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400 | The $\bal$\ and $\bcl$\ Bailey Transform and Lemma | math.CA | We announce a higher-dimensional generalization of the Bailey Transform,
Bailey Lemma, and iterative ``Bailey chain'' concept in the setting of basic
hypergeometric series very well-poised on unitary $A_{\ell}$ or symplectic
$C_{\ell}$ groups. The classical case, corresponding to $A_1$ or equivalently
$\roman U(2)$, co... | math |
401 | Convolution polynomials | math.CA | The polynomials that arise as coefficients when a power series is raised to
the power $x$ include many important special cases, which have surprising
properties that are not widely known. This paper explains how to recognize and
use such properties, and it closes with a general result about approximating
such polynomia... | math |
402 | Johann Faulhaber and sums of powers | math.CA | Early 17th-century mathematical publications of Johann Faulhaber contain some
remarkable theorems, such as the fact that the $r$-fold summation of
$1^m,2^m,...,n^m$ is a polynomial in $n(n+r)$ when $m$ is a positive odd
number. The present paper explores a computation-based approach by which
Faulhaber may well have dis... | math |
403 | Singularities of the Radon transform | math.CA | Singularities of the Radon transform of a piecewise smooth function $f(x)$,
$x\in R^n$, $n\geq 2$, are calculated. If the singularities of the Radon
transform are known, then the equations of the surfaces of discontinuity of
$f(x)$ are calculated by applying the Legendre transform to the functions,
which appear in the ... | math |
404 | Best uniform rational approximation of $x^α$ on $[0,1]$ | math.CA | A strong error estimate for the uniform rational approximation of $x^\alpha$
on $[0,1]$ is given, and its proof is sketched. Let $E_{nn}(x^\alpha,[0,1])$
denote the minimal approximation error in the uniform norm. Then it is shown
that $$\lim_{n\to\infty}e^{2\pi\sqrt{\alpha n}}E_{nn}(x^\alpha,[0,1]) =
4^{1+\alpha}|\sin... | math |
405 | On weighted transplantation and multipliers for Laguerre expansions | math.CA | Using the standard square--function method (based on the Poisson semigroup),
multiplier conditions of H\"ormander type are derived for Laguerre expansions
in $L^p$--spaces with power weights in the $A_p$-range; this result can be
interpreted as an ``upper end point'' multiplier criterion which is fairly good
for $p$ ne... | math |
406 | Associated Stieltjes-Carlitz polynomials and a generalization of Heun's differential equation. | math.CA | The generating function of Stieltjes-Carlitz polynomials is a solution of
Heun's differential equation and using this relation Carlitz was the first to
get exact closed forms for some Heun functions. Similarly the associated
Stieltjes-Carlitz polynomials lead to a new differential equation which we call
associated Heun... | math |
407 | A high-school algebra wallet-sized proof, of the Bieberbach conjecture After L. Weinstein] | math.CA | Weinstein's[2] brilliant short proof of de Branges'[1] theorem can be made
yet much shorter(modulo routine calculations), completely elementary (modulo
L\"owner theory), self contained(no need for the esoteric Legendre polynomials'
addition theorem), and motivated(ditto), as follows. | math |
408 | Using sums of squares to prove that certain entire functions have only real zeros | math.CA | It is shown how sums of squares of real valued functions can be used to give
new proofs of the reality of the zeros of the Bessel functions $J_\alpha (z)$
when $\alpha \ge -1,$ confluent hypergeometric functions ${}_0F_1(c\/; z)$ when
$c>0$ or $0>c>-1$, Laguerre polynomials $L_n^\alpha(z)$ when $\alpha \ge -2,$
and Jac... | math |
409 | On necessary multiplier conditions for Laguerre expansions | math.CA | The necessary multiplier conditions for Laguerre expansions derived in Gasper
and Trebels \cite{laguerre} are supplemented and modified. This allows us to
place Markett's Cohen type inequality \cite{cohen} (up to the $\log $--case) in
the general framework of necessary conditions. | math |
410 | Some integrals involving Bessel functions | math.CA | A number of new definite integrals involving Bessel functions are presented.
These have been derived by finding new integral representations for the product
of two Bessel functions of different order and argument in terms of the
generalized hypergeometric function with subsequent reduction to special cases.
Connection ... | math |
411 | Jacobi polynomials of type BC, Jack polynomials, limit transitions and O(\infty) | math.CA | This is an extended abstract of a lecture held at the Conference ``Fourier
and Radon transformations on symmetric spaces'' in honor of Professor S.
