orthogonality relation

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11: 10 Bessel Functions

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12: 1 Algebraic and Analytic Methods

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13: 18.2 General Orthogonal Polynomials

… ►The orthogonality relations (18.2.1)–(18.2.3) each determine the polynomials $p_n(x)$ uniquely up to constant factors, which may be fixed by suitable standardizations. … ►

§18.2(iv) Recurrence Relations

… ►are OP’s with orthogonality relation … ►However, if OP’s have an orthogonality relation on a bounded interval, then their orthogonality measure is unique, up to a positive constant factor. … ►For OP’s $p_n$ with $h_n$ and orthogonality relation as in (18.2.5) and (18.2.5_5), the Poisson kernel is defined by …

14: Wolter Groenevelt

… ►Groenevelt’s research interests is in special functions and orthogonal polynomials and their relations with representation theory and interacting particle systems. …

15: 15.9 Relations to Other Functions

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§15.9(i) Orthogonal Polynomials

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Krawtchouk

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15.9.10 $$P_n^{(\lambda )}(x;\varphi )=\frac{{\left(2\lambda \right)}_n}{n!}{\mathrm{e}}^{n\mathrm{i}\varphi }F(\genfrac{}{}{0pt}{}{-n,\lambda +\mathrm{i}x}{2\lambda };1-{\mathrm{e}}^{-2\mathrm{i}\varphi }).$$

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Symbols:

$F(a,b;c;z)$ or $F(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}F_1^{}(a,b;c;z)$ Gauss’ hypergeometric function, $P_n^{(\lambda )}(x;\varphi )$: Meixner–Pollaczek polynomial, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $\mathrm{i}$: imaginary unit, $x$: real variable and $n$: integer

Permalink:

http://dlmf.nist.gov/15.9.E10

Encodings:

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See also:

Annotations for §15.9(i), §15.9(i), §15.9 and Ch.15

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16: 18.20 Hahn Class: Explicit Representations

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§18.20(ii) Hypergeometric Function and Generalized Hypergeometric Functions

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17: 13.6 Relations to Other Functions

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§13.6(v) Orthogonal Polynomials

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Charlier Polynomials

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18: Bibliography I

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Two families of orthogonal polynomials related to Jacobi polynomials. Rocky Mountain J. Math. 21 (1), pp. 359–375.

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External Links:

ISSN 0035-7596, MathReview (Joaquin Bustoz), ZentralBlatt

Referenced by:

§18.30(i).

Encodings:

BibTeX

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19: 18.5 Explicit Representations

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§18.5(iii) Finite Power Series, the Hypergeometric Function, and Generalized Hypergeometric Functions

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20: 18.26 Wilson Class: Continued

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§18.26(i) Representations as Generalized Hypergeometric Functions and Dualities

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§18.26(ii) Limit Relations

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§18.26(iii) Difference Relations

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§18.26(iv) Generating Functions

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