basic hypergeometric functions

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11: 17.7 Special Cases of Higher ${}_r{}^{}\varphi _s^{}$ Functions

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17.7.2 $${}_2{}^{}\varphi _2^{}(\genfrac{}{}{0pt}{}{a^2,b^2}{ab{q}^{\frac{1}{2}},-ab{q}^{\frac{1}{2}}};q,-q)=\frac{{(a^{2}q,b^{2}q;q^2)}_{\mathrm{\infty }}}{{(q,a^2{b}^{2}q;q^2)}_{\mathrm{\infty }}}.$$

ⓘ

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 and $q$: complex base

Permalink:

http://dlmf.nist.gov/17.7.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §17.7(i), §17.7(i), §17.7 and Ch.17

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17.7.4 $${}_3{}^{}\varphi _2^{}(\genfrac{}{}{0pt}{}{a,b,q^{-n}}{c,ab{q}^{1-n}/c};q,q)=\frac{{(c/a,c/b;q)}_n}{{(c,c/(ab);q)}_n}.$$

ⓘ

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, $q$: complex base and $n$: nonnegative integer

Permalink:

http://dlmf.nist.gov/17.7.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §17.7(ii), §17.7(ii), §17.7 and Ch.17

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17.7.12 $${}_4{}^{}\varphi _3^{}(\genfrac{}{}{0pt}{}{a,aq,b^2{q}^{2n},q^{-2n}}{b,bq,a^2{q}^2};q^2,q^2)=\frac{a^n{(-q,b/a;q)}_n}{{(-aq,b;q)}_n}.$$

ⓘ

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, $q$: complex base and $n$: nonnegative integer

Permalink:

http://dlmf.nist.gov/17.7.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §17.7(iii), §17.7(iii), §17.7 and Ch.17

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17.7.13 $${}_4{}^{}\varphi _3^{}(\genfrac{}{}{0pt}{}{a,aq,b^2{q}^{2n-2},q^{-2n}}{b,bq,a^2};q^2,q^2)=\frac{a^n{(-q,b/a;q)}_n(1-b{q}^{n-1})}{{(-a,b;q)}_n(1-b{q}^{2n-1})}.$$

ⓘ

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, $q$: complex base and $n$: nonnegative integer

Permalink:

http://dlmf.nist.gov/17.7.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §17.7(iii), §17.7(iii), §17.7 and Ch.17

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17.7.19 $${}_{r+1}{}^{}\varphi _r^{}(\genfrac{}{}{0pt}{}{a,b_1{q}^{m_1},\mathrm{\dots },b_r{q}^{m_r}}{b_1,\mathrm{\dots },b_r};q,a^{-1}{q}^{1-(m_1+\mathrm{\cdots }+m_r)})=0,$$

ⓘ

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, $q$: complex base, $m$: nonnegative integer and $r$: nonnegative integer

Permalink:

http://dlmf.nist.gov/17.7.E19

Encodings:

pMML, png, TeX

See also:

Annotations for §17.7(iii), §17.7(iii), §17.7 and Ch.17

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12: Bibliography C

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Basic hypergeometric functions and orthogonal Laurent polynomials. Proc. Amer. Math. Soc. 140 (6), pp. 2075–2089.

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External Links:

ISSN 0002-9939, Link, MathReview (Roelof Koekoek), ZentralBlatt

Referenced by:

§18.33(iv).

Encodings:

BibTeX

…

13: 17.10 Transformations of ${}_r{}^{}\psi _r^{}$ Functions

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17.10.1 $${}_2{}^{}\psi _2^{}(\genfrac{}{}{0pt}{}{a,b}{c,d};q,z)=\frac{{(az,d/a,c/b,dq/(abz);q)}_{\mathrm{\infty }}}{{(z,d,q/b,cd/(abz);q)}_{\mathrm{\infty }}}{}_2{}^{}\psi _2^{}(\genfrac{}{}{0pt}{}{a,abz/d}{az,c};q,\frac{d}{a}),$$

ⓘ

Symbols:

