Step-λ Type-A Lauricella Matrix Hypergeometric Functions: Scaling Equivalence and Analytic Consequences

We consider the step-λ parametrization of the type A Lauricella matrix hypergeometric function FA,λ, defined by replacing the classical matrix Pochhammer symbol with (A)m,λ=A(A+λI)⋯A+(m−1)λI,λ>0. For scalar parameters, this is the step-λ factorial used in the degenerate Gauss hypergeometric matrix function, not the Gamma-ratio convention also termed a degenerate Pochhammer symbol. For fixed λ>0, the scaling identity FA,λ[A,Bi;Ci;xi]=FA[A/λ,Bi/λ;Ci/λ;λxi] shows that the function is a reparametrization of the classical Lauricella matrix function. Accordingly, the classical sufficient convergence region, differential relations, coupled matrix PDE system, Euler transformation, and Euler-type representations recorded below are consequences of the classical theory under this rescaling; they are not claimed as independent new results. The material not obtained merely by applying the scaling identity consists of the direct coefficient recursion and its path-independence criterion, the fixed-parameter zero-step analysis, and the accompanying truncation and numerical checks. Under the right-update compatibility condition, the coefficient recursion is path-independent even when A need not commute with the right-hand parameter family. With the parameters fixed as λ→0+, the general result is coefficientwise convergence along the admissible set, together with finite-order error estimates; locally uniform convergence is proved only under the stronger simultaneous-diagonalization and positivity assumptions. The numerical section includes a genuinely noncommuting check of the coefficient recursion, a separate commuting Jordan-type comparison with Euler quadrature, and a certified Cauchy truncation bound.