The spectral analysis of three families of exceptional Laguerre polynomials

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• 作者：Constanze Liaw^a ; ^{Constanze_Liaw@baylor.edu" class="auth_mail" title="E-mail the corresponding author} ; Lance L. Littlejohn^a ; ^{Lance_Littlejohn@baylor.edu" class="auth_mail" title="E-mail the corresponding author} ; Robert Milson^b ; ^{robert.milson@dal.ca" class="auth_mail" title="E-mail the corresponding author} ; Jessica Stewart^a ; ^{Jessica_Stewart@baylor.edu" class="auth_mail" title="E-mail the corresponding author}
• 关键词：primary ; 33C45 ; 34B24 ; 42C05 ; secondary ; 33C47 ; 47B25
• 刊名：Journal of Approximation Theory
• 出版年：2016
• 出版时间：February 2016
• 年：2016
• 卷：202
• 期：Complete
• 页码：5-41
• 全文大小：336 K

文摘

The Bochner Classification Theorem (1929) characterizes the polynomial sequences class="mathmlsrc">title="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021904515001598&_mathId=si1.gif&_user=111111111&_pii=S0021904515001598&_rdoc=1&_issn=00219045&md5=6cca6d261bbe401e9b5a352eba2b46dd">class="imgLazyJSB inlineImage" height="15" width="46" alt="View the MathML source" style="margin-top: -5px; vertical-align: middle" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0021904515001598-si1.gif">class="mathContainer hidden">class="mathCode">${\left\{p_n\right\}}_{n=0}^{\infty }$, with class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021904515001598&_mathId=si2.gif&_user=111111111&_pii=S0021904515001598&_rdoc=1&_issn=00219045&md5=723a17cafe05f624fea5245b0474fe19" title="Click to view the MathML source">degp_n=nclass="mathContainer hidden">class="mathCode">$deg{p}_n=n$ that simultaneously form a complete set of eigenstates for a second-order differential operator and are orthogonal with respect to a positive Borel measure having finite moments of all orders. Indeed, up to a complex linear change of variable, only the classical Hermite, Laguerre, and Jacobi polynomials, with certain restrictions on the polynomial parameters, satisfy these conditions. In 2009, Gómez-Ullate, Kamran, and Milson found that for sequences class="mathmlsrc">title="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021904515001598&_mathId=si3.gif&_user=111111111&_pii=S0021904515001598&_rdoc=1&_issn=00219045&md5=9537b63341f5d44c59ccb1c5535393d9">class="imgLazyJSB inlineImage" height="15" width="46" alt="View the MathML source" style="margin-top: -5px; vertical-align: middle" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0021904515001598-si3.gif">class="mathContainer hidden">class="mathCode">${\left\{p_n\right\}}_{n=1}^{\infty }$, class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021904515001598&_mathId=si2.gif&_user=111111111&_pii=S0021904515001598&_rdoc=1&_issn=00219045&md5=723a17cafe05f624fea5245b0474fe19" title="Click to view the MathML source">degp_n=nclass="mathContainer hidden">class="mathCode">$deg{p}_n=n$ (without the constant polynomial), the only such sequences satisfying these conditions are the exceptional  class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021904515001598&_mathId=si5.gif&_user=111111111&_pii=S0021904515001598&_rdoc=1&_issn=00219045&md5=8ba8069a2ad35c81c6fb318a69fd3429" title="Click to view the MathML source">X₁class="mathContainer hidden">class="mathCode">$X_1$-Laguerre and class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021904515001598&_mathId=si5.gif&_user=111111111&_pii=S0021904515001598&_rdoc=1&_issn=00219045&md5=8ba8069a2ad35c81c6fb318a69fd3429" title="Click to view the MathML source">X₁class="mathContainer hidden">class="mathCode">$X_1$-Jacobi polynomials. Subsequently, during the past five years, several mathematicians and physicists have discovered and studied other exceptional orthogonal polynomials class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021904515001598&_mathId=si7.gif&_user=111111111&_pii=S0021904515001598&_rdoc=1&_issn=00219045&md5=071128957dc21be1f5674db307f3484f" title="Click to view the MathML source">{p_n}_n∈N0⧵Aclass="mathContainer hidden">class="mathCode">${\left\{p_n\right\}}_{n\in {\mathbb{N}}_0⧵A}$, where class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021904515001598&_mathId=si13.gif&_user=111111111&_pii=S0021904515001598&_rdoc=1&_issn=00219045&md5=20c0814113700bbe73fd23dc6a513425" title="Click to view the MathML source">Aclass="mathContainer hidden">class="mathCode">$A$ is a finite subset of the non-negative integers class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021904515001598&_mathId=si19.gif&_user=111111111&_pii=S0021904515001598&_rdoc=1&_issn=00219045&md5=701b26bf7d6a6746a58b15ade358975e" title="Click to view the MathML source">N₀class="mathContainer hidden">class="mathCode">${\mathbb{N}}_0$ and where class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021904515001598&_mathId=si2.gif&_user=111111111&_pii=S0021904515001598&_rdoc=1&_issn=00219045&md5=723a17cafe05f624fea5245b0474fe19" title="Click to view the MathML source">degp_n=nclass="mathContainer hidden">class="mathCode">$deg{p}_n=n$ for all class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021904515001598&_mathId=si11.gif&_user=111111111&_pii=S0021904515001598&_rdoc=1&_issn=00219045&md5=b4faf92a40979fcf2f1e9580a2be2d6e" title="Click to view the MathML source">n∈N₀⧵Aclass="mathContainer hidden">class="mathCode">$n\in {\mathbb{N}}_0⧵A$. We call such a sequence an exceptional polynomial sequence of codimension class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021904515001598&_mathId=si12.gif&_user=111111111&_pii=S0021904515001598&_rdoc=1&_issn=00219045&md5=3e2bf59cb01110987e6eaa4721d232fa" title="Click to view the MathML source">|A|class="mathContainer hidden">class="mathCode">$\left|A\right|$, where the latter denotes the cardinality of class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021904515001598&_mathId=si13.gif&_user=111111111&_pii=S0021904515001598&_rdoc=1&_issn=00219045&md5=20c0814113700bbe73fd23dc6a513425" title="Click to view the MathML source">Aclass="mathContainer hidden">class="mathCode">$A$. All exceptional sequences with a non singular weight, found to date, have the remarkable feature that they form a complete orthogonal set in their natural Hilbert space setting.

Among the exceptional sets already known are two types of exceptional Laguerre polynomials, called the Type I and Type II exceptional Laguerre polynomials, each omitting class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021904515001598&_mathId=si14.gif&_user=111111111&_pii=S0021904515001598&_rdoc=1&_issn=00219045&md5=71acef71fbfd3d13184e1a8cac86b86a" title="Click to view the MathML source">mclass="mathContainer hidden">class="mathCode">$m$ polynomials. In this paper, we briefly discuss these polynomials and construct the self-adjoint operators generated by their corresponding second-order differential expressions in the appropriate Hilbert spaces. In addition, we present a novel derivation of the Type III family of exceptional Laguerre polynomials along with a detailed disquisition of its properties. We include several representations of these polynomials, orthogonality, norms, completeness, the location of their local extrema and roots, root asymptotics, as well as a complete spectral study of the second-order Type III exceptional Laguerre differential expression.