Validation of the power-series eigen-function for f(t) = √(6 + t)

Establish whether the Taylor-series-constructed function φ with φ(0) = 3 and φ'(0) = 1 indeed satisfies the eigen-function identity 6 + φ(θ/α) = φ(θ)^2 with α = 1/6, i.e., determine whether the constructed series converges and equals a genuine solution of the functional equation on a neighborhood of θ = 0.

Background

The authors construct φ via recursive determination of its Taylor coefficients intending to satisfy the eigen-function equation 6 + φ(θ/α) = φ(θ)^2 for f(t) = √(6 + t) with α = 1/6. They obtain successive coefficients and numerical evidence, but raise a fundamental question about exact satisfaction of the functional identity by the constructed series.

This problem asks for a rigorous validation (or refutation) that the formally constructed power series corresponds to an analytic φ solving the functional equation, thereby confirming the eigen-function approach yields an explicit candidate sequence limit in this case.