On Hal and Lah Numbers: Random Factorial Reflections, Rigidity, and Moment Recovery

Hal and Lah numbers describe changes of factorial polynomial bases. We use this viewpoint to study the matrix R(a, h) representing the reflection x → a −x in a factorial basis of step h, and then allow both parameters to be random. A covariance identity for the second subdiagonal of E[R]^2 −I yields rigidity at degree four and a stability inequality with optimal constant 13/3. We also recover all mixed moments of a prescribed total degree from one subdiagonal of the mean matrix. At degree three, the mean matrix is an involution exactly when the sum of the two parameters is almost surely constant. For compactly supported parameter distributions, the complete matrix determines the joint law. The presentation develops the requisite factorial identities and connections with Laguerre polynomials, probability, interpolation, and differential operators. Examples, counterexamples, and complete proofs distinguish established background from the mixture results, whose bibliographical priority remains to be assessed.