Helgason's 65th birthday, Roskilde, Denmark, Sept. 10--12, 1992. | math |
412 | Some results on co-recursive associated Laguerre and Jacobi polynomials | math.CA | We present results on co-recursive associated Laguerre and Jacobi polynomials
which are of interest for the solution of the Chapman-Kolmogorov equations of
some birth and death processes with or without absorption. Explicit forms,
generating functions, and absolutely continuous part of the spectral measures
are given. ... | math |
413 | Painlevé-type differential equations for the recurrence coefficients of semi-classical orthogonal polynomials. | math.CA | Recurrence coefficients of semi-classical orthogonal polynomials (orthogonal
polynomials related to a weight function $w$ such that $w'/w$ is a rational
function) are shown to be solutions of non linear differential equations with
respect to a well-chosen parameter, according to principles established by D.
G. Chudnovs... | math |
414 | The impact of Stieltjes' work on continued fractions and orthogonal polynomials | math.CA | 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. | math |
415 | Generalized Hermite polynomials and the Bose-like oscillator calculus | math.CA | This paper studies a suitably normalized set of generalized Hermite
polynomials and sets down a relevant Mehler formula, Rodrigues formula, and
generalized translation operator. Weighted generalized Hermite polynomials are
the eigenfunctions of a generalized Fourier transform which satisfies an F. and
M. Riesz theorem ... | math |
416 | Uniform multi-parameter limit transitions in the Askey tableau | math.CA | Extended abstract for the Proceedings of the Conference ``Modern developments
in complex analysis and related topics'' (on the occasion of the 70th birthday
of prof.\ dr.\ J. Korevaar), University of Amsterdam, January 27--29, 1993. | math |
417 | A right inverse of the Askey-Wilson operator | math.CA | We establish an integral representation of a right inverse of the
Askey-Wilson finite difference operator on $L^2$ with weight $(1-x^2)^{-1/2}$.
The kernel of this integral operator is $\vartheta'_4/\vartheta_4$ and is the
Riemann mapping function that maps the open unit disc conformally onto the
interior of an ellipse... | math |
418 | Orthogonal matrix polynomials and higher order recurrence relations | math.CA | It is well-known that orthogonal polynomials on the real line satisfy a
three-term recurrence relation and conversely every system of polynomials
satisfying a three-term recurrence relation is orthogonal with respect to some
positive Borel measure on the real line. In this paper we extend this result
and show that ever... | math |
419 | Diagonalization of certain integral operators II | math.CA | We establish an integral representations of a right inverses of the
Askey-Wilson finite difference operator in an $L^2$ space weighted by the
weight function of the continuous $q$-Jacobi polynomials. We characterize the
eigenvalues of this integral operator and prove a $q$-analog of the expansion
of $e^{ixy}$ in Jacobi... | math |
420 | Asymptotic approximations for symmetric elliptic integrals | math.CA | Symmetric elliptic integrals, which have been used as replacements for
Legendre's integrals in recent integral tables and computer codes, are
homogeneous functions of three or four variables. When some of the variables
are much larger than the others, asymptotic approximations with error bounds
are presented. In most c... | math |
421 | Some basic bilateral sums and integrals | math.CA | By splitting the real line into intervals of unit length a doubly infinite
integral of the form $\Int F(q^x)\,dx,\; 0<q<1$, can clearly be expressed as
$\Integ \Sum F(q^{x+n})\,dx$, provided $F$ satisfies the appropriate
conditions. This simple idea is used to prove Ramanujan's integral analogues of
his \ph{1}{1} sum a... | math |
422 | From Schrödinger spectra to orthogonal polynomials, via a functional equation | math.CA | The main difference between certain spectral problems for linear
Schr\"odinger operators, e.g. the almost Mathieu equation, and three-term
recurrence relations for orthogonal polynomials is that in the former the index
ranges across $\ZZ$ and in the latter only across $\Zp$. We present a technique
that, by a mixture of... | math |
423 | Solutions to the associated q-Askey-Wilson polynomial recurrence relation | math.CA | A $\tphin$ contiguous relation is used to derive contiguous relations for a
very-well-poised $\ephis$. These in turn yield solutions to the associated
$q$-Askey-Wilson polynomial recurrence relation, expressions for the associated
continued fraction, the weight function and a $q$-analogue of a generalized
