کد مقاله کد نشریه سال انتشار مقاله انگلیسی نسخه تمام متن
4606884 1631408 2016 37 صفحه PDF دانلود رایگان
عنوان انگلیسی مقاله ISI
The spectral analysis of three families of exceptional Laguerre polynomials
موضوعات مرتبط
مهندسی و علوم پایه ریاضیات آنالیز ریاضی
پیش نمایش صفحه اول مقاله
The spectral analysis of three families of exceptional Laguerre polynomials
چکیده انگلیسی

The Bochner Classification Theorem (1929) characterizes the polynomial sequences {pn}n=0∞, with degpn=ndegpn=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 {pn}n=1∞, degpn=ndegpn=n (without the constant polynomial), the only such sequences satisfying these conditions are the exceptional  X1X1-Laguerre and X1X1-Jacobi polynomials. Subsequently, during the past five years, several mathematicians and physicists have discovered and studied other exceptional orthogonal polynomials {pn}n∈N0⧵A{pn}n∈N0⧵A, where AA is a finite subset of the non-negative integers N0N0 and where degpn=ndegpn=n for all n∈N0⧵An∈N0⧵A. We call such a sequence an exceptional polynomial sequence of codimension |A||A|, where the latter denotes the cardinality of AA. 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 mm 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.

ناشر
Database: Elsevier - ScienceDirect (ساینس دایرکت)
Journal: Journal of Approximation Theory - Volume 202, February 2016, Pages 5–41
نویسندگان
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