with other orthogonal polynomials

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11: 18.35 Pollaczek Polynomials

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18.35.7 $${(1-z{\mathrm{e}}^{\mathrm{i}\theta })}^{-\lambda +\mathrm{i}{\tau }_{a,b}(\theta )}{(1-z{\mathrm{e}}^{-\mathrm{i}\theta })}^{-\lambda -\mathrm{i}{\tau }_{a,b}(\theta )}=\sum _{n=0}^{\mathrm{\infty }}{P}_n^{(\lambda )}(\cos \theta ;a,b)z^n,$$ , .

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Symbols:

$P_n^{(\lambda )}(x;a,b)$: Pollaczek polynomial, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $z$: complex variable, $n$: nonnegative integer and ${\tau }_{a,b}(\theta )$

Proved:

Ismail (2009, (5.4.8))(proved)

Permalink:

http://dlmf.nist.gov/18.35.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §18.35(iii), §18.35 and Ch.18

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12: 18.38 Mathematical Applications

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13: 13.6 Relations to Other Functions

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§13.6(v) Orthogonal Polynomials

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14: Richard A. Askey

… ►Over his career his primary research areas were in Special Functions and Orthogonal Polynomials, but also included other topics from Classical Analysis and related areas. …

15: 18.23 Hahn Class: Generating Functions

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Hahn

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16: 18.27 $q$-Hahn Class

§18.27 $q$-Hahn Class

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18.27.26 $$\underset{q\to 1}{lim}\frac{{\tilde{h}}_n({(1-q^2)}^{\frac{1}{2}}x;q)}{{\left(1-q^2\right)}^{n/2}}=2^{-n}{H}_n\left(x\right).$$

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Symbols:

$H_n\left(x\right)$: Hermite polynomial, ${\tilde{h}}_n(x;q)$: discrete $q$-Hermite II polynomial, $q$: real variable, $n$: nonnegative integer and $x$: real variable

Source:

Koekoek et al. (2010, (14.29.14))

Referenced by:

§18.27(vii), §18.27(vii), Erratum (V1.2.0) Section 18.27

Permalink:

http://dlmf.nist.gov/18.27.E26

Encodings:

pMML, png, TeX

Addition (effective with 1.2.0):

This equation was added.

See also:

Annotations for §18.27(vii), §18.27(vii), §18.27 and Ch.18

17: 3.5 Quadrature

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18: 18.28 Askey–Wilson Class

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§18.28(ii) Askey–Wilson Polynomials

… ►Genest et al. (2016) showed that these polynomials coincide with the nonsymmetric Wilson polynomials in Groenevelt (2007).

19: 18.30 Associated OP’s

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§18.30(i) Associated Jacobi Polynomials

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20: 18.40 Methods of Computation

… ►For applications in which the OP’s appear only as terms in series expansions (compare §18.18(i)) the need to compute them can be avoided altogether by use instead of Clenshaw’s algorithm (§3.11(ii)) and its straightforward generalization to OP’s other than Chebyshev. …