Dynamical Symmetry Algebras for F01, Φ2, and Φ3

We develop a unified Lie-theoretic treatment of the confluent hypergeometric limit function F01 and the Humbert functions Φ2 and Φ3. For each family, we specify an analytic representation space, determine differential generators, prove closure to a finite-dimensional solvable Lie algebra, and give a faithful matrix realization. The associated one-parameter subgroup actions yield recurrence, generating, and translation formulas under explicit analytic hypotheses. For the Humbert functions, the entire translation identities also provide reproducible local continuation schemes. The main contribution is the structural closure and identification of the symmetry algebras, together with a unified operator derivation of the resulting formulas.