 by 1 to confirm this check rejects an error.
This differential identity does not alone fix integration constants and is
not represented as a uniquely normalizing recurrence.
Numerical checks use direct complex Arb U(n,0,2*pi*i), scaled by -4 Gamma(n),
which is the dominant contribution in AM (4.5), (5.3). The real part is
compared with the prescribed sine expansion for several n and truncations.
U(1,2,2)=1/2 is the nonzero known-answer control. These are truncation-error
experiments, not certified remainder bounds for the asymptotic series or
independent extraction of all 100 digits of each high-order coefficient.
The coefficients themselves are certified by their finite exact expressions
and Arb evaluation. No asymptotic remainder is used to justify their digits.
# Source discrepancy resolved before submission
In arXiv:2402.14216v1, the statement of Corollary 5.6 gives the constant
coefficient as H(1,ell)*2^ell. Its proof gives H(1,ell)*2^(-2ell), as does
equation (5.2) with k=0. The first printed coefficient has constant 3/16,
which agrees with the proof and (5.2) and disagrees with the statement.
The generator uses equation (5.2). The first six printed polynomials and the
constant term H(1,ell)/4^ell are checked separately throughout the range.
This is a localized typographical inconsistency, not a choice between two
viable definitions. The family convention is unchanged.
The leading coefficient of C_ell(x) is 1/(ell! 3^ell), also verified through
ell=60. Thus for ell>=1 C_ell(pi^2) is transcendental, because a nonconstant
rational polynomial at a transcendental number cannot be algebraic. Only
C_0=1 is rational in this selection and it is stored exactly.