${}_r{}^{}\psi _s^{}(a_1,\mathrm{\dots },a_r;b_1,\mathrm{\dots },b_s;q,z)$ or ${}_r{}^{}\psi _s^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_r}{b_1,\mathrm{\dots },b_s};q,z)$: bilateral basic hypergeometric (or bilateral $q$-hypergeometric) function, ${(a_1,a_2,\mathrm{\dots },a_r;q)}_n$: multiple $q$-Pochhammer symbol, $q$: complex base and $z$: complex variable

Permalink:

http://dlmf.nist.gov/17.10.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §17.10, §17.10 and Ch.17

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17.10.2 $${}_2{}^{}\psi _2^{}(\genfrac{}{}{0pt}{}{a,b}{c,d};q,z)=\frac{{(az,bz,cq/(abz),dq/(abz);q)}_{\mathrm{\infty }}}{{(q/a,q/b,c,d;q)}_{\mathrm{\infty }}}{}_2{}^{}\psi _2^{}(\genfrac{}{}{0pt}{}{abz/c,abz/d}{az,bz};q,\frac{cd}{abz}).$$

ⓘ

Symbols:

${}_r{}^{}\psi _s^{}(a_1,\mathrm{\dots },a_r;b_1,\mathrm{\dots },b_s;q,z)$ or ${}_r{}^{}\psi _s^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_r}{b_1,\mathrm{\dots },b_s};q,z)$: bilateral basic hypergeometric (or bilateral $q$-hypergeometric) function, ${(a_1,a_2,\mathrm{\dots },a_r;q)}_n$: multiple $q$-Pochhammer symbol, $q$: complex base and $z$: complex variable

Permalink:

http://dlmf.nist.gov/17.10.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §17.10, §17.10 and Ch.17

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17.10.3 $${}_8{}^{}\psi _8^{}(\genfrac{}{}{0pt}{}{q{a}^{\frac{1}{2}},-q{a}^{\frac{1}{2}},c,d,e,f,a{q}^{-n},q^{-n}}{a^{\frac{1}{2}},-a^{\frac{1}{2}},aq/c,aq/d,aq/e,aq/f,q^{n+1},a{q}^{n+1}};q,\frac{a^2{q}^{2n+2}}{cdef})=\frac{{(aq,q/a,aq/(cd),aq/(ef);q)}_n}{{(q/c,q/d,aq/e,aq/f;q)}_n}{}_4{}^{}\psi _4^{}(\genfrac{}{}{0pt}{}{e,f,a{q}^{n+1}/(cd),q^{-n}}{aq/c,aq/d,q^{n+1},ef/(a{q}^n)};q,q),$$

ⓘ

Symbols:

${}_r{}^{}\psi _s^{}(a_1,\mathrm{\dots },a_r;b_1,\mathrm{\dots },b_s;q,z)$ or ${}_r{}^{}\psi _s^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_r}{b_1,\mathrm{\dots },b_s};q,z)$: bilateral basic hypergeometric (or bilateral $q$-hypergeometric) function, ${(a_1,a_2,\mathrm{\dots },a_r;q)}_n$: multiple $q$-Pochhammer symbol, $q$: complex base and $n$: nonnegative integer

Permalink:

http://dlmf.nist.gov/17.10.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §17.10, §17.10 and Ch.17

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17.10.5 $$\frac{{(aq/b,aq/c,aq/d,aq/e,q/(ab),q/(ac),q/(ad),q/(ae);q)}_{\mathrm{\infty }}}{{(fa,ga,f/a,g/a,q{a}^2,q/a^2;q)}_{\mathrm{\infty }}}{}_8{}^{}\psi _8^{}(\genfrac{}{}{0pt}{}{qa,-qa,ba,ca,da,ea,fa,ga}{a,-a,aq/b,aq/c,aq/d,aq/e,aq/f,aq/g};q,\frac{q^2}{bcdefg})=\frac{{(q,q/(bf),q/(cf),q/(df),q/(ef),qf/b,qf/c,qf/d,qf/e;q)}_{\mathrm{\infty }}}{{(fa,q/(fa),aq/f,f/a,g/f,fg,q{f}^2;q)}_{\mathrm{\infty }}}{}_8{}^{}\varphi _7^{}(\genfrac{}{}{0pt}{}{f^2,qf,-qf,fb,fc,fd,fe,fg}{f,-f,fq/b,fq/c,fq/d,fq/e,fq/g};q,\frac{q^2}{bcdefg})+\mathrm{idem}(f;g).$$

