An Introduction to Orthogonal Polynomials
Assuming no further prerequisites than a first undergraduate course in real analysis, this concise introduction covers general elementary theory related to orthogonal polynomials. It includes necessary background material of the type not usually found in the standard mathematics curriculum. Suitable for advanced undergraduate and graduate courses, it is also appropriate for independent study.
Topics include the representation theorem and distribution functions, continued fractions and chain sequences, the recurrence formula and properties of orthogonal polynomials, special functions, and some specific systems of orthogonal polynomials. Numerous examples and exercises, an extensive bibliography, and a table of recurrence formulas supplement the text.
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According to Theorem Al-Salam Amer An+l Bessel polynomials Carlitz chain sequence Charlier polynomials Chihara class of orthogonal classical orthogonal polynomials continued fraction converges COROLLARY corresponding monic deﬁned deﬁnition denote determined differential equation distribution function Erdélyi example ﬁnite ﬁrst follows Gauss quadrature formula Hahn Hamburger moment problem hence Hermite polynomials I-Ex I-Theorem inequality integral interval of orthogonality Jacobi polynomials kernel polynomials Laguerre polynomials leading coefﬁcient Legendre polynomials limit point linear MATHEMATICS maximal parameter sequence Meixner polynomials monic OPS monic polynomials non-decreasing obtain OPS with respect orthogonal polynomials orthogonality relation orthonormal polynomial pn(x Pollaczek polynomial of degree polynomial sequence positive-deﬁnite moment functional positive-deﬁnite OPS properties Prove Qn(x quasi-deﬁnite quasi-orthogonal polynomial real numbers recurrence formula satisﬁes Show speciﬁc spectrum Stieltjes-Wigert polynomials sufﬁcient condition supporting set symmetric Szego Tchebichef polynomials Theorem 3.2 theory true interval veriﬁed Wall polynomials weight function write