Dougall's the... | math |
424 | Watson's basic analogue of Ramanujan's entry 40 and its generalization | math.CA | We generalize Watson's $ q $-analogue of Ramanujan's Entry 40 continued
fraction by deriving solutions to a $ {}_{10} \phi_9 $ series contiguous
relation and applying Pincherle's theorem. Watson's result is recovered as a
special terminating case, while a limit case yields a new continued fraction
associated with an $ ... | math |
425 | Criterion for the resolvent set of nonsymmetric tridiagonal operators | math.CA | We study nonsymmetric tridiagonal operators acting in the Hilbert space
$\ell^2$ and describe the spectrum and the resolvent set of such operators in
terms of a continued fraction related to the resolvent. In this way we
establish a connection between Pad\'e approximants and spectral properties of
nonsymmetric tridiago... | math |
426 | Weak convergence of orthogonal polynomials | math.CA | The weak convergence of orthogonal polynomials is given under conditions on
the asymptotic behaviour of the coefficients in the three-term recurrence
relation. The results generalize known results and are applied to several
systems of orthogonal polynomials, including orthogonal polynomials on a finite
set of points. | math |
427 | A characterization of the Rogers q-Hermite polynomials | math.CA | In this paper we characterize the Rogers q-Hermite polynomials as the only
orthogonal polynomial set which is also ${\cal D}_q$-Appell where ${\cal D}_q $
is the Askey-Wilson finite difference operator. | math |
428 | Bracket notation for the `coefficient of' operator | math.CA | When $G(z)$ is a power series in $z$, many authors now write `$[z^n] G(z)$'
for the coefficient of $z^n$ in $G(z)$, using a notation introduced by Goulden
and Jackson in [\GJ, p. 1]. More controversial, however, is the proposal of the
same authors [\GJ, p. 160] to let `$[z^n/n!] G(z)$' denote the coefficient of
$z^n/n!... | math |
429 | Relative asymptotics for polynomials orthogonal with respect to a discrete Sobolev inner product | math.CA | We investigate the asymptotic properties of orthogonal polynomials for a
class of inner products including the discrete Sobolev inner products $\langle
h,g \rangle = \int hg\, d\mu + \sum_{j=1}^m \sum_{i=0}^{N_j} M_{j,i}
h^{(i)}(c_j) g^{(i)}(c_j)$, where $\mu$ is a certain type of complex measure on
the real line, and ... | math |
430 | Asymptotics for the simplest generalized Jacobi polynomials recurrence coefficients from Freud's equations: numerical explorations. | math.CA | Generalized Jacobi polynomials are orthogonal polynomials related to a weight
function which is smooth and positive on the whole interval of orthogonality up
to a finite number of points, where algebraic singularities occur. The
influence of these singular points on the asymptotic behaviour of the
recurrence coefficien... | math |
431 | q-Special functions, a tutorial | math.CA | A tutorial introduction is given to q-special functions and to q-analogues of
the classical orthogonal polynomials, up to the level of Askey-Wilson
polynomials. | math |
432 | Formal power series | math.CA | In this article we will describe the \Maple\ implementation of an algorithm
presented in~\cite{Koe92}--\cite{Koeortho} which computes an {\em exact\/}
formal power series (FPS) of a given function. This procedure will enable the
user to reproduce most of the results of the extensive bibliography on
series~\cite{Han}. W... | math |
433 | Algorithmic work with orthogonal polynomials and special functions | math.CA | In this article we present a method to implement orthogonal polynomials and
many other special functions in Computer Algebra systems enabling the user to
work with those functions appropriately, and in particular to verify different
types of identities for those functions. Some of these identities like
differential equ... | math |
434 | Spaces of functions satisfying simple differential equations | math.CA | In \cite{Koe92}--\cite{Koe93c} the first author published an algorithm for
the conversion of analytic functions for which derivative rules are given into
their representing power series $\sum\limits_{k=0}^{\infty}a_{k}z^{k}$ at the
origin and vice versa, implementations of which exist in {\sc Mathematica}
\cite{Wol}, (... | math |
435 | Biorthogonal polynomials and zero-mapping transformations | math.CA | The authors have presented in \cite{IN2} a technique to generate
transformations $\cal T$ of the set ${\Bbb P}_n$ of $n$th degree polynomials to
itself such that if $p\in{\Bbb P}_n$ has all its zeros in $(c,d)$ then ${\cal
T}\{p\}$ has all its zeros in $(a,b)$, where $(a,b)$ and $(c,d)$ are given real
intervals. The te... | math |
