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In mathematics , the reciprocal difference of a finite sequence of numbers
(
x
0
,
x
1
,
.
.
.
,
x
n
)
{\displaystyle (x_{0},x_{1},...,x_{n})}
f
(
x
)
{\displaystyle f(x)}
ρ
1
(
x
1
,
x
2
)
=
x
1
−
x
2
f
(
x
1
)
−
f
(
x
2
)
{\displaystyle \rho _{1}(x_{1},x_{2})={\frac {x_{1}-x_{2}}{f(x_{1})-f(x_{2})}}}
ρ
2
(
x
1
,
x
2
,
x
3
)
=
x
1
−
x
3
ρ
1
(
x
1
,
x
2
)
−
ρ
1
(
x
2
,
x
3
)
+
f
(
x
2
)
{\displaystyle \rho _{2}(x_{1},x_{2},x_{3})={\frac {x_{1}-x_{3}}{\rho _{1}(x_{1},x_{2})-\rho _{1}(x_{2},x_{3})}}+f(x_{2})}
ρ
n
(
x
1
,
x
2
,
…
,
x
n
+
1
)
=
x
1
−
x
n
+
1
ρ
n
−
1
(
x
1
,
x
2
,
…
,
x
n
)
−
ρ
n
−
1
(
x
2
,
x
3
,
…
,
x
n
+
1
)
+
ρ
n
−
2
(
x
2
,
…
,
x
n
)
{\displaystyle \rho _{n}(x_{1},x_{2},\ldots ,x_{n+1})={\frac {x_{1}-x_{n+1}}{\rho _{n-1}(x_{1},x_{2},\ldots ,x_{n})-\rho _{n-1}(x_{2},x_{3},\ldots ,x_{n+1})}}+\rho _{n-2}(x_{2},\ldots ,x_{n})}
