(2000). Recounting the Rationals. The American Mathematical Monthly: Vol. 107, No. 4, pp. 360-363.

/pdf/recounting-the-rationals-1m7h3brntu.pdf

Recounting the Rationals

Asymptotic Properties of Zeros of Hypergeometric Polynomials

Scientific computation on mathematical problems and conjectures

We consider rational approximations of the form
 
$$\left\{ {(1 + z)^{\alpha n + 1} \frac{{p_{cn} (z)}}{{q_n (z)}}} \right\}$$

 in certain natural regions in the complex plane wherep
cn andq
n are polynomials of degreecn andn, respectively. In particular we construct natural maximal regions (as a function of α andc) where the collection of such rational functions is dense in the analytical functions. So from this point of view we have rather complete analog theorems to the results concerning incomplete polynomials on an interval. The analysis depends on an examination of the zeros and poles of the Pade approximants to (1+z)αn+1. This is effected by an asymptotic analysis of certain integrals. In this sense it mirrors the well-known results of Saff and Varga on the zeros and poles of the Pade approximant to exp. Results that, in large measure, we recover as a limiting case. In order to make the asymptotic analysis as painless as possible we prove a fairly general result on the behavior, inn, of integrals of the form
 
$$\int_0^1 {[t(1 - t)f_z (t)]^n {\text{ }}dt,}$$

 wheref
z
(t) is analytic inz and a polynomial int. From this we can and do analyze automatically (by computer) the limit curves and regions that we need.

http://www.cecm.sfu.ca/personal/pborwein/PAPERS/P105.pdf

Incomplete rational approximation in the complex plane

For certain ranges of parameters, it is shown that the hypergeometric function F(a, b; b+1; z) has no zeros in a specified half-plane. It is also shown that the zeros of the hypergeometric polynomials 

$$F(-n,\ kn\ + \ell\ + 1;\ kn\ + \ell\ + 2;\ z)$$

cluster on one loop of a specified lemniscate. Other results then follow from quadratic relations.

Zeros of Hypergeometric Functions

The operator lm is defined as ra-fold indefinite integration with zero constants of integration. The zero distribution of Im(p) for polynomials p is studied in general, and for two special classes of polynomials in detail. The main results are: (i) The zeros of In(Pn), where Pn(z) is the n-th Legendre polynomial, converge to a certain algebraic curve; (ii) the zeros of Y%Ln+\\{nz) /k\\ (c > 2 an integer) converge to pieces of a circle and of two \"Szegô curves\".

/pdf/zeros-of-iterated-integrals-of-polynomials-1e0bw1mwej.pdf

Zeros of Iterated Integrals of Polynomials

Protein post-translational modifications (PTMs), which include covalent addition of functional groups to proteins and removal of specific amino acid(s) from proteins after synthesis, are often regulated by enzymes and linked to various cellular processes. Identification of enzyme substrates and their modification sites can assist in understanding the control of PTM enzymes on cellular processes, which is important for comprehending physiopathological processes. Due to the rapid advancement of analytical techniques, mass spectrometry (MS)-based proteomic approach has been one of the dominant methods for detecting enzyme-substrate interactions and modification site preferences in biological samples. Here, the development and optimization of MS-based proteomic methodologies in profiling substrates of PTM enzymes are briefly reviewed. 

Mass spectrometry-based candidate substrate and site identification of PTM enzymes

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Papers

Incomplete rational approximation in the complex plane

Zeros of Iterated Integrals of Polynomials

Mass spectrometry-based candidate substrate and site identification of PTM enzymes