Narrow Results

A recent asymptotic expansion for the positive zeros $x=j_{\nu ,m}$ ($m=1,2,3,\dots$) of the Bessel function of the first kind $J_{\nu }(x)$ is studied, where the order $\nu$ is positive. Unlike previous well-known expansions in the literature, this is uniformly valid for one or both $m$ and $\nu$ unbounded, namely $m=1,2,3,\dots$ and $1\le \nu <\infty$. Explicit and simple lower and upper error bounds are derived for the difference between $j_{\nu ,m}$ and the first three terms of the expansion. The bounds are sharp in the sense they are close to the value of the fourth term of the expansion (i.e. the first neglected term).

For a sequence ${\{M_n\}}_{n=1}^{\infty }$ of expanding matrices with $M_n\in M_d(\mathbb{Z})$ and a sequence ${\{D_n\}}_{n=1}^{\infty }$ of finite digit sets with $D_n\subset {\mathbb{Z}}^d$, the Moran measure ${\mu }_{\{M_n\},\{D_n\}}$ is defined by the infinite convolution

In studying various quantum-billiard configurations, R. L. Liboff (J. Math. Phys.35 (1994) 2218), was led to investigate the vanishing of f(ν)=j_2ν,1 - j_ν2, where j_μk is the kth positive zero of the Bessel function J_μ(x). Here we show that the even more general function f_α(ν)=c_αν,k - c_ν,k+l is increasing and vanishes once (and only once) in 0<ν<∞, provided α≥π/2 and , k, l=1,2,3,…. As usual, c_μn is the nth positive zero of the cylinder function C_μ(x)=J_μ (x)cosθ - Y_μ(x)sinθ. Specialized to Liboff's case, f(ν), this yields not only the existence of a zero of f(ν) but also its uniqueness.

We derive representations for some entire q-functions and use it to derive asymptotics and closed form expressions for large zeros of a class of entire functions including the Ramanujan function, and q-Bessel functions.

We study the uniform asymptotics of a system of polynomials orthogonal on [-1, 1] with weight function w(x) = exp{-1/(1 - x²)^μ}, 0 < μ < 1/2, via the Riemann–Hilbert approach. These polynomials belong to the Szegö class. In some earlier literature involving Szegö class weights, Bessel-type parametrices at the endpoints ±1 are used to study the uniform large degree asymptotics. Yet in the present investigation, we show that the original endpoints ±1 of the orthogonal interval are to be shifted to the MRS numbers ±β_n, depending on the polynomial degree n and serving as turning points. The parametrices at ±β_n are constructed in shrinking neighborhoods of size 1 - β_n, in terms of the Airy function. The polynomials exhibit a singular behavior as compared with the classical orthogonal polynomials, in aspects such as the location of the extreme zeros, and the approximation away from the orthogonal interval. The singular behavior resembles that of the typical non-Szegö class polynomials, cf. the Pollaczek polynomials. Asymptotic approximations are obtained in overlapping regions which cover the whole complex plane. Several large-n asymptotic formulas for π_n(1), i.e. the value of the nth monic polynomial at 1, and for the leading and recurrence coefficients, are also derived.

We obtain uniform asymptotic approximations for the monic Meixner–Sobolev polynomials S_n(x). These approximations for n → ∞, are uniformly valid for x/n restricted to certain intervals, and are in terms of Airy functions. We also give asymptotic approximations for the location of the zeros of S_n(x), especially the small and the large zeros are discussed. As a limit case, we also give a new asymptotic approximation for the large zeros of the classical Meixner polynomials.

The method is based on an integral representation in which a hypergeometric function appears in the integrand. After a transformation, the hypergeometric functions can be uniformly approximated by unity, and all that remains are simple integrals for which standard asymptotic methods are used. As far as we are aware, this is the first time that standard uniform asymptotic methods have been used for the Sobolev-class of orthogonal polynomials.

The pseudo-ultraspherical polynomial of degree n can be defined by where is the ultraspherical polynomial. It is known that when λ < -n, the finite set is orthogonal on (-∞, ∞) with respect to the weight function (1 + x²)^λ-½ and when λ < 1 - n, the polynomial has exclusively real and simple zeros. Here, we undertake a deeper study of the zeros of these polynomials including bounds, numbers of real zeros, monotonicity and interlacing properties. Our methods include the Sturm comparison theorem, recurrence relations, and the explicit expression for the polynomials.

In this paper, we study the asymptotic behavior of the Wilson polynomials $W_n(x;a,b,c,d)$ as their degree tends to infinity. These polynomials lie on the top level of the Askey scheme of hypergeometric orthogonal polynomials. Infinite asymptotic expansions are derived for these polynomials in various cases, for instance, (i) when the variable $x$ is fixed and (ii) when the variable is rescaled as $x=n^{2}t$ with $t\ge 0$. Case (ii) has two subcases, namely, (a) zero-free zone ($t>1$) and (b) oscillatory region $(0<t<1)$. Corresponding results are also obtained in these cases (iii) when $t$ lies in a neighborhood of the transition point $t=1$, and (iv) when $t$ is in the neighborhood of the transition point $t=0$. The expansions in the last two cases hold uniformly in $t$. Case (iv) is also the only unsettled case in a sequence of works on the asymptotic analysis of linear difference equations.

This paper considers the well-known Erdös–Lax and Turán-type inequalities that relate the sup-norm of a univariate complex coefficient polynomial and its derivative, when there is a restriction on its zeros. The obtained results produce inequalities that are sharper than the previous ones. Moreover, a numerical example is presented, showing that in some situations, the bounds obtained by our results can be considerably sharper than the ones previously known.

In this paper, some generalizations of lower bound estimates for the maximum modulus of the $t$th polar derivative $D_{{\xi }_t}{D}_{{\xi }_{t-1}}\dots D_{{\xi }_2}{D}_{{\xi }_1}$, where ${\xi }_t\in ℂ$, $1\le t<n$ of polynomials are established under the assumption that one of the underlying polynomials $p$ and $q$ does not vanish in $\left|z\right|>k,k\le 1$. The obtained estimates include several known lower bounds for polar derivatives as special cases.