Let $\lbrace(x_0,y_0),(x_1,y_1),\,\dots,\,(x_n,y_n)\ |\ x_i\ne x_j\rbrace\subset\mathbb{R}^2$
Let $P$ and $Q$ be the polynomials that interpolate $\lbrace(x_0,y_0),(x_1,y_1),\,\dots,\,(x_{n-1},y_{n-1})\rbrace$, resp. $\lbrace(x_1,y_1),\,\dots,\,(x_n,y_n)\rbrace$
Question:
is there a name for the operation $\frac{Q-P}{(x-x_1)*,\dots,\,*(x-x_{n-1})}$
and/or its scalar value $c\in\mathbb{R}$
here is some python code that produces some examples of these constants:
from sympy.polys.polyfuncs import interpolate
from sympy import simplify
from sympy.abc import a, b, c, d, e, f, g, h, x
p = interpolate([(0, a)], x)
q = interpolate([(1, b)], x)
print(f'{simplify((p-q)/(1))}')
p = interpolate([(0, a), (1, b)], x)
q = interpolate([(1, b), (2, c)], x)
print(f'{simplify((p-q)/((x-1)))}')
p = interpolate([(0, a), (1, b), (2, c)], x)
q = interpolate([(1, b), (2, c), (3, d)], x)
print(f'{simplify((p-q)/((x-1)*(x-2)))}')
p = interpolate([(0, a), (1, b), (2, c), (3,d)], x)
q = interpolate([(1, b), (2, c), (3, d), (4,e)], x)
print(f'{simplify((p-q)/((x-1)*(x-2)*(x-3)))}')
p = interpolate([(0, a), (1, b), (2, c), (3,d), (4,e)], x)
q = interpolate([(1, b), (2, c), (3, d), (4,e), (5,f)], x)
print(f'{simplify((p-q)/((x-1)*(x-2)*(x-3)*(x-4)))}')
which generates for polynomial degrees 1 to 5:
1: a - b
2: -a + 2*b - c
3: a/2 - 3*b/2 + 3*c/2 - d/2
4: -a/6 + 2*b/3 - c + 2*d/3 - e/6
5: a/24 - 5*b/24 + 5*c/12 - 5*d/12 + 5*e/24 - f/24
