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21: 18.27 $q$-Hahn Class

… ►The $q$-hypergeometric OP’s comprise the $q$-Hahn class (or $q$-linear lattice class) OP’s and the Askey–Wilson class (or $q$-quadratic lattice class) OP’s (§18.28). … ►For other formulas, including $q$-difference equations, recurrence relations, duality formulas, special cases, and limit relations, see Koekoek et al. (2010, Chapter 14). … ►

18.27.4_1 $$h_n=\frac{{\left(\alpha q\right)}^{nN}}{1-\alpha \beta q^{2n+1}}\frac{{(\alpha \beta q^{n+1};q)}_{N+1}{(\beta q;q)}_n}{{\left[\genfrac{}{}{0.0pt}{}{N}{n}\right]}_q{(\alpha q;q)}_n}.$$

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Symbols:

${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), ${\left[\genfrac{}{}{0.0pt}{}{n}{m}\right]}_q$: $q$-binomial coefficient (or Gaussian polynomial), $N$: positive integer, $q$: real variable, $n$: nonnegative integer and $h_n$

Referenced by:

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

Permalink:

http://dlmf.nist.gov/18.27.E4_1

Encodings:

pMML, png, TeX

Addition (effective with 1.2.0):

This equation was added.

See also:

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

►

18.27.4_2 $$\underset{q\to 1}{lim}{Q}_n(q^{-x};q^{\alpha },q^{\beta },N;q)=Q_n(x;\alpha ,\beta ,N).$$

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Symbols:

$Q_n(x;\alpha ,\beta ,N)$: Hahn polynomial, $Q_n(x;\alpha ,\beta ,N;q)$: $q$-Hahn polynomial, $N$: positive integer, $q$: real variable, $n$: nonnegative integer and $x$: real variable

Referenced by:

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

Permalink:

http://dlmf.nist.gov/18.27.E4_2

Encodings:

pMML, png, TeX

Addition (effective with 1.2.0):

This equation was added.

See also:

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

… ►

18.27.6_5 $$P_n(x;a,b,c,d;q)=P_n(qa{c}^{-1}x;a,b,-a{c}^{-1}d;q).$$

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Symbols:

$P_n(x;a,b,c;q)$: big $q$-Jacobi polynomial, $q$: real variable, $n$: nonnegative integer, $x$: real variable and $P_n^{(\alpha ,\beta )}(x;c,d;q)$

Source:

Koekoek et al. (2010, p. 442)

Referenced by:

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

Permalink:

http://dlmf.nist.gov/18.27.E6_5

Encodings:

pMML, png, TeX

Addition (effective with 1.2.0):

This equation was added.

See also:

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

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