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1: 5.18 $q$-Gamma and $q$-Beta Functions

§5.18 $q$-Gamma and $q$-Beta Functions

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§5.18(iii) $q$-Beta Function

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5.18.11 $${\mathrm{B}}_q(a,b)=\frac{{\mathrm{\Gamma }}_q\left(a\right){\mathrm{\Gamma }}_q\left(b\right)}{{\mathrm{\Gamma }}_q\left(a+b\right)}.$$

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Defines:

${\mathrm{B}}_q(a,b)$: $q$-beta function

Symbols:

${\mathrm{\Gamma }}_q\left(z\right)$: $q$-gamma function, $q$: real or complex variable, $a$: real or complex variable and $b$: real or complex variable

Permalink:

http://dlmf.nist.gov/5.18.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §5.18(iii), §5.18 and Ch.5

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5.18.12 $${\mathrm{B}}_q(a,b)={\int }_0^1\frac{t^{a-1}{(tq;q)}_{\mathrm{\infty }}}{{(t{q}^b;q)}_{\mathrm{\infty }}}{d}_{q}t,$$ , $\mathrm{\Re }a>0$, $\mathrm{\Re }b>0$.

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Symbols:

$\int$: integral, ${\mathrm{B}}_q(a,b)$: $q$-beta function, ${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), $d_{q}x$: $q$-differential, $\mathrm{\Re }$: real part, $q$: real or complex variable, $a$: real or complex variable and $b$: real or complex variable

Permalink:

http://dlmf.nist.gov/5.18.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §5.18(iii), §5.18 and Ch.5

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2: 5.21 Methods of Computation

… ►For the computation of the $q$-gamma and $q$-beta functions see Gabutti and Allasia (2008).

3: Bibliography I

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$q$-Hermite polynomials, biorthogonal rational functions, and $q$-beta integrals. Trans. Amer. Math. Soc. 346 (1), pp. 63–116.

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External Links:

ISSN 0002-9947, FullText, MathReview (R. A. Askey), ZentralBlatt

Referenced by:

§18.28(vii).

Encodings:

BibTeX

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4: Bibliography

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A $q$-beta integral on the unit circle and some biorthogonal rational functions. Proc. Amer. Math. Soc. 121 (2), pp. 553–561.

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External Links:

ISSN 0002-9939, FullText, MathReview (S. P. Goyal), ZentralBlatt

Referenced by:

§18.33(v).

Encodings:

BibTeX

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5: Bibliography P

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Tables of Digamma and Trigamma Functions. In Tracts for Computers, No. 1, K. Pearson (Ed.),

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External Links:

ZentralBlatt

Referenced by:

§5.1, Notations F.

Encodings:

BibTeX

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Exponential asymptotics of the Mittag-Leffler function. Proc. Roy. Soc. London Ser. A 458, pp. 3041–3052.

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External Links:

ISSN 1364-5021, Document, MathReview (Roderick Wong), ZentralBlatt

Referenced by:

§10.46.

Encodings:

BibTeX

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Orthogonal polynomials and some $q$-beta integrals of Ramanujan. J. Math. Anal. Appl. 112 (2), pp. 517–540.

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External Links:

ISSN 0022-247X, Document, MathReview (Mourad E. H. Ismail), ZentralBlatt

Referenced by:

§18.33(iv), §18.33(v).

Encodings:

BibTeX

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Tables of the Incomplete Beta-function. 2nd edition, Published for the Biometrika Trustees at the Cambridge University Press, Cambridge.

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External Links:

MathReview, ZentralBlatt

Referenced by:

1st item.

Encodings:

BibTeX

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A note on the computation of the incomplete beta function. Adv. Eng. Software 12 (1), pp. 39–44.

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Notes:

Includes Basic code.

External Links:

Document

Referenced by:

§8.28(iv).