436 | Explicit representations of biorthogonal polynomials | math.CA | Given a parametrised weight function $\omega(x,\mu)$ such that the quotients
of its consecutive moments are M\"obius maps, it is possible to express the
underlying biorthogonal polynomials in a closed form \cite{IN2}. In the present
paper we address ourselves to two related issues. Firstly, we demonstrate that,
subject... | math |
437 | Q-Hermite polynomials and classical orthogonal polynomials | math.CA | We use generating functions to express orthogonality relations in the form of
$q$-beta integrals. The integrand of such a $q$-beta integral is then used as a
weight function for a new set of orthogonal or biorthogonal | math |
438 | Characterizations of generalized Hermite and sieved ultraspherical polynomials | math.CA | A new characterization of the generalized Hermite polynomials and of the
orthogonal polynomials with respect to the maesure $|x|^\g (1-x^2)^{\a-1/2}dx$
is derived which is based on a "reversing property" of the coefficients in the
corresponding recurrence formulas and does not use the representation in terms
of general... | math |
439 | A note on some peculiar nonlinear extremal phenomena of the Chebyshev polynomials | math.CA | We consider the problem of maximizing the sum of squares of the leading
coefficients of polynomials $P_{i_1}(x),\ldots ,P_{i_m}(x)$ (where $P_j(x)$ is
a polynomial of degree $j$) under the restriction that the sup-norm of
$\sum_{j=1}^m P_{i_j}^2(x)$ is bounded on the interval $[-b,b]$ ($b>0$). A
complete solution of th... | math |
440 | New bounds for Hahn and Krawichouk polynomials | math.CA | For the Hahn and Krawtchouk polynomials orthogonal on the set $\{0,
\ldots,N\}$ new identities for the sum of squares are derived which generalize
the trigonometric identity for the Chebyshev polynomials of the first and
second kind. These results are applied in order to obtain conditions (on the
degree of the polynomi... | math |
441 | Some new asymptotic properties for the zeros of Jacobi, Laguerre and Hermite polynomials | math.CA | For the generalized Jacobi, Laguerre and Hermite polynomials $P_n^{(\alpha_n,
\beta_n)} (x), L_n^{(\alpha_n)} (x),$\break $H_n^{(\gamma_n)} (x)$ the limit
distributions of the zeros are found, when the sequences $\alpha_n$ or
$\beta_n$ tend to infinity with a larger order than $n$. The derivation uses
special propertie... | math |
442 | Ladder operators for Szegő polynomials and related biorthogonal rational functions | math.CA | We find the raising and lowering operators for orthogonal polynomials on the
unit circle introduced by Szeg\H{o} and for their four parameter generalization
to ${}_4\phi_3$ biorthogonal rational functions on the unit circle. | math |
443 | Generalized orthogonality and continued fractions | math.CA | The connection between continued fractions and orthogonality which is
familiar for $J$-fractions and $T$-fractions is extended to what we call
$R$-fractions of type I and II. These continued fractions are associated with
recurrence relations that correspond to multipoint rational interpolants. A
Favard type theorem is ... | math |
444 | The Askey-Wilson polynomials and q-Sturm-Lioville problems | math.CA | We find the adjoint of the Askey-Wilson divided difference operator with
respect to the inner procuct on L^2(-1,1,(1-x^2)^-1/2 dx) defined as a Cauchy
principle value and show that the Askey-Wilson polynomials are solutions of a
q-Sturm-Liouville problem. From these facts we deduce various properties of the
polynomials... | math |
445 | Fractional integration for Laguerre expansions | math.CA | The aim of this note is to provide a fractional integration theorem in the
framework of Laguerre expansions. The method of proof consists of establishing
an asymptotic estimate for the involved kernel and then applying a method of
Hedberg \cite{pro}. We combine this result with sufficient $(p,p)$ multiplier
criteria of... | math |
446 | On a restriction problem of de Leeuw type for Laguerre multipliers | math.CA | In 1965 K. de Leeuw \cite{deleeuw} proved among other things in the Fourier
transform setting: {\it If a continuous function $m(\xi _1, \ldots ,\xi _n)$ on
${\bf R}^n$ generates a bounded transformation on $L^p({\bf R}^n),\; 1\le p \le
\infty ,$ then its trace $\tilde{m}(\xi _1, \ldots ,\xi _m)=m(\xi _1, \ldots
,\xi _m... | math |
447 | Numerical computation of real or complex elliptic integrals | math.CA | Algorithms for numerical computation of symmetric elliptic integrals of all
three kinds are improved in several ways and extended to complex values of the
variables (with some restrictions in the case of the integral of the third
kind). Numerical check values, consistency checks, and relations to Legendre's
