Dynamical Symmetry Algebras for F0 1, Φ ₂ , and Φ ₃

We develop a unified Lie‐theoretic treatment of the confluent hypergeometric limit function and the Humbert functions Φ ₂ and Φ ₃ . 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.