q-zAl-Salam--Chihara polynomials

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31: 18.10 Integral Representations

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Ultraspherical

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Legendre

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Jacobi

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Ultraspherical

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Laguerre

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32: 18.5 Explicit Representations

§18.5 Explicit Representations

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Laguerre

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Hermite

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33: 18.35 Pollaczek Polynomials

§18.35 Pollaczek Polynomials

… ►There are 3 types of Pollaczek polynomials: … ►For the monic polynomials … ► … ► …

34: 24.16 Generalizations

§24.16 Generalizations

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Polynomials and Numbers of Integer Order

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Nörlund Polynomials

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§24.16(ii) Character Analogs

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§24.16(iii) Other Generalizations

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35: 18.18 Sums

§18.18 Sums

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Ultraspherical

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Legendre

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Hermite

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36: Bibliography M

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Symmetric Functions and Orthogonal Polynomials. University Lecture Series, Vol. 12, American Mathematical Society, Providence, RI.

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External Links:

ISBN 0-8218-0770-6, MathReview (Angèle M. Hamel), ZentralBlatt

Referenced by:

§17.14.

Encodings:

BibTeX

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Orthogonal polynomials associated with root systems. Sém. Lothar. Combin. 45, pp. Art. B45a, 40 pp. (electronic).

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External Links:

ISSN 1286-4889, FullText, MathReview, ZentralBlatt

Referenced by:

§18.37(iii).

Encodings:

BibTeX

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Affine Hecke Algebras and Orthogonal Polynomials. Cambridge Tracts in Mathematics, Vol. 157, Cambridge University Press, Cambridge.

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External Links:

ISBN 0-521-82472-9, MathReview (Eric M. Opdam), ZentralBlatt

Referenced by:

§18.37(iii).

Encodings:

BibTeX

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The supports of measures associated with orthogonal polynomials and the spectra of the related selfadjoint operators. Rocky Mountain J. Math. 21 (1), pp. 501–527.

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External Links:

ISSN 0035-7596, Document, Link, MathReview (T. S. Chihara)

Referenced by:

§18.2(xi).

Encodings:

BibTeX

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Exceptional orthogonal polynomials.

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Notes:

Lecture at ICMAT School on Orthogonal Polynomials in Approximation Theory and Mathematical Physics, available at https://www.icmat.es/RT/optrim/school/lectures/Milson/icmat17.pdf

Referenced by:

§18.36(vi), §18.38(iii).

Encodings:

BibTeX

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37: 29.20 Methods of Computation

… ►These matrices are the same as those provided in §29.15(i) for the computation of Lamé polynomials with the difference that $n$ has to be chosen sufficiently large. … ►A fourth method is by asymptotic approximations by zeros of orthogonal polynomials of increasing degree. … ►

§29.20(ii) Lamé Polynomials

… ►The corresponding eigenvectors yield the coefficients in the finite Fourier series for Lamé polynomials. … ►

§29.20(iii) Zeros

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38: 18.34 Bessel Polynomials

§18.34 Bessel Polynomials

… ►Often only the polynomials (18.34.2) are called Bessel polynomials, while the polynomials (18.34.1) and (18.34.3) are called generalized Bessel polynomials. … … ►

§18.34(ii) Orthogonality

… ►expressed in terms of Romanovski–Bessel polynomials, Laguerre polynomials or Whittaker functions, we have …

39: Errata

… ►We have significantly expanded the section on associated orthogonal polynomials, including expanded properties of associated Laguerre, Hermite, Meixner–Pollaczek, and corecursive orthogonal and numerator and denominator orthogonal polynomials. …In regard to orthogonal polynomials on the unit circle, we now discuss monic polynomials, Verblunsky’s Theorem, and Szegő’s theorem. We also discuss non-classical Laguerre polynomials and give much more details and examples on exceptional orthogonal polynomials. We have also completely expanded our discussion on applications of orthogonal polynomials in the physical sciences, and also methods of computation for orthogonal polynomials. … ►

18.28.8 $$\frac{1}{2\pi }{\int }_0^{\pi }{Q}_n(\cos \theta ;a,b|q)Q_m(\cos \theta ;a,b|q){\left|\frac{{({\mathrm{e}}^{2i\theta };q)}_{\mathrm{\infty }}}{{(a{\mathrm{e}}^{\mathrm{i}\theta },b{\mathrm{e}}^{\mathrm{i}\theta };q)}_{\mathrm{\infty }}}\right|}^{2}d\theta =\frac{{\delta }_{n,m}}{{(q^{n+1},ab{q}^n;q)}_{\mathrm{\infty }}},$$ $a,b\in ℝ$ or $a=\overline{b}$; $ab\ne 1$; $|a|,|b|\le 1$

The constraint which originally stated that “” has been updated to be “$ab\ne 1$”.

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40: 18.17 Integrals

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Jacobi

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Laguerre

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Ultraspherical

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Legendre

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Hermite

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