integrals a... | math |
448 | Painlevé equations for semi-classical recurrence coefficients | math.CA | The title says it all. | math |
449 | The last of the hypergeometric continued fractions | math.CA | A contiguous relation for complementry pairs of very well poised balanced
${}_{10}\phi_9$ basic hypergeometric functions is used to derive an explict
expression for the associated continued fraction. This generalizes the
continued fraction results associated with both Ramanujan's Entry 40 and
Askey-Wilson polynomials w... | math |
450 | On Jacobi and continuous Hahn polynomials | math.CA | Jacobi polynomials are mapped onto the continuous Hahn polynomials by the
Fourier transform and the orthogonality relations for the continuous Hahn
polynomials then follow from the orthogonality relations for the Jacobi
polynomials and the Parseval formula. In a special case this relation dates
back to work by Bateman ... | math |
451 | Yet another basic analogue of Graf's addition formula | math.CA | An identity involving basic Bessel functions and Al-Salam--Chihara
polynomials is proved for which we recover Graf's addition formula for the
Bessel function as the base $q$ tends to $1$. The corresponding product formula
is derived. Some known identities for Jackson's $q$-Bessel functions are
obtained as limiting case... | math |
452 | The quadratic formula made hard: A less radical approach to solving equations | math.CA | It appears that, along with many of my friends and colleagues, I had been
brainwashed by the great and tragic lives of Abel and Galois to believe that no
general formulas are possible for roots of equations higher than quartic. This
seemed to be confirmed by the brilliant and arduous solution of the general
quintic by ... | math |
453 | Contiguous relations, basic hypergeometric functions, and orthogonal polynomials : III. associated continuous dual q-Hahn polynomials | math.CA | Explicit solutions for the three-term recurrence satisfied by associated
continuous dual $q$-Hahn polynomials are obtained. A minimal solution is
identified and an explicit expression for the related continued fraction is
derived. The absolutely continuous component of the spectral measure is
obtained. Eleven limit cas... | math |
454 | Extensions and results from a method for evaluating fractional integrals | math.CA | We present a method derived from Laplace transform theory that enables the
evaluation of fractional integrals. This method is adapted and extended in a
variety of ways to demonstrate its utility in deriving alternative
representations for other classes of integrals. We also use the method in
conjunction with several di... | math |
455 | Addition formula for 2-parameter family of Askey-Wilson polynomials | math.CA | For a two parameter family of Askey-Wilson polynomials, that can be regarded
as basic analogues of the Legendre polynomials, an addition formula is derived.
The addition formula is a two-parameter extension of Koornwinder's addition
formula for the little $q$-Legendre polynomials. A corresponding product
formula is der... | math |
456 | Algebraische Darstellung transzendenter Funktionen | math.CA | Ich m\"ochte in diesem Bericht algorithmische Methoden vorstellen, die im
wesentlichen in diesem Jahrzehnt Einzug in die Computeralgebra gefunden haben.
Die haupts\"achlichen Ideen gehen auf Stanley \cite{Sta} und Zeilberger
\cite{Zei1}--\cite{Zei4} zur\"uck, vgl.\ die Beschreibung \cite{Strehl1}, und
haben ihre Wurzel... | math |
457 | Algorithms for the indefinite and definite summation | math.CA | The celebrated Zeilberger algorithm which finds holonomic recurrence
equations for definite sums of hypergeometric terms $F(n,k)$ is extended to
certain nonhypergeometric terms. An expression $F(n,k)$ is called a
hypergeometric term if both $F(n+1,k)/F(n,k)$ and $F(n,k+1)/F(n,k)$ are
rational functions. Typical example... | math |
458 | REDUCE package for the indefinite and definite summation | math.CA | This article describes the REDUCE package ZEILBERG implemented by Gregor
St\"olting and the author.
The REDUCE package ZEILBERG is a careful implementation of the Gosper and
Zeilberger algorithms for indefinite, and definite summation of hypergeometric
terms, respectively. An expression $a_k$ is called a {\sl hyperge... | math |
459 | Speed of convergence of two-dimensional Fourier integrals | math.CA | Recently we found necessary and sufficient conditions for the convergence at
a preassigned point of the spherical partial sums of the Fourier integral in a
class of piecewise smooth functions in Euclidean space. These yield elementary
examples of divergent Fourier integrals in three dimensions and higher.
