A note on Hermite interpolation

Let x ₀ , x ₁ , . . ., x _n ∈ ℝ, be pairwise disjoint, and let θ ₀ , θ ₁ , . . ., θ _n ∈ Set θ := Σ _ν=0 ⁿ θ _ν . For each pair j, p such that 0 ≤ j ≤ n and 0 ≤ p ≤ θ _j -1, let y _j,p be a complex number. Then there is a unique polynomial, H(x), of degree θ - 1, such that H ^(p) (x _j ) = y _j,p , for 0 ≤ p ≤ θ _j - 1, 0 ≤ j ≤ n. In particular, there is a unique fundamental Hermite polynomial, T _j,p (x), of degree θ - 1, such that T _j,p ^(r) (x _s ) = δ _j,s δ _p,r , 0 ≤ r ≤ θ _s - 1, 0 ≤ s ≤ n, δ being Kronecker's delta, and we have the representation. We give an explicit representation of the polynomials T _j,p (x).