The One-Parameter Binomial Family: Pascal Matrices, Nilpotent Generators, Fractional Powers, and Translation Operators

This paper studies the one-parameter family of lower-triangular binomial matrices Lₙ(a), defined by
Lₙ(a)[r,c] = aʳ⁻ᶜ · C(r−1, c−1) for r ≥ c, with 0 otherwise.

The central result is the exact additive law
Lₙ(a)Lₙ(b) = Lₙ(a+b),
which shows that a ↦ Lₙ(a) is a one-parameter subgroup of the unit lower-triangular group. The family admits a nilpotent generator Nₙ with
Lₙ(a) = eᵃᴺⁿ,
giving a canonical logarithm-based definition of fractional and complex powers:
Lₙ(a)ᵗ = Lₙ(ta).

A second main theorem identifies right multiplication by Lₙ(a) with translation on polynomial coefficient rows:

if p(x) = ∑ₘ₌₀ⁿ⁻¹ uₘxᵐ,

then uLₙ(a) is exactly the coefficient row of p(x+a). In this model the generator Nₙ acts as differentiation. The paper also develops row and column generating functions, a divided-power normalization in which the family becomes conjugate to

eᵃᴶⁿ for a single Jordan block Jₙ, the exact unipotent similarity type for a ≠ 0, the rank filtration rank((Lₙ(a)−I)ᵐ) = n−m, total nonnegativity for a > 0, and an explicit checkerboard sign law for minors when a < 0.

The result is a compact finite-dimensional framework linking Pascal matrices, nilpotent exponentials, translation operators, Jordan theory, and structured positivity.

Code on request

code verifies the additive law, exponential representation, derivative action, translation theorem, and consistency of fractional powers on finite sections.