Meanwhile, se... | math |
460 | How many zeros of a random polynomial are real? | math.CA | We provide an elementary geometric derivation of the Kac integral formula for
the expected number of real zeros of a random polynomial with independent
standard normally distributed coefficients. We show that the expected number of
real zeros is simply the length of the moment curve $(1,t,\ldots,t^n)$
projected onto th... | math |
461 | Wiener's Tauberian theorem in L^1(G//K) and harmonic functions in the unit disk | math.CA | Our main result is to give necessary and sufficient conditions, in terms of
Fourier transforms, on a closed ideal $I$ in $\loneg$, the space of radial
integrable functions on $G=SU(1,1)$, so that $I=\loneg$ or $I=\lonez$---the
ideal of $\loneg$ functions whose integral is zero. This is then used to prove
a generalizati... | math |
462 | Orthogonal polynomials and Laurent polynomials related to the Hahn-Exton q-Bessel function | math.CA | Laurent polynomials related to the Hahn-Exton $q$-Bessel function, which are
$q$-analogues of the Lommel polynomials, have been introduced by Koelink and
Swarttouw. The explicit strong moment functional with respect to which the
Laurent $q$-Lommel polynomials are orthogonal is given. The strong moment
functional gives ... | math |
463 | Special non uniform lattice ($snul$) orthogonal polynomials on discrete dense sets of points. | math.CA | Difference calculus compatible with polynomials (i.e., such that the divided
difference operator of first order applied to any polynomial must yield a
polynomial of lower degree) can only be made on special lattices well known in
contemporary $q-$calculus. Orthogonal polynomials satisfying difference
relations on such ... | math |
464 | Principal pairs for oscillatory second order linear differential equations | math.CA | Nonoscillatory second order differential equations always admit ``special'',
principal solutions. For a certain type of oscillatory equation principal pairs
of solutions were introduced by \'A. Elbert, F. Neuman and J. Vosmansk\'y, {\em
Diff. Int. Equations} {\bf 5} (1992), 945--960. In this paper, the notion of
princi... | math |
465 | A Riemann--Lebesgue lemma for Jacobi expansions | math.CA | A Lemma of Riemann--Lebesgue type for Fourier--Jacobi coefficients is
derived. Via integral representations of Dirichlet--Mehler type for Jacobi
polynomials its proof directly reduces to the classical Riemann--Lebesgue Lemma
for Fourier coefficients. Other proofs are sketched. Analogous results are also
derived for Lag... | math |
466 | Ultraspherical multipliers revisited | math.CA | Sufficient ultraspherical multiplier criteria are refined in such a way that
they are comparable with necessary multiplier conditions. Also new necessary
conditions for Jacobi multipliers are deduced which, in particular, imply known
Cohen type inequalities. Muckenhoupt's transplantation theorem is used in an
essential... | math |
467 | Weighted norm inequalities for polynomial expansions associated to some measures with mass points | math.CA | Fourier series in orthogonal polynomials with respect to a measure $\nu$ on
$[-1,1]$ are studied when $\nu$ is a linear combination of a generalized Jacobi
weight and finitely many Dirac deltas in $[-1,1]$. We prove some weighted norm
inequalities for the partial sum operators $S_n$, their maximal operator $S^*$
and th... | math |
468 | Lecture notes for an introductory minicourse on q-series | math.CA | These lecture notes were written for a mini-course that was designed to
introduce students and researchers to {\it $q$-series,} which are also called
{\it basic hypergeometric series} because of the parameter $q$ that is used as
a base in series that are ``{\it over, above or beyond}'' the {\it geometric
series}. We st... | math |
469 | Perturbation of orthogonal polynomials on an arc of the unit circle | math.CA | Orthogonal polynomials on the unit circle are completely determined by their
reflection coefficients through the Szeg\H{o} recurrences. We assume that the
reflection coefficients converge to some complex number a with 0 < |a| < 1. The
polynomials then live essentially on the arc {e^(i theta): alpha <= theta <= 2
pi - a... | math |
470 | Schur functions and orthogonal polynomials on the unit circle | math.CA | We apply a theorem of Geronimus to derive some new formulas connecting Schur
functions with orthogonal polynomials on the unit circle. The applications
include the description of the associated measures and a short proof of Boyd's
result about Schur functions. We also give a simple proof for the above
mentioned theorem... | math |
471 | Upward extension of the Jacobi matrix for orthogonal polynomials | math.CA | Orthogonal polynomials on the real line always satisfy a three-term
recurrence relation. The recurrence coefficients determine a tridiagonal
semi-infinite matrix (Jacobi matrix) which uniquely characterizes the
orthogonal polynomials. We investigate new orthogonal polynomials by adding to
the Jacobi matrix $r$ new rows... | math |
472 | Compact Jacobi matrices: from Stieltjes to Krein and M(a,b) | math.CA | In a note at the end of his paper {\it Recherches sur les fractions
continues}, Stieltjes gave a necessary and sufficient condition when a
continued fraction is represented by a meromorphic function. This result is
related to the study of compact Jacobi matrices. We indicate how this notion
was developped and used sinc... | math |
473 | Contiguous relations, continued fractions and orthogonality | math.CA | We examine a special linear combination of balanced very-well-poised $\tphia$
basic hypergeometric series that is known to satisfy a transformation. We call
this $\Phi$ and show that it satisfies certain three-term contiguous relations.