ⓘ

Symbols:

$\mathrm{idem}({\chi }_1;{\chi }_2,\mathrm{\dots },{\chi }_n)$: $\mathrm{idem}$ 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, ${}_r{}^{}\psi _s^{}(a_1,\mathrm{\dots },a_r;b_1,\mathrm{\dots },b_s;q,z)$ or ${}_r{}^{}\psi _s^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_r}{b_1,\mathrm{\dots },b_s};q,z)$: bilateral basic hypergeometric (or bilateral $q$-hypergeometric) function, ${(a_1,a_2,\mathrm{\dots },a_r;q)}_n$: multiple $q$-Pochhammer symbol and $q$: complex base

Permalink:

http://dlmf.nist.gov/17.10.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §17.10, §17.10 and Ch.17

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17.10.6 $$\frac{{(aq/b,aq/c,aq/d,aq/e,aq/f,q/(ab),q/(ac),q/(ad),q/(ae),q/(af);q)}_{\mathrm{\infty }}}{{(ag,ah,ak,g/a,h/a,k/a,q{a}^2,q/a^2;q)}_{\mathrm{\infty }}}{}_{10}{}^{}\psi _{10}^{}(\genfrac{}{}{0pt}{}{qa,-qa,ba,ca,da,ea,fa,ga,ha,ka}{a,-a,aq/b,aq/c,aq/d,aq/e,aq/f,aq/g,aq/h,aq/k};q,\frac{q^2}{bcdefghk})=\frac{{(q,q/(bg),q/(cg),q/(dg),q/(eg),q/(fg),qg/b,qg/c,qg/d,qg/e,qg/f;q)}_{\mathrm{\infty }}}{{(gh,gk,h/g,ag,q/(ag),g/a,aq/g,q{g}^2;q)}_{\mathrm{\infty }}}{}_{10}{}^{}\varphi _9^{}(\genfrac{}{}{0pt}{}{g^2,qg,-qg,gb,gc,gd,ge,gf,gh,gk}{g,-g,qg/b,qg/c,qg/d,qg/e,qg/f,qg/h,qg/k};q,\frac{q^2}{bcdefghk})+\mathrm{idem}(g;h,k).$$

ⓘ

Symbols:

$\mathrm{idem}({\chi }_1;{\chi }_2,\mathrm{\dots },{\chi }_n)$: $\mathrm{idem}$ 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, ${}_r{}^{}\psi _s^{}(a_1,\mathrm{\dots },a_r;b_1,\mathrm{\dots },b_s;q,z)$ or ${}_r{}^{}\psi _s^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_r}{b_1,\mathrm{\dots },b_s};q,z)$: bilateral basic hypergeometric (or bilateral $q$-hypergeometric) function, ${(a_1,a_2,\mathrm{\dots },a_r;q)}_n$: multiple $q$-Pochhammer symbol, $k$: nonnegative integer and $q$: complex base

Permalink:

http://dlmf.nist.gov/17.10.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §17.10, §17.10 and Ch.17

14: 17.8 Special Cases of ${}_r{}^{}\psi _r^{}$ Functions

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17.8.2 $${}_1{}^{}\psi _1^{}(\genfrac{}{}{0pt}{}{a}{b};q,z)=\frac{{(q,b/a,az,q/(az);q)}_{\mathrm{\infty }}}{{(b,q/a,z,b/(az);q)}_{\mathrm{\infty }}}.$$

ⓘ

Symbols:

${}_r{}^{}\psi _s^{}(a_1,\mathrm{\dots },a_r;b_1,\mathrm{\dots },b_s;q,z)$ or ${}_r{}^{}\psi _s^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_r}{b_1,\mathrm{\dots },b_s};q,z)$: bilateral basic hypergeometric (or bilateral $q$-hypergeometric) function, ${(a_1,a_2,\mathrm{\dots },a_r;q)}_n$: multiple $q$-Pochhammer symbol, $q$: complex base and $z$: complex variable

Permalink:

http://dlmf.nist.gov/17.8.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §17.8, §17.8 and Ch.17

