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We describe the asymptotic behavior of the multivariate BC-type Jacobi polynomials as the number of variables and the Young diagram indexing the polynomial go to infinity. In particular, our results describe the approximation of the…
Comparison of approximate solutions that were obtained by using different asymptotic methods of solutions of difference equations with the exact solution is presented. Results show that for the studied equation the method of transformation…
We propose a formula for finding the horizontal, oblique or curvilinear asymptote of any rational polynomial function of any positive degree, as a sum of matrix determinants formed directly from the coefficients of the terms in the given…
We study the uniform asymptotics for the orthogonal polynomials with respect to weights composed of both absolutely continuous measure and discrete measure, by taking a special class of the sieved Pollazek Polynomials as an example. The…
In this paper we obtain the formal asymptotic expansion of the logarithms $\ln p_s(\alpha)$ of $p_s(\alpha)$, which are canonical continuations of polynomials of binomial type $p_n(\alpha)$. Our approach is based on linear methods which do…
We consider the number of partitions of $n$ whose Young diagrams fit inside an $m \times \ell$ rectangle; equivalently, we study the coefficients of the $q$-binomial coefficient $\binom{m+\ell}{m}_q$. We obtain sharp asymptotics throughout…
The remedian uses a $k\times b$ matrix to approximate the median of $n\leq b^{k}$ streaming input values by recursively replacing buffers of $b$ values with their medians, thereby ignoring its $200(\lceil b/2\rceil / b)^{k}%$ most extreme…
We study the asymptotic distribution of zeros for the random polynomials $P_n(z) = \sum_{k=0}^n A_k B_k(z)$, where $\{A_k\}_{k=0}^{\infty}$ are non-trivial i.i.d. complex random variables. Polynomials $\{B_k\}_{k=0}^{\infty}$ are…
We consider several aspects of conjugating symmetry methods, including the method of invariants, with an asymptotic approach. In particular we consider how to extend to the stochastic setting several ideas which are well established in the…
We develop asymptotic approximations that can be applied to sequential estimation and inference problems, adaptive randomized controlled trials, and related settings. In batched adaptive settings where the decision at one stage can affect…
The Darwin-Fowler method in combination with the steepest descent approach is a common tool in the asymptotic description of many models arising from statistical physics. In this work, we focus rather on the non-asymptotic behavior of the…
In this lecture notes we try to familiarize the audience with the theory of Bernoulli polynomials; we study their properties, and we give, with proofs and references, some of the most relevant results related to them. Several applications…
We use a non-linear characterization of orthonormal polynomials due to Saff in order to show that the behavior of orthonormal polynomials is determined only by its leading coefficient and its normalization. Several applications of this…
This is a survey article on an old topic in classical analysis. We present some new developments in asymptotics in the last fifty years. We start with the classical method of Darboux and its generalizations, including an uniformity…
We study the small-mass asymptotic behavior of so-called angular integrals, appearing in phase-space calculations in perturbative quantum field theory. For this purpose we utilize the strategy of expansion by regions, which is a universal…
In this paper we study the asymptotic analysis of the orthogonal trigonometric polynomials by the Riemann-Hilbert problem for the periodic analytic functions.
The numerical evaluation of an individual Bessel or Hankel function of large order and large argument is a notoriously problematic issue in physics. Recurrence relations are inefficient when an individual function of high order and argument…
We use the Legendre polynomials and the Hermite polynomials as two examples to illustrate a simple and systematic technique on deriving asymptotic formulas for orthogonal polynomials via recurrence relations. Another application of this…
The problem of integer partitions is addressed using the microcanonical approach which is based on the analogy between this problem in the number theory and the calculation of microstates of a many-boson system. For ordinary…
For any real numbers $B \ge 1$ and $\delta \in (0, 1)$ and function $f: [0, B] \rightarrow \mathbb{R}$, let $d_{B; \delta} (f) \in \mathbb{Z}_{> 0}$ denote the minimum degree of a polynomial $p(x)$ satisfying $\sup_{x \in [0, B]} \big| p(x)…