From two sets of contiguous relations for $\Phi$ we obtain fifty-six pairwise
linea... | math |
474 | Transformation and summation formulas for Kampe de Feriet series. | math.CA | The double hypergeometric Kamp\'e de F\'eriet series $F^{0:3}_{1:1}(1,1)$
depends upon 9 complex parameters. We present three cases with 2 relations
between those 9 parameters, and show that under these circumstances
$F^{0:3}_{1:1}(1,1)$ can be written as a ${}_4F_3(1)$ series. Some limiting
cases of these transformati... | math |
475 | Extremal solutions of the two-dimensional $L$-problem of moments, II | math.CA | All extremal solutions of the truncated $L$-problem of moments in two real
variables , with support contained in a given compact set, are described as
characteristic functions of semi-algebraic sets given by a single polynomial
inequality. An exponential kernel, arising as the determinantal function of a
naturally asso... | math |
476 | Approximation by analytic matrix functions. The four block problem | math.CA | We study the problem of finding a superoptimal solution to the four block
problem. Given a bounded block matrix function
$\left(\begin{array}{cc}\Phi_{11}
&\Phi_{12}\\\Phi_{21}&\Phi_{22}\end{array}\right)$ on the unit circle the four
block problem is to minimize the $L^\infty$ norm of $\left(\begin{array}{cc}
\Phi_{11}... | math |
477 | On the inversion of $y^αe^y$ in terms of associated Stirling numbers | math.CA | The function $y=\Phi_\alpha(x)$, the solution of $y^\alpha e^y=x$ for $x$ and
$y$ large enough, has a series expansion in terms of $\ln x$ and $\ln\ln x$,
with coefficients given in terms of Stirling cycle numbers. It is shown that
this expansion converges for $x>(\alpha e)^\alpha$ for $\alpha \ge 1$. It is
also shown ... | math |
478 | Estimates for Jacobi-Sobolev type orthogonal polynomials | math.CA | Let the Sobolev-type inner product <f,g> = \int fg d mu_0+ int f' g' d mu_1
with mu_0 = w + M delta_c, mu_1= N delta_c where w is the Jacobi weight, c is
either 1 or -1 and M, N >= 0. We obtain estimates and asymptotic properties on
[-1,1] for the polynomials orthonormal with respect to <.,.> and their kernels.
We also... | math |
479 | The Askey-scheme of hypergeometric orthogonal polynomials and its q-analogue | math.CA | We list the so-called Askey-scheme of hypergeometric orthogonal polynomials.
In chapter 1 we give the definition, the orthogonality relation, the three term
recurrence relation and generating functions of all classes of orthogonal
polynomials in this scheme. In chapeter 2 we give all limit relation between
different cl... | math |
480 | The identification problem for transcendental functions | math.CA | In this article algorithmic methods are presented that have essentially been
introduced into computer algebra systems like Mathematica within the last
decade. The main ideas are due to Stanley and Zeilberger. Some of them had
already been discovered in the last century by Beke, but because of their
complexity the under... | math |
481 | On the De Branges theorem | math.CA | Recently, Todorov and Wilf independently realized that de Branges' original
proof of the Bieberbach and Milin conjectures and the proof that was later
given by Weinstein deal with the same special function system that de Branges
had introduced in his work.
In this article, we present an elementary proof of this state... | math |
482 | Weinstein's functions and the Askey-Gasper identity | math.CA | In his 1984 proof of the Bieberbach and Milin conjectures de Branges used a
positivity result of special functions which follows from an identity about
Jacobi polynomial sums that was found by Askey and Gasper in 1973, published in
1976.
In 1991 Weinstein presented another proof of the Bieberbach and Milin
conjecture... | math |
483 | Uniform asymptotics for the incomplete gamma functions starting from negative values of the parameters | math.CA | We consider the asymptotic behavior of the incomplete gamma functions
gamma(-a,-z) and Gamma(-a,-z) as a goes to infinity. Uniform expansions are
needed to describe the transition area z~a in which case error functions are
used as main approximants. We use integral representations of the incomplete
gamma functions and ... | math |
484 | Hankel Multipliers And Transplantation Operators | math.CA | Connections between Hankel transforms of different order for $L^p$-functions
are examined. Well known are the results of Guy [Guy] and Schindler [Sch].
Further relations result from projection formulae for Bessel functions of
different order. Consequences for Hankel multipliers are exhibited and
implications for radial... | math |
485 | Preud's equations for orthogonal polynomials as discrete Painlevé equations | math.CA | We consider orthogonal polynomials p_n with respect to an exponential weight
function w(x) = exp(-P(x)). The related equations for the recurrence
coefficients have been explored by many people, starting essentially with
Laguerre [49], in order to study special continued fractions, recurrence
relations, and various asym... | math |
486 | Algorithms for classical orthogonal polynomials | math.CA | In this article explicit formulas for the recurrence equation
p_{n+1}(x) = (A_n x + B_n) p_n(x) - C_n p_{n-1}(x)
and the derivative rules
sigma(x) p'_n(x) = alpha_n p_{n+1}(x) + beta_n p_n(x) + gamma_n p_{n-1}(x)
and
sigma(x) p'_n(x) = (alpha_n-tilde x + beta_n-tilde) p_n(x) + gamma_n-tilde
p_{n-1}(x)
respe... | math |
487 | On a problem of Koornwinder | math.CA | In this note we solve a problem about the rational representablility of
hupergeometric terms which represent hypergeometric sums. This problem was
proposed by Koornwinder in [4]. | math |
488 | On the zeros of the Hahn-Exton q-Bessel function and associated q-Lommel polynomials | math.CA | For the Bessel function \begin{equation} \label{bessel} J_{\nu}(z) =
\sum\limits_{k=0}^{\infty} \frac{(-1)^k \left( \frac{z}{2} \right)^{\nu+2k}}{k!
