3,863 research outputs found
Recursion Rules for the Hypergeometric Zeta Functions
The hypergeometric zeta function is defined in terms of the zeros of the
Kummer function M(a, a + b; z). It is established that this function is an
entire function of order 1. The classical factorization theorem of Hadamard
gives an expression as an infinite product. This provides linear and quadratic
recurrences for the hypergeometric zeta function. A family of associated
polynomials is characterized as Appell polynomials and the underlying
distribution is given explicitly in terms of the zeros of the associated
hypergeometric function. These properties are also given a probabilistic
interpretation in the framework of Beta distributions
S-Restricted Compositions Revisited
An S-restricted composition of a positive integer n is an ordered partition
of n where each summand is drawn from a given subset S of positive integers.
There are various problems regarding such compositions which have received
attention in recent years. This paper is an attempt at finding a closed- form
formula for the number of S-restricted compositions of n. To do so, we reduce
the problem to finding solutions to corresponding so-called interpreters which
are linear homogeneous recurrence relations with constant coefficients. Then,
we reduce interpreters to Diophantine equations. Such equations are not in
general solvable. Thus, we restrict our attention to those S-restricted
composition problems whose interpreters have a small number of coefficients,
thereby leading to solvable Diophantine equations. The formalism developed is
then used to study the integer sequences related to some well-known cases of
the S-restricted composition problem
What is good mathematics?
Some personal thoughts and opinions on what ``good quality mathematics'' is,
and whether one should try to define this term rigorously. As a case study, the
story of Szemer\'edi's theorem is presented.Comment: 12 pages, no figures. To appear, Bull. Amer. Math. So
From asymptotics to spectral measures: determinate versus indeterminate moment problems
In the field of orthogonal polynomials theory, the classical Markov theorem
shows that for determinate moment problems the spectral measure is under
control of the polynomials asymptotics. The situation is completely different
for indeterminate moment problems, in which case the interesting spectral
measures are to be constructed using Nevanlinna theory. Nevertheless it is
interesting to observe that some spectral measures can still be obtained from
weaker forms of Markov theorem. The exposition will be illustrated by
orthogonal polynomials related to elliptic functions: in the determinate case
by examples due to Stieltjes and some of their generalizations and in the
indeterminate case by more recent examples.Comment: Lecture given at the International Mediterranean Congress of
Mathematics, Almeria, 6-10 june 2005, latex2e, 16 page
Associated polynomials and birth-death processes
We consider sequences of orthogonal polynomials with positive zeros, and pursue the question of how (partial) knowledge of the orthogonalizing measure for the {\it associated polynomials} can lead to information about the orthogonalizing measure for the original polynomials, with a view to applications in the setting of birth-death processes. In particular, we relate the supports of the two measures, and their moments of positive and negative orders. Our results indicate how the prevalence of recurrence or -recurrence in a birth-death process can be recognized from certain properties of an associated measure. \u
On associated polynomials and decay rates for birth-death processes
We consider sequences of orthogonal polynomials and pursue the question of how (partial) knowledge of the orthogonalizing measure for the {\it associated polynomials} can lead to information about the orthogonalizing measure for the original polynomials. In particular, we relate the supports of the two measures, and their moments. As an application we analyse the relation between two decay rates connected with a birth-death process. \u
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