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17.8.4 $${}_2{}^{}\psi _2^{}(b,c;aq/b,aq/c;q,-aq/(bc))=\frac{{(aq/(bc);q)}_{\mathrm{\infty }}{(a{q}^2/b^2,a{q}^2/c^2,q^2,aq,q/a;q^2)}_{\mathrm{\infty }}}{{(aq/b,aq/c,q/b,q/c,-aq/(bc);q)}_{\mathrm{\infty }}},$$

ⓘ

Symbols:

${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), ${}_r{}^{}\psi _s^{}(a_1,\mathrm{\dots },a_r;b_1,\mathrm{\dots },b_s;q,z)$ or ${}_r{}^{}\psi _s^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_r}{b_1,\mathrm{\dots },b_s};q,z)$: bilateral basic hypergeometric (or bilateral $q$-hypergeometric) function, ${(a_1,a_2,\mathrm{\dots },a_r;q)}_n$: multiple $q$-Pochhammer symbol and $q$: complex base

Permalink:

http://dlmf.nist.gov/17.8.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §17.8, §17.8 and Ch.17

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17.8.5 $${}_3{}^{}\psi _3^{}(\genfrac{}{}{0pt}{}{b,c,d}{q/b,q/c,q/d};q,\frac{q}{bcd})=\frac{{(q,q/(bc),q/(bd),q/(cd);q)}_{\mathrm{\infty }}}{{(q/b,q/c,q/d,q/(bcd);q)}_{\mathrm{\infty }}},$$

ⓘ

Symbols:

${}_r{}^{}\psi _s^{}(a_1,\mathrm{\dots },a_r;b_1,\mathrm{\dots },b_s;q,z)$ or ${}_r{}^{}\psi _s^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_r}{b_1,\mathrm{\dots },b_s};q,z)$: bilateral basic hypergeometric (or bilateral $q$-hypergeometric) function, ${(a_1,a_2,\mathrm{\dots },a_r;q)}_n$: multiple $q$-Pochhammer symbol and $q$: complex base

Permalink:

http://dlmf.nist.gov/17.8.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §17.8, §17.8 and Ch.17

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17.8.6 $${}_4{}^{}\psi _4^{}(\genfrac{}{}{0pt}{}{-q{a}^{\frac{1}{2}},b,c,d}{-a^{\frac{1}{2}},aq/b,aq/c,aq/d};q,\frac{q{a}^{\frac{3}{2}}}{bcd})=\frac{{(aq,aq/(bc),aq/(bd),aq/(cd),q{a}^{\frac{1}{2}}/b,q{a}^{\frac{1}{2}}/c,q{a}^{\frac{1}{2}}/d,q,q/a;q)}_{\mathrm{\infty }}}{{(aq/b,aq/c,aq/d,q/b,q/c,q/d,q{a}^{\frac{1}{2}},q{a}^{-\frac{1}{2}},q{a}^{\frac{3}{2}}/(bcd);q)}_{\mathrm{\infty }}},$$

ⓘ

Symbols:

${}_r{}^{}\psi _s^{}(a_1,\mathrm{\dots },a_r;b_1,\mathrm{\dots },b_s;q,z)$ or ${}_r{}^{}\psi _s^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_r}{b_1,\mathrm{\dots },b_s};q,z)$: bilateral basic hypergeometric (or bilateral $q$-hypergeometric) function, ${(a_1,a_2,\mathrm{\dots },a_r;q)}_n$: multiple $q$-Pochhammer symbol and $q$: complex base

Permalink:

http://dlmf.nist.gov/17.8.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §17.8, §17.8 and Ch.17

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17.8.7 $${}_6{}^{}\psi _6^{}(\genfrac{}{}{0pt}{}{q{a}^{\frac{1}{2}},-q{a}^{\frac{1}{2}},b,c,d,e}{a^{\frac{1}{2}},-a^{\frac{1}{2}},aq/b,aq/c,aq/d,aq/e};q,\frac{q{a}^2}{bcde})=\frac{{(aq,aq/(bc),aq/(bd),aq/(be),aq/(cd),aq/(ce),aq/(de),q,q/a;q)}_{\mathrm{\infty }}}{{(aq/b,aq/c,aq/d,aq/e,q/b,q/c,q/d,q/e,q{a}^2/(bcde);q)}_{\mathrm{\infty }}}.$$