\Gamma(\nu+1+k)} \end{equation} there exist several $q$-analogues. The oldest
$q$-analogues of the Bessel function were introduced by F. H. Jackson at the
beginning of thi... | math |
489 | Representations of orthogonal polynomials | math.CA | Zeilberger's algorithm provides a method to compute recurrence and
differential equations from given hypergeometric series representations, and an
adaption of Almquist and Zeilberger computes recurrence and differential
equations for hyperexponential integrals. Further versions of this algorithm
allow the computation o... | math |
490 | Errata, updates of the references, etc., for the book Basic Hypergeometric Series | math.CA | Here are the latest errata, etc., to the Gasper and Rahman "Basic
Hypergeometric Series" book. Any additional errata will be added to the end of
the last list. | math |
491 | Basic analog of Fourier series on a {\large $\que$}-quadratic grid | math.CA | We prove orthogonality relations for some analogs of trigonometric functions
on a $q$-quadratic grid and introduce the corresponding $q$-Fourier series. We
also discuss several other properties of this basic trigonometric system and
the $q$-Fourier series. | math |
492 | Some orthogonal very-well-poised $_8\varphi_7$-functions that generalize Askey-Wilson polynomials | math.CA | In a recent paper Ismail, Masson, and Suslov have established a continuous
orthogonality relation and some other properties of a $_2\varphi_1$-Bessel
function on a $q$-quadratic grid. Dick Askey suggested that the ``Bessel-type
orthogonality'' at the $_2\varphi_1$-level has really a general character and
can be extende... | math |
493 | Correlation between pole location and asymptotic behavior for Painlevé I solutions | math.CA | We extend the technique of asymptotic series matching to exponential
asymptotics expansions (transseries) and show that the extension provides a
method of finding singularities of solutions of nonlinear differential
equations, using asymptotic information. This transasymptotic matching method
is applied to Painlev\'e's... | math |
494 | On sums of powers of zeros of polynomials | math.CA | Due to Girard's (sometimes called Waring's) formula the sum of the $r-$th
power of the zeros of every one variable polynomial of degree $N$, $P_{N}(x)$,
can be given explicitly in terms of the coefficients of the monic ${\tilde
P}_{N}(x)$ polynomial. This formula is closely related to a known \par
\noindent $N-1$ varia... | math |
495 | On the h-function | math.CA | The paper is devoted to study the $H$-function defined by the Mellin-Barnes
integral
$$H^{m,n}_{\thinspace p,q}(z)={\frac1{2\pi i}}\int_{\Lss}
\HHs^{m,n}_{\thinspace p,q}(s)z^{-s}ds,$$
where the function $\HH^{m,n}_{\thinspace p,q}(s)$ is a certain ratio of
products of Gamma functions with the argument $s$ and the ... | math |
496 | Matrix-variate growth-decay models | math.CA | Input-output, growth-decay, production-consumption type situations abound in
many practical problems. When the input and output variables are independently
gamma distributed, various aspects of the residual effect are already tackled
by the author. Matrix-variate analogues, their connections to quadratic and
bilinear f... | math |
497 | Expansions of _4F_3 when the upper parameters differ by integers | math.CA | In this article three expansion formulas for a generalized hypergeometric
function $_4F_3$ are derived, when its upper parameters differ by integers.
Though the results are special cases of a general continuation formula for
$_pF_q$, they are sufficiently general and unify a number of known results. | math |
498 | On inversion of H-Transform in $\eufb114_{ν,r}$-space | math.CA | The paper is devoted to study the inversion of the integral transform
$$(\mbox{\boldmath$H$}f)(x)=\int^\infty_0H^{m,n}_{\thinspace p,q}
\left[xt\left|\begin{array}{c}(a_i,\alpha_i)_{1,p}\\[1mm](b_j,\beta_j)_{1,q}
\end{array}\right.\right]f(t)dt$$
involving the $H$-function as the kernel in the space $\euf114_{\nu ,... | math |
499 | On the existence of doubling measures with certain regularity properties | math.CA | Given a compact pseudo-metric space, we associate to it upper and lower
dimensions, depending only on the metric. Then we construct a doubling metric
for which the measure of a dillated ball is closely related to these
dimensions. | math |
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