Everett's Formula

$$f_p=(1-p)f_0+pf_1+E_2\delta_0^2+F_2\delta_1^2+E_4\delta_0^4+F_4\delta_1^4+E_6\delta_0^6+F_6\delta_1^6+\ldots,$$ (1)

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

$$E_{2n} \equiv G_{2n}-G_{2n+1}\equiv B_{2n}-B_{2n+1}$$ (2)

$$F_{2n} \equiv G_{2n+1}\equiv B_{2n}+B_{2n+1},$$ (3)

where $G_k$ are the Coefficients from Gauss's Backward Formula and Gauss's Forward Formula and $B_k$ are the Coefficients from Bessel's Finite Difference Formula. The $E_k$s and $F_k$s also satisfy

$$E_{2n}(p) = F_{2n}(q)$$ (4)

$$F_{2n}(p) = E_{2n}(q),$$ (5)

for

$$q\equiv 1-p.$$ (6)

See also Bessel's Finite Difference Formula

References

Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 880-881, 1972.

Acton, F. S. Numerical Methods That Work, 2nd printing. Washington, DC: Math. Assoc. Amer., pp. 92-93, 1990.

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