Laplace, Legendre, and Gauss

Abstract

Laplace’s efforts in our field appear both in his work on celestial mechanics and on probability theory. Moreover these contributions, which overlap each other and those of Gauss very much, are central to the interests of Laplace. One of his chief tools was that of the generating function. He was perhaps the first to exploit fully the generating function of a sequence y₀, y₁, y₂,... . He wrote the function as

$$u={{y}_{0}}+{{y}_{1}}t+{{y}_{2}}{{t}^{2}}+{{y}_{3}}{{t}^{3}}+...+{{y}_{x}}{{t}^{x}}+{{y}_{x+1}}{{t}^{x+1}}+...+{{y}^{\infty }}{{t}_{\infty }}$$

without any consideration of convergence. Let us accept his formalistic approach and see how the generating function tool, which he used so powerfully, could produce various interpolation formulas.¹