Asymptotic expansions for probability distributions. II (Q1061409)

[For part I see ibid. 23, No.3, 196-213(1983; Zbl 0536.60033); English translation in Lith. Math. 7, 23, 341-353 (1983).] Local limit theorems for a lattice variable whose cumulants satisfy a condition of Bernstein-Statulevičius type are proved. In terms of Chebyshev-Hermite polynomials linear combinations of the asymptotic expansions are constructed, the Gaussian distribution being the limit one. The expansions involve the parameter h, free at some region. The choice of the parameter makes it possible to vary the form of expansions.