ⓘ

Symbols:

${}_r{}^{}\psi _s^{}(a_1,\mathrm{\dots },a_r;b_1,\mathrm{\dots },b_s;q,z)$ or ${}_r{}^{}\psi _s^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_r}{b_1,\mathrm{\dots },b_s};q,z)$: bilateral basic hypergeometric (or bilateral $q$-hypergeometric) function, ${(a_1,a_2,\mathrm{\dots },a_r;q)}_n$: multiple $q$-Pochhammer symbol and $q$: complex base

Permalink:

http://dlmf.nist.gov/17.8.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §17.8, §17.8 and Ch.17

…

15: 17.11 Transformations of $q$-Appell Functions

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17.11.1 $${\mathrm{\Phi }}^{(1)}(a;b,b^{\prime };c;q;x,y)=\frac{{(a,bx,b^{\prime }y;q)}_{\mathrm{\infty }}}{{(c,x,y;q)}_{\mathrm{\infty }}}{}_3{}^{}\varphi _2^{}(\genfrac{}{}{0pt}{}{c/a,x,y}{bx,b^{\prime }y};q,a),$$

ⓘ

Symbols:

${\mathrm{\Phi }}^{(1)}(a;b,b^{\prime };c;q;x,y)$: first $q$-Appell 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, ${(a_1,a_2,\mathrm{\dots },a_r;q)}_n$: multiple $q$-Pochhammer symbol, $q$: complex base, $x$: real variable and $y$: real variable

Referenced by:

§17.11, Erratum (V1.0.10) for Section 17.1

Permalink:

http://dlmf.nist.gov/17.11.E1

Encodings:

pMML, png, TeX

Correction (effective with 1.0.10):

The notation for ${\mathrm{\Phi }}^{(1)}$ has been updated to that of Gasper and Rahman (2004) to explicitly include the $q$ argument.

See also:

Annotations for §17.11 and Ch.17

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17.11.4 $$\sum _{m_1,\mathrm{\dots },m_n\geqq 0}\frac{{(a;q)}_{m_1+m_2+\mathrm{\cdots }+m_n}{(b_1;q)}_{m_1}{(b_2;q)}_{m_2}\mathrm{\cdots }{(b_n;q)}_{m_n}{x}_1^{m_1}{x}_2^{m_2}\mathrm{\cdots }{x}_n^{m_n}}{{(q;q)}_{m_1}{(q;q)}_{m_2}\mathrm{\cdots }{(q;q)}_{m_n}{(c;q)}_{m_1+m_2+\mathrm{\cdots }+m_n}}=\frac{{(a,b_1{x}_1,b_2{x}_2,\mathrm{\dots },b_n{x}_n;q)}_{\mathrm{\infty }}}{{(c,x_1,x_2,\mathrm{\dots },x_n;q)}_{\mathrm{\infty }}}{}_{n+1}{}^{}\varphi _n^{}(\genfrac{}{}{0pt}{}{c/a,x_1,x_2,\mathrm{\dots },x_n}{b_1{x}_1,b_2{x}_2,\mathrm{\dots },b_n{x}_n};q,a).$$

ⓘ

Symbols:

${(a;q)}_n$: $q$-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, ${(a_1,a_2,\mathrm{\dots },a_r;q)}_n$: multiple $q$-Pochhammer symbol, $q$: complex base, $m$: nonnegative integer, $n$: nonnegative integer and $x$: real variable

Permalink:

http://dlmf.nist.gov/17.11.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §17.11 and Ch.17

16: 18.28 Askey–Wilson Class

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18.28.1_5 $$R_n(z)=R_n(z;a,b,c,d|q)=\frac{p_n(\frac{1}{2}(z+z^{-1});a,b,c,d|q)}{a^{-n}{(ab,ac,ad;q)}_n}={}_4{}^{}\varphi _3^{}(\genfrac{}{}{0pt}{}{q^{-n},abcd{q}^{n-1},az,a{z}^{-1}}{ab,ac,ad};q,q).$$

ⓘ

Symbols:

$p_n(x;a,b,c,d|q)$: Askey–Wilson polynomial, ${}_{r+1}{}^{}\varphi _s^{}(a_0,\mathrm{\dots },a_r;b_1,\mathrm{\dots },b_s;q,z)$ 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, $z$: complex variable, $q$: real variable and $n$: nonnegative integer

Source:

Koornwinder and Mazzocco (2018, (1))

Referenced by:

(18.28.26), §18.28(ii), §18.28(ix), §18.28(x), §18.38(iii), Erratum (V1.2.0) Section 18.27

Permalink:

http://dlmf.nist.gov/18.28.E1_5

Encodings:

pMML, png, TeX

Addition (effective with 1.2.0):

This equation was added.

