Integral Exponential Series: Fibonacci Weights and Implicit Equations

We apply classical prime-power congruence criteria to two families of exponential series. For the Fibonacci numbers Fₙ, we characterize the polynomials \(P∈[T]\) for which exp(∑n≥1P(Fₙ)xⁿ/n) has integer coefficients: this holds if and only if P is even and P(0)≡ P(1)≡ P(2)5. In particular, $5$ is the least positive multiplier for each positive even power of Fₙ, and the series with weights 5Fₙ²ᵏ has degree 5ᵏ⁻¹ over \((x)\) for k≥1. For \(R_n(y)∈[y]\) satisfying Rₚₙ(y)≡ Rₙ(y) p for every prime p and every n≥1, we prove that H(x,y)=exp(∑n≥1Rₙ(y)ⁿ xⁿ/n) belongs to \([y][[x]]\) and that $A(x)=H(x,A(x))$ has a unique integral solution. For \(q∈ \), taking Rₙ(y)=qⁿ+y proves the conjecture in OEIS A163138; the Fibonacci result proves the conjecture in A207969 and A207970. For the implicit family with indeterminate q, the coefficient of xⁿ has degree n², leading coefficient $1/n$, and nonnegative integer coefficients in the binomial basis, for every n≥1.