Lambda-deformed pascal matrices: Operator deformation, conditioning stabilization, and structural properties

We introduce a one-parameter deformation of the classical Pascal matrix based on a rationally modified lowering operator acting on the λ -rising-factorial basis. The resulting matrices preserve a Pascal-type lower-triangular structure while exhibiting substantially improved conditioning. Explicit entry formulas, an inverse relation, and uniform norm bounds are obtained. For every fixed λ > 0, both the infinity-norm and two-norm condition numbers remain bounded as the matrix size increases and converge to the same finite limit. Numerical experiments support these results and illustrate the improved finite-precision behaviour of the deformed matrices.