Gauss's Forward Formula

$$f_p=f_0+p\delta_{1/2}+G_2\delta_0^2+G_3\delta_{1/2}^3+G_4\delta_0^4+G_5\delta_{1/2}^5+\ldots,$$

for $p\in[0,1]$, where $\delta$ is the Central Difference and

$$G_{2n} = {p+n-1\choose 2n}$$

$$G_{2n+1} = {p+n\choose 2n+1},$$

where ${n\choose k}$ is a Binomial Coefficient.

See also Central Difference, Gauss's Backward Formula

References

Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 433, 1987.