The Riemann-Hilbert approach to strong asymptotics for orthogonal polynomials on [-1,1]

Large parameter cases of the Gauss hypergeometric function

A liquid and a powder made from a herbal complex consisting of Uncariae Ramulus cum Uncis, Agrimoniae Herba, Sanguisorbae Radix, Eclipta Prostrate Herba, Pulsatillae Radix, Sophorae Flavescentis Radix, Rehmanniae Radix and Glycyrrhizae Radix were studied for their anticoccidial activities in chickens. Chickens were administered with herbal liquid, powder, diclazuril or without medication during the study and challenged with oocysts of Eimeria tenella. Results indicated that the birds medicated showed less bloody faeces than those without medication. The intestinal lesion was mild in the chicks medicated with herbal liquid without significantly different lesion score when compared with uninfected chicks. The birds with medication had significantly higher body weight gains than birds without medication. Therefore, the herbal complex used in this study was effective against E. tenella infection in chickens.

Effects of a Herbal Complex Against Eimeria tenella Infection in Chickens

The subject of special functions is often presented as a collection of disparate results, rarely organized in a coherent way. This book emphasizes general principles that unify and demarcate the subjects of study. The authors' main goals are to provide clear motivation, efficient proofs, and original references for all of the principal results. The book covers standard material, but also much more. It shows how much of the subject can be traced back to two equations - the hypergeometric equation and confluent hypergeometric equation - and it details the ways in which these equations are canonical and special. There is extended coverage of orthogonal polynomials, including connections to approximation theory, continued fractions, and the moment problem, as well as an introduction to new asymptotic methods. There are also chapters on Meijer G-functions and elliptic functions. The final chapter introduces Painleve transcendents, which have been termed the 'special functions of the twenty-first century'.

Special Functions and Orthogonal Polynomials

An alternative proof is given for the connection between a system of continuous Hahn polynomials and identities for symmetric elements in the Heisenberg algebra, which was first observed by Bender, Mead, and Pinsky [Phys. Rev. Lett. 56, 2445 (1986); J. Math. Phys. 28, 509 (1987)]. The continuous Hahn polynomials turn out to be Meixner–Pollaczek polynomials. Use is made of the connection between Laguerre polynomials and Meixner–Pollaczek polynomials, the Rodrigues formula for Laguerre polynomials, an operational formula involving Meixner–Pollaczek polynomials, and the Schrodinger model for the irreducible unitary representations of the three‐dimensional Heisenberg group.

Meixner-Pollaczek polynomials and the Heisenberg algebra

An infinite asymptotic expansion is derived for the Meixner—Pollaczek polynomials M
 n
 
(nα;δ, η) as n→∞ , which holds uniformly for -M≤α≤ M , where M can be any positive number. This expansion involves the parabolic cylinder function and its derivative. If α

 n, s
 denotes the s th zero of M
 n
 
(nα;δ, η) , counted from the right, and if α˜

 n,s
 denotes its s th zero counted from the left, then for each fixed s , three-term asymptotic approximations are obtained for both α

 n,s
 and α˜

 n,s
 as n→∞ .

X. Li

Papers

On the Asymptotics of the Meixner—Pollaczek Polynomials and Their Zeros