Encodings:

BibTeX

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6: 18.28 Askey–Wilson Class

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18.28.13 $$C_n(\cos \theta ;\beta |q)=\sum _{\mathrm{\ell }=0}^n\frac{{(\beta ;q)}_{\mathrm{\ell }}{(\beta ;q)}_{n-\mathrm{\ell }}}{{(q;q)}_{\mathrm{\ell }}{(q;q)}_{n-\mathrm{\ell }}}{\mathrm{e}}^{\mathrm{i}(n-2\mathrm{\ell })\theta }=\frac{{(\beta ;q)}_n}{{(q;q)}_n}{\mathrm{e}}^{\mathrm{i}n\theta }{}_2\varphi _1(\genfrac{}{}{0pt}{}{q^{-n},\beta }{{\beta }^{-1}{q}^{1-n}};q,{\beta }^{-1}q{\mathrm{e}}^{-2\mathrm{i}\theta }).$$

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Defines:

$C_n(x;\beta |q)$: continuous $q$-ultraspherical polynomial

Symbols:

$\cos z$: cosine function, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, ${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), ${}_{r+1}\varphi _s(a_0,\mathrm{\dots },a_r;b_1,\mathrm{\dots },b_s;q,z)$ 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, $q$: real variable, $\mathrm{\ell }$: nonnegative integer, $n$: nonnegative integer and $x$: real variable

Permalink:

http://dlmf.nist.gov/18.28.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §18.28(v), §18.28 and Ch.18

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18.28.14 $$C_n(\cos \theta ;\beta |q)=\frac{{({\beta }^2;q)}_n}{{(q;q)}_n{\beta }^{\frac{1}{2}n}}{}_4\varphi _3(\genfrac{}{}{0pt}{}{q^{-n},{\beta }^2{q}^n,{\beta }^{\frac{1}{2}}{\mathrm{e}}^{\mathrm{i}\theta },{\beta }^{\frac{1}{2}}{\mathrm{e}}^{-\mathrm{i}\theta }}{\beta q^{\frac{1}{2}},-\beta ,-\beta q^{\frac{1}{2}}};q,q).$$

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Symbols:

$\cos z$: cosine function, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, ${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), ${}_{r+1}\varphi _s(a_0,\mathrm{\dots },a_r;b_1,\mathrm{\dots },b_s;q,z)$ 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, $C_n(x;\beta |q)$: continuous $q$-ultraspherical polynomial, $q$: real variable and $n$: nonnegative integer

Permalink:

http://dlmf.nist.gov/18.28.E14

Encodings:

pMML, png, TeX

See also:

Annotations for §18.28(v), §18.28 and Ch.18

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18.28.15 $$\frac{1}{2\pi }{\int }_0^{\pi }{C}_n(\cos \theta ;\beta |q)C_m(\cos \theta ;\beta |q){\left|\frac{{({\mathrm{e}}^{2\mathrm{i}\theta };q)}_{\mathrm{\infty }}}{{(\beta {\mathrm{e}}^{2\mathrm{i}\theta };q)}_{\mathrm{\infty }}}\right|}^{2}d\theta =\frac{{(\beta ,\beta q;q)}_{\mathrm{\infty }}}{{({\beta }^2,q;q)}_{\mathrm{\infty }}}\frac{(1-\beta ){({\beta }^2;q)}_n}{(1-\beta q^n){(q;q)}_n}{\delta }_{n,m},$$ .

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Symbols:

${\delta }_{j,k}$: Kronecker delta, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\int$: integral, ${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), ${(a_1,a_2,\mathrm{\dots },a_r;q)}_n$: multiple $q$-Pochhammer symbol, $C_n(x;\beta |q)$: continuous $q$-ultraspherical polynomial, $q$: real variable, $m$: nonnegative integer and $n$: nonnegative integer

Permalink:

http://dlmf.nist.gov/18.28.E15

Encodings:

pMML, png, TeX

See also:

Annotations for §18.28(v), §18.28 and Ch.18

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18.28.19 $$R_n(x)=R_n(x;\alpha ,\beta ,\gamma ,\delta |q)=\sum _{\mathrm{\ell }=0}^n\frac{q^{\mathrm{\ell }}{(q^{-n},\alpha \beta q^{n+1};q)}_{\mathrm{\ell }}}{{(\alpha q,\beta \delta q,\gamma q,q;q)}_{\mathrm{\ell }}}\prod _{j=0}^{\mathrm{\ell }-1}(1-q^{j}x+\gamma \delta q^{2j+1})={}_4\varphi _3(\genfrac{}{}{0pt}{}{q^{-n},\alpha \beta q^{n+1},q^{-y},\gamma \delta q^{y+1}}{\alpha q,\beta \delta q,\gamma q};q,q),$$ $\alpha q$, $\beta \delta q$, or $\gamma q=q^{-N}$; $n=0,1,\mathrm{\dots },N$.

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Defines:

$R_n(x;\alpha ,\beta ,\gamma ,\delta |q)$: $q$-Racah polynomial and $R_n(x)$ (locally)

Symbols:

${}_{r+1}\varphi _s(a_0,\mathrm{\dots },a_r;b_1,\mathrm{\dots },b_s;q,z)$ 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, ${(a_1,a_2,\mathrm{\dots },a_r;q)}_n$: multiple $q$-Pochhammer symbol, $y$: real variable, $q$: real variable, $\mathrm{\ell }$: nonnegative integer, $n$: nonnegative integer, $N$: positive integer, $\delta$: arbitary small positive constant and $x$: real variable

Permalink:

http://dlmf.nist.gov/18.28.E19

Encodings:

pMML, png, TeX

See also:

Annotations for §18.28(viii), §18.28 and Ch.18

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7: Errata

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A sentence was added recommending §27.14(ii) for properties of ${(q;q)}_{\mathrm{\infty }}$.

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18.27.6 $$P_n^{(\alpha ,\beta )}(x;c,d;q)=\frac{c^n{q}^{-(\alpha +1)n}{(q^{\alpha +1},-q^{\alpha +1}{c}^{-1}d;q)}_n}{{(q,-q;q)}_n}{P}_n(q^{\alpha +1}{c}^{-1}x;q^{\alpha },q^{\beta },-q^{\alpha }{c}^{-1}d;q)$$

Originally the first argument to the big $q$-Jacobi polynomial on the right-hand side was written incorrectly as $q^{\alpha +1}{c}^{-1}dx$.

Reported 2017-09-27 by Tom Koornwinder.

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The title was changed from Transformations of Higher ${}_r\varphi _r$ Functions to Further Transformations of ${}_{r\mathrm{+}\mathrm{1}}\varphi _r$ Functions.

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The notation used for the $q$-Appell functions in Equations (17.4.5), (17.4.6),(17.4.7), (17.4.8), (17.11.1), (17.11.2) and (17.11.3) was updated to explicitly include the argument $q$, as used in Gasper and Rahman (2004).

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17.13.3 $${\int }_0^{\mathrm{\infty }}{t}^{\alpha -1}\frac{{(-t{q}^{\alpha +\beta };q)}_{\mathrm{\infty }}}{{(-t;q)}_{\mathrm{\infty }}}dt=\frac{\mathrm{\Gamma }\left(\alpha \right)\mathrm{\Gamma }\left(1-\alpha \right){\mathrm{\Gamma }}_q\left(\beta \right)}{{\mathrm{\Gamma }}_q\left(1-\alpha \right){\mathrm{\Gamma }}_q\left(\alpha +\beta \right)}$$

Originally the differential was identified incorrectly as $d_{q}t$; the correct differential is $dt$.

Reported 2011-04-08.

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8: 18.27 $q$-Hahn Class

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§18.27(ii) $q$-Hahn Polynomials

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§18.27(iii) Big $q$-Jacobi Polynomials

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§18.27(iv) Little $q$-Jacobi Polynomials

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§18.27(v) $q$-Laguerre Polynomials

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Discrete $q$-Hermite II

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