See also:

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

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18.28.9 $$Q_n(\frac{1}{2}(a{q}^{-y}+a^{-1}{q}^y);a,b|q^{-1})={(-1)}^n{b}^n{q}^{-\frac{1}{2}n(n-1)}{({(ab)}^{-1};q)}_n{}_3{}^{}\varphi _1^{}(\genfrac{}{}{0pt}{}{q^{-n},q^{-y},a^{-2}{q}^y}{{(ab)}^{-1}};q,q^{n}a{b}^{-1}).$$

ⓘ

Defines:

$Q_n(x;a,b|q^{-1})$: $q^{-1}$-Al-Salam–Chihara polynomial

Symbols:

${(a;q)}_n$: $q$-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, $y$: real variable, $q$: real variable, $n$: nonnegative integer and $x$: real variable

Permalink:

http://dlmf.nist.gov/18.28.E9

Encodings:

pMML, png, TeX

See also:

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

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

ⓘ

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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18.28.16 $$H_n\left(\cos \theta |q\right)=\sum _{\mathrm{\ell }=0}^n\frac{{(q;q)}_n{\mathrm{e}}^{\mathrm{i}(n-2\mathrm{\ell })\theta }}{{(q;q)}_{\mathrm{\ell }}{(q;q)}_{n-\mathrm{\ell }}}={\mathrm{e}}^{\mathrm{i}n\theta }{}_2{}^{}\varphi _0^{}(\genfrac{}{}{0pt}{}{q^{-n},0}{-};q,q^n{\mathrm{e}}^{-2\mathrm{i}\theta }).$$

ⓘ

Defines:

$H_n\left(x|q\right)$: continuous $q$-Hermite 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.E16

Encodings:

pMML, png, TeX

See also:

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

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18.28.18 $$h_n\left(\sinh t|q\right)=\sum _{\mathrm{\ell }=0}^n{q}^{\frac{1}{2}\mathrm{\ell }(\mathrm{\ell }+1)}\frac{{(q^{-n};q)}_{\mathrm{\ell }}}{{(q;q)}_{\mathrm{\ell }}}{\mathrm{e}}^{(n-2\mathrm{\ell })t}={\mathrm{e}}^{nt}{}_1{}^{}\varphi _1^{}(\genfrac{}{}{0pt}{}{q^{-n}}{0};q,-q{\mathrm{e}}^{-2t})={\mathrm{i}}^{-n}{H}_n\left(\mathrm{i}\sinh t|q^{-1}\right).$$

ⓘ

Defines:

$h_n\left(x|q\right)$: continuous $q^{-1}$-Hermite polynomial

Symbols:

$H_n\left(x|q\right)$: continuous $q$-Hermite polynomial, $\mathrm{e}$: base of natural logarithm, $\sinh z$: hyperbolic sine function, $\mathrm{i}$: 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, $t$: real variable, $q$: real variable, $\mathrm{\ell }$: nonnegative integer, $n$: nonnegative integer and $x$: real variable

Permalink:

http://dlmf.nist.gov/18.28.E18

Encodings:

pMML, png, TeX

See also:

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

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17: 18.27 $q$-Hahn Class

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18.27.3 $$Q_n(x)=Q_n(x;\alpha ,\beta ,N;q)={}_3{}^{}\varphi _2^{}(\genfrac{}{}{0pt}{}{q^{-n},\alpha \beta q^{n+1},x}{\alpha q,q^{-N}};q,q),$$ $n=0,1,\mathrm{\dots },N$.

