limit relations

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1: 18.27 $q$-Hahn Class

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From Big $q$-Jacobi to Jacobi

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From Big $q$-Jacobi to Little $q$-Jacobi

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From Little $q$-Jacobi to Jacobi

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From Little $q$-Laguerre to Laguerre

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Limit Relations

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2: 18.21 Hahn Class: Interrelations

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§18.21(ii) Limit Relations and Special Cases

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3: 18.7 Interrelations and Limit Relations

§18.7 Interrelations and Limit Relations

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§18.7(iii) Limit Relations

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4: 18.19 Hahn Class: Definitions

… ►In addition to the limit relations in §18.7(iii) there are limit relations involving the further families in the Askey scheme, see §§18.21(ii) and 18.26(ii). …

5: 18.26 Wilson Class: Continued

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§18.26(ii) Limit Relations

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6: 18.28 Askey–Wilson Class

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§18.28(x) Limit Relations

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7: 17.4 Basic Hypergeometric Functions

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17.4.2 $$\underset{q\to 1-}{lim}{}_{r+1}{}^{}\varphi _s^{}(\genfrac{}{}{0pt}{}{q^{a_0},q^{a_1},\mathrm{\dots },q^{a_r}}{q^{b_1},\mathrm{\dots },q^{b_s}};q,{(q-1)}^{s-r}z)={}_{r+1}{}^{}F_s^{}(\genfrac{}{}{0pt}{}{a_0,a_1,\mathrm{\dots },a_r}{b_1,\mathrm{\dots },b_s};z).$$

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Symbols:

${}_p{}^{}F_q^{}(a_1,\mathrm{\dots },a_p;b_1,\mathrm{\dots },b_q;z)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q};z)$: alternatively ${}_p{}^{}F_q^{}(𝐚;𝐛;z)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{𝐚}{𝐛};z)$
generalized hypergeometric function, ${}_{r+1}{}^{}\varphi _s^{}(a_0,\mathrm{\dots },a_r;b_1,\mathrm{\dots },b_s;q,z)$ or ${}_{r+1}{}^{}\varphi _s^{}(\genfrac{}{}{0pt}{}{a_0,\mathrm{\dots },a_r}{b_1,\mathrm{\dots },b_s};q,z)$: basic hypergeometric (or $q$-hypergeometric) function, $q$: complex base, $r$: nonnegative integer, $s$: nonnegative integer and $z$: complex variable

Source:

Koekoek et al. (2010, p. 15)

Referenced by:

Erratum (V1.1.11) for Equation (17.4.2)

Permalink:

http://dlmf.nist.gov/17.4.E2

Encodings:

pMML, png, TeX

Generalization (effective with 1.1.11):

This limitrelation which was previously accurate for ${}_{r+1}{}^{}\varphi _r^{}$ was updated to be accurate for the more general ${}_{r+1}{}^{}\varphi _s^{}$.

See also:

Annotations for §17.4(i), §17.4 and Ch.17

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8: Bibliography C

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Asymptotic behaviour of the zeros of the (generalized) Laguerre polynomial $L_n^{\alpha }(x)$ as the index $\alpha \to \mathrm{\infty }$ and limiting formula relating Laguerre polynomials of large index and large argument to Hermite polynomials. Lett. Nuovo Cimento (2) 23 (3), pp. 101–102.

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External Links:

ISSN 0024-1318, Document, MathReview (R. A. Askey)

Referenced by:

§18.7(iii).

Encodings:

BibTeX

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9: Errata

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17.4.2 $$\underset{q\to 1-}{lim}{}_{r+1}{}^{}\varphi _s^{}(\genfrac{}{}{0pt}{}{q^{a_0},q^{a_1},\mathrm{\dots },q^{a_r}}{q^{b_1},\mathrm{\dots },q^{b_s}};q,{(q-1)}^{s-r}z)={}_{r+1}{}^{}F_s^{}(\genfrac{}{}{0pt}{}{a_0,a_1,\mathrm{\dots },a_r}{b_1,\mathrm{\dots },b_s};z)$$

This limit relation, which was previously accurate for ${}_{r+1}{}^{}\varphi _r^{}$, has been updated to be accurate for ${}_{r+1}{}^{}\varphi _s^{}$.

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10: 28.34 Methods of Computation

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Representations for $w_{\text{I}}(\pi ;a,\pm q)$ with limit formulas for special solutions of the recurrence relations §28.4(ii) for fixed $a$ and $q$; see Schäfke (1961a).

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