# How to write mathematica code for Newton's Forward Difference Formula?

I want to write the Newton's Forward Difference Formula in mathematica:

I know there many functions to interpolate data in mathematica, but I really need Newton's Forward Difference Formula.

Thanks

The $$\Delta$$ operator in Mathematica is DifferenceDelta. Here is Newton's formula:

Sum[Binomial[a, n] DifferenceDelta[f[x], {x, n}], {n, 0, Infinity}]


Examples:

Block[{a = 1, f = Exp},
Sum[Binomial[a, n] DifferenceDelta[f[x], {x, n}], {n, 0, Infinity}]
]
(*  E^(1 + x)  *)

Block[{a = 2, f = TrigToExp@*Sin},
Sum[Binomial[a, n] DifferenceDelta[f[x], {x, n}], {n, 0, Infinity}]
] // FullSimplify
(*  Sin[2 + x]  *)

• Dear Michael, many thanks for your response, suppose now you have the following data: {{0., 1.}, {0.1, 1.10517}, {0.2, 1.2214}, {0.3, 1.34986}, {0.4, 1.49182}, {0.5, 1.64872}, {0.6, 1.82212}, {0.7, 2.01375}, {0.8, 2.22554}, {0.9, 2.4596}, {1., 2.71828}}, how can we find the f(x) which fit the data using Newton forward formula? Commented Apr 20, 2019 at 7:38
• @user62716 The formula you cite is an infinite series, but it sounds like you might want mathworld.wolfram.com/… Commented Apr 20, 2019 at 15:51
• Yes Michael, I cited the link in my question, can you help to sort this....thanks Commented Apr 20, 2019 at 16:04
• @user62716 Don't you need an infinite number of data points to do an infinite sum? I suppose you might want to truncate the sum and just compute a finite partial sum? Please clarify. Commented Apr 20, 2019 at 17:21
• Dear Michael, yes please it is not infinite sum, just finite sum for the data: {{0., 1.}, {0.1, 1.10517}, {0.2, 1.2214}, {0.3, 1.34986}, {0.4, 1.49182}, {0.5, 1.64872}, {0.6, 1.82212}, {0.7, 2.01375}, {0.8, 2.22554}, {0.9, 2.4596}, {1., 2.71828}} Commented Apr 20, 2019 at 18:32