Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Using $\D$-operators to construct orthogonal polynomials satisfying
  higher order difference or differential equations   We introduce the concept of $\D$-operators associated to a sequence of
polynomials $(p_n)_n$ and an algebra $\A$ of operators acting in the linear
space of polynomials. In this paper, we show that this concept is a powerful
tool to generate families of orthogonal polynomials which are eigenfunctions of
a higher order difference or differential operator. Indeed, given a classical
discrete family $(p_n)_n$ of orthogonal polynomials (Charlier, Meixner,
Krawtchouk or Hahn), we form a new sequence of polynomials $(q_n)_n$ by
considering a linear combination of two consecutive $p_n$:
$q_n=p_n+\beta_np_{n-1}$, $\beta_n\in \RR$. Using the concept of $\D$-operator,
we determine the structure of the sequence $(\beta_n)_n$ in order that the
polynomials $(q_n)_n$ are common eigenfunctions of a higher order difference
operator. In addition, we generate sequences $(\beta_n)_n$ for which the
polynomials $(q_n)_n$ are also orthogonal with respect to a measure. The same
approach is applied to the classical families of Laguerre and Jacobi
polynomials.

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The categories of this paper are: atom-ph
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The categories of this paper are: nucl-th
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The categories of this paper are: funct-an
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The categories of this paper are: adap-org
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The categories of this paper are: solv-int
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The categories of this paper are: q-alg
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The categories of this paper are: gr-qc
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The categories of this paper are: dg-ga
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The categories of this paper are: acc-phys
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The categories of this paper are: hep-ex
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The categories of this paper are: bayes-an
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The categories of this paper are: plasm-ph
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The categories of this paper are: nucl-ex
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The categories of this paper are: chao-dyn
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The categories of this paper are: q-fin
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The categories of this paper are: chem-ph