For some families of one-dimensional locally infinitely divisible Markov processes xet 0≤t≤T with frequent small jumps, large deviation expansions for expectations are proved: as epsilon ↓ 0 Ee expe-1F xe =expe -1Ff0 -Sf0 0≤i≤ s/2Ki˙ei+o&parl0; es/2&parr0; where s is a positive integer, S is the normalized action functional, constants Ki are expressed through derivatives of the smooth functional F, and &phis;0 is the unique maximizer of F -- S The proof of above large deviation expansions relies on asymptotic expansions for expectations of a smooth functional G of stochastic processes etaepsilon = epsilon--1/2(xi epsilon -- &phis;0) : as epsilon ↓ 0 EeGhe =EGh +e1/2EA1Gh +˙˙˙+es/2EAsG h+o&parl0;es/2&parr0; for some Gaussian diffusion eta and suitable differential operators Ai