Fefferman [22] initiated a program of expressing the asymptotic expansion of the boundary singularity of the Bergman kernel for strictly pseudoconvex domains explicitly in terms of boundary invariants. Hirachi, Komatsu and Nakazawa [30] carried out computations of the first few terms of Fefferman’s asymptotic expansion building partly on Graham’s work on CR invariants and Nakazawa’s work on the...

In [Temme N.M., Special functions. An introduction to the classical functions of mathematical physics, A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York, 1996, Section 11.3.3.1] a uniform asymptotic expansion for the incomplete beta function was derived. It was not obvious from those results that the expansion is actually an asymptotic expansion. We derive a remainder estimate...

pupose: this research aims to investigate how users expand their queries when encountered with irrelevant retrievals and through examine the degree of match between users' expanded subject queries and the list of persian subject headings. furthermore, it compares the patterns of query expansion in two subject areas: 1) agriculture (applied sciences); 2) education and psychology (humanities). me...

We express a certain complex-valued solution of Legendre’s differential equation as the product of an oscillatory exponential function and an integral involving only nonoscillatory elementary functions. By calculating the logarithmic derivative of this solution, we show that Legendre’s differential equation admits a nonoscillatory phase function. Moreover, we derive from our expression an asymp...

We define a type of generalized asymptotic series called v-asymptotic. We show that every func­ tion with moderate growth at infinity has a v-asymptotic expansion. We also describe the set of v-asymptotic series, where a given function with moderate growth has a unique v-asymptotic ex­ pansion. As an application to random matrix theory we calculate the coefficients and establish the uniqueness ...

We give a systematic view of the asymptotic expansion of two well-known sequences, the central binomial coefficients and the Catalan numbers. The main point is explanation of the nature of the best shift in variable n, in order to obtain “nice” asymptotic expansions. We also give a complete asymptotic expansion of partial sums of these sequences.

We present a method for the efficient approximation of integrals with highly oscillatory vector-valued kernels, such as integrals involving Airy functions or Bessel functions. We construct a vector-valued version of the asymptotic expansion, which allows us to determine the asymptotic order of a Levin-type method. Levin-type methods are constructed using collocation, and choosing a basis based ...

We establish a large n complete asymptotic expansion for q-Laguerre polynomials and a complete asymptotic expansion for a q-Bessel function of large argument. These expansions are needed in our study of an exactly solvable random transfer matrix model for disordered electronic systems. We also give a new derivation of an asymptotic formula due to Littlewood (1907).

For the free models of statistical mechanics on torus, exact asymptotic expansion of the free energy in the vicinity of the critical point is found. It is shown that there is direct relation between the terms of the expansion and the Kronecker's double series. The latter can be expressed in terms of the elliptic θ-functions in all orders of the asymptotic expansion.