Search: MSC category 33C45
( Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.) [See also 42C05 for general orthogonal polynomials and functions] )
1. CMB Online first
 Awonusika, Richard; Taheri, Ali

A spectral identity on Jacobi polynomials and its analytic implications
The Jacobi coefficients $c^{\ell}_{j}(\alpha,\beta)$ ($1\leq
j\leq \ell$, $\alpha,\beta\gt 1$) are linked to the Maclaurin
spectral expansion of the Schwartz kernel of functions of the
Laplacian on a compact rank one symmetric space. It
is proved that these coefficients can be computed by transforming
the even derivatives of the the Jacobi polynomials $P_{k}^{(\alpha,\beta)}$ ($k\geq 0, \alpha,\beta\gt 1$) into a spectral sum associated with
the Jacobi operator. The first few coefficients are explicitly
computed and a direct trace
interpretation of the Maclaurin coefficients is presented.
Keywords:Jacobi coefficient, LaplaceBeltrami operator, symmetric space, Maclaurin expansion, Jacobi polynomial Categories:33C05, 33C45, 35A08, 35C05, 35C10, 35C15 

2. CMB 2015 (vol 58 pp. 877)
 Zaatra, Mohamed

Generating Some Symmetric Semiclassical Orthogonal Polynomials
We show that if $v$ is a regular semiclassical form
(linear functional), then the symmetric form $u$ defined by the
relation
$x^{2}\sigma u = \lambda v$,
where $(\sigma f)(x)=f(x^{2})$ and the odd
moments of $u$ are $0$, is also
regular and semiclassical form for every
complex $\lambda $ except for a discrete set of numbers depending
on $v$. We give explicitly the threeterm recurrence relation
and the
structure relation coefficients of the orthogonal polynomials
sequence associated with $u$ and the class of the form $u$ knowing
that of $v$. We conclude with an illustrative example.
Keywords:orthogonal polynomials, quadratic decomposition, semiclassical forms, structure relation Categories:33C45, 42C05 

3. CMB 2015 (vol 58 pp. 423)
 Yamagishi, Masakazu

Resultants of Chebyshev Polynomials: The First, Second, Third, and Fourth Kinds
We give an explicit formula for the resultant of Chebyshev polynomials of the
first, second, third, and fourth kinds.
We also compute the resultant of modified cyclotomic polynomials.
Keywords:resultant, Chebyshev polynomial, cyclotomic polynomial Categories:11R09, 11R18, 12E10, 33C45 

4. CMB 2008 (vol 51 pp. 561)
5. CMB 2002 (vol 45 pp. 567)
 De Sole, Alberto; Kac, Victor G.

Subalgebras of $\gc_N$ and Jacobi Polynomials
We classify the subalgebras of the general Lie conformal algebra
$\gc_N$ that act irreducibly on $\mathbb{C} [\partial]^N$ and that
are normalized by the sl$_2$part of a Virasoro element. The
problem turns out to be closely related to classical Jacobi
polynomials $P_n^{(\sigma,\sigma)}$, $\sigma \in \mathbb{C}$. The
connection goes both wayswe use in our classification some
classical properties of Jacobi polynomials, and we derive from the
theory of conformal algebras some apparently new properties of
Jacobi polynomials.
Categories:17B65, 17B68, 17B69, 33C45 

6. CMB 2000 (vol 43 pp. 496)
 Xu, Yuan

Harmonic Polynomials Associated With Reflection Groups
We extend Maxwell's representation of harmonic polynomials to $h$harmonics
associated to a reflection invariant weight function $h_k$. Let $\CD_i$,
$1\le i \le d$, be Dunkl's operators associated with a reflection group.
For any homogeneous polynomial $P$ of degree $n$, we prove the
polynomial $\xb^{2 \gamma +d2+2n}P(\CD)\{1/\xb^{2 \gamma +d2}\}$ is
a $h$harmonic polynomial of degree $n$, where $\gamma = \sum k_i$ and
$\CD=(\CD_1,\ldots,\CD_d)$. The construction yields a basis for
$h$harmonics. We also discuss selfadjoint operators acting on the
space of $h$harmonics.
Keywords:$h$harmonics, reflection group, Dunkl's operators Categories:33C50, 33C45 
