760 research outputs found
The impact of Stieltjes' work on continued fractions and orthogonal polynomials
Stieltjes' work on continued fractions and the orthogonal polynomials related
to continued fraction expansions is summarized and an attempt is made to
describe the influence of Stieltjes' ideas and work in research done after his
death, with an emphasis on the theory of orthogonal polynomials
Some classical multiple orthogonal polynomials
Recently there has been a renewed interest in an extension of the notion of
orthogonal polynomials known as multiple orthogonal polynomials. This notion
comes from simultaneous rational approximation (Hermite-Pade approximation) of
a system of several functions. We describe seven families of multiple
orthogonal polynomials which have he same flavor as the very classical
orthogonal polynomials of Jacobi, Laguerre and Hermite. We also mention some
open research problems and some applications
Orthogonal polynomial expansions for the Riemann xi function
We study infinite series expansions for the Riemann xi function in
three specific families of orthogonal polynomials: (1) the Hermite polynomials;
(2) the symmetric Meixner-Pollaczek polynomials ; and (3)
the continuous Hahn polynomials . The first expansion was discussed in
earlier work by Tur\'an, and the other two expansions are new. For each of the
three expansions, we derive formulas for the coefficients, show that they
appear with alternating signs, derive formulas for their asymptotic behavior,
and derive additional interesting properties and relationships. We also apply
some of the same techniques to prove a new asymptotic formula for the Taylor
coefficients of the Riemann xi function.
Our results continue and expand the program of research initiated in the
1950s by Tur\'an, who proposed using the Hermite expansion of the Riemann xi
function as a tool to gain insight into the location of the Riemann zeta zeros.
We also uncover a connection between Tur\'an's ideas and the separate program
of research involving the so-called De Bruijn-Newman constant. Most
significantly, the phenomena associated with the new expansions in the
Meixner-Pollaczek and continuous Hahn polynomial families suggest that those
expansions may be even more natural tools than the Hermite expansion for
approaching the Riemann hypothesis and related questions.Comment: Changes from previous version: typo corrections, added references and
other minor improvements to Chapter 4, formattin
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