ⓘ

Defines:

$Q_n(x;\alpha ,\beta ,N;q)$: $q$-Hahn polynomial

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, $N$: positive integer, $q$: real variable, $n$: nonnegative integer and $x$: real variable

Referenced by:

§18.27(ii)

Permalink:

http://dlmf.nist.gov/18.27.E3

Encodings:

pMML, png, TeX

See also:

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

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18.27.5 $$P_n(x;a,b,c;q)={}_3{}^{}\varphi _2^{}(\genfrac{}{}{0pt}{}{q^{-n},ab{q}^{n+1},x}{aq,cq};q,q).$$

ⓘ

Defines:

$P_n(x;a,b,c;q)$: big $q$-Jacobi polynomial

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, $q$: real variable, $n$: nonnegative integer and $x$: real variable

Referenced by:

(18.27.12_5), (18.27.14_3), (18.28.26)

Permalink:

http://dlmf.nist.gov/18.27.E5

Encodings:

pMML, png, TeX

See also:

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

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18.27.13 $$p_n(x)=p_n(x;a,b;q)={}_2{}^{}\varphi _1^{}(\genfrac{}{}{0pt}{}{q^{-n},ab{q}^{n+1}}{aq};q,qx)={(-b)}^{-n}{q}^{-n(n+1)/2}\frac{{(qb;q)}_n}{{(qa;q)}_n}{}_3{}^{}\varphi _2^{}(\genfrac{}{}{0pt}{}{q^{-n},ab{q}^{n+1},qbx}{qb,0};q,q).$$

ⓘ

Defines:

$p_n(x;a,b;q)$: little $q$-Jacobi polynomial

Symbols:

${(a;q)}_n$: $q$-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, $p_n(x)$: polynomial of degree $n$ , $q$: real variable, $n$: nonnegative integer and $x$: real variable

Referenced by:

(18.28.27), (18.28.28), Erratum (V1.2.0) for Equation (18.27.13)

Permalink:

http://dlmf.nist.gov/18.27.E13

Encodings:

pMML, png, TeX

Addition (effective with 1.2.0):

The ${}_3{}^{}\varphi _2^{}$ representation was added.

See also:

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

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18.27.14_5 $$p_n(x;a,0;q)={}_2{}^{}\varphi _1^{}(\genfrac{}{}{0pt}{}{q^{-n},0}{aq};q,qx).$$

ⓘ

Symbols:

$p_n(x;a,b;q)$: little $q$-Jacobi polynomial, ${}_{r+1}{}^{}\varphi _s^{}(a_0,\mathrm{\dots },a_r;b_1,\mathrm{\dots },b_s;q,z)$ 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, $n$: nonnegative integer and $x$: real variable

Source:

Koekoek et al. (2010, (14.12.12), (14.20.1))

Referenced by:

§18.27(iv)

Permalink:

http://dlmf.nist.gov/18.27.E14_5

Encodings:

pMML, png, TeX

See also:

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

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18.27.15 $$L_n^{(\alpha )}(x;q)=\frac{{(q^{\alpha +1};q)}_n}{{(q;q)}_n}{}_1{}^{}\varphi _1^{}(\genfrac{}{}{0pt}{}{q^{-n}}{q^{\alpha +1}};q,-x{q}^{n+\alpha +1}).$$

ⓘ

Defines:

$L_n^{(\alpha )}(x;q)$: $q$-Laguerre polynomial

Symbols:

${(a;q)}_n$: $q$-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, $n$: nonnegative integer and $x$: real variable

Permalink:

http://dlmf.nist.gov/18.27.E15

Encodings:

pMML, png, TeX

See also:

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

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18: 35.7 Gaussian Hypergeometric Function of Matrix Argument

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§35.7(ii) Basic Properties

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19: Richard A. Askey

… ►His well-known book Special Functions (with G. …One of his most influential papers Some basic hypergeometric orthogonal polynomials that generalize Jacobi polynomials (with J. …Published in 1985 in the Memoirs of the American Mathematical Society, it also introduced the directed graph of hypergeometric orthogonal polynomials commonly known as the Askey scheme. … ►

5 Gamma Function

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16 Generalized Hypergeometric Functions & Meijer G-Function

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20: 16.4 Argument Unity

… ► … ► … ► … ► … ►Denote, formally, the bilateral hypergeometric function …