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11: 31.6 Path-Multiplicative Solutions

§31.6 Path-Multiplicative Solutions

►A further extension of the notation (31.4.1) and (31.4.3) is given by …These solutions are called path-multiplicative. …

12: 17.12 Bailey Pairs

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$${(\frac{aq}{{\rho }_1},\frac{aq}{{\rho }_2};q)}_n{\alpha }_n^{\prime }={({\rho }_1,{\rho }_2;q)}_n{\left(\frac{aq}{{\rho }_1{\rho }_2}\right)}^n{\alpha }_n$$

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$${(\frac{aq}{{\rho }_1},\frac{aq}{{\rho }_2};q)}_n{\beta }_n^{\prime }=\sum _{j=0}^n{({\rho }_1,{\rho }_2;q)}_j{(\frac{aq}{{\rho }_1{\rho }_2};q)}_{n-j}{\left(\frac{aq}{{\rho }_1{\rho }_2}\right)}^j\frac{{\beta }_j}{{(q;q)}_{n-j}}$$

…

13: 18.28 Askey–Wilson Class

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18.28.1 $$p_n(x)=p_n(x;a,b,c,d|q)=a^{-n}\sum _{\mathrm{\ell }=0}^n{q}^{\mathrm{\ell }}{(ab{q}^{\mathrm{\ell }},ac{q}^{\mathrm{\ell }},ad{q}^{\mathrm{\ell }};q)}_{n-\mathrm{\ell }}\frac{{(q^{-n},abcd{q}^{n-1};q)}_{\mathrm{\ell }}}{{(q;q)}_{\mathrm{\ell }}}\prod _{j=0}^{\mathrm{\ell }-1}(1-2a{q}^{j}x+a^2{q}^{2j}),$$

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Defines:

$p_n(x;a,b,c,d|q)$: Askey–Wilson polynomial

Symbols:

${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), ${(a_1,a_2,\mathrm{\dots },a_r;q)}_n$: multiple $q$-Pochhammer symbol, $p_n(x)$: polynomial of degree $n$ , $q$: real variable, $\mathrm{\ell }$: nonnegative integer, $n$: nonnegative integer and $x$: real variable

Referenced by:

(18.28.29), §18.28(ii), §18.38(iii), Erratum (V1.2.0) for Equation (18.28.1)

Permalink:

http://dlmf.nist.gov/18.28.E1

Encodings:

pMML, png, TeX

See also:

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

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18.28.4 $$h_0=\frac{{(abcd;q)}_{\mathrm{\infty }}}{{(q,ab,ac,ad,bc,bd,cd;q)}_{\mathrm{\infty }}},$$

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Symbols:

${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), ${(a_1,a_2,\mathrm{\dots },a_r;q)}_n$: multiple $q$-Pochhammer symbol, $q$: real variable and $h_n$

Permalink:

http://dlmf.nist.gov/18.28.E4

Encodings:

pMML, png, TeX

See also:

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

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18.28.5 $$h_n=h_0\frac{(1-abcd{q}^{n-1}){(q,ab,ac,ad,bc,bd,cd;q)}_n}{(1-abcd{q}^{2n-1}){(abcd;q)}_n},$$ $n=1,2,\mathrm{\dots }$.

ⓘ

Symbols:

${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), ${(a_1,a_2,\mathrm{\dots },a_r;q)}_n$: multiple $q$-Pochhammer symbol, $q$: real variable, $n$: nonnegative integer and $h_n$

Permalink:

http://dlmf.nist.gov/18.28.E5

Encodings:

pMML, png, TeX

See also:

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

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18.28.21 $${\omega }_y=\frac{{(\alpha q,\beta \delta q,\gamma q,\gamma \delta q;q)}_y}{{(q,\frac{\gamma \delta }{\alpha }q,\frac{\gamma }{\beta }q,\delta q;q)}_y}\frac{1-\gamma \delta q^{2y+1}}{{\left(\alpha \beta q\right)}^y},$$

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Symbols:

${(a_1,a_2,\mathrm{\dots },a_r;q)}_n$: multiple $q$-Pochhammer symbol, $y$: real variable, $\delta$: arbitrary small positive constant and $q$: real variable

Referenced by:

§18.28(viii), Erratum (V1.2.0) Section 18.28

Permalink:

http://dlmf.nist.gov/18.28.E21

Encodings:

pMML, png, TeX

See also:

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

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18.28.22 $$h_n=\frac{{\left(\alpha \beta \right)}^{n+1}{q}^{{\left(n+1\right)}^2}}{\alpha \beta q^{2n+1}-1}\frac{{(q;q)}_n}{{(\alpha q,\beta \delta q,\gamma q;q)}_n}\frac{{(\frac{\gamma }{\alpha \beta }{q}^{-n},\frac{\delta }{\alpha }{q}^{-n},\frac{1}{\beta }{q}^{-n},\gamma \delta q;q)}_{\mathrm{\infty }}}{{(\frac{1}{\alpha \beta }{q}^{-n},\frac{\gamma \delta }{\alpha }q,\frac{\gamma }{\beta }q,\delta q;q)}_{\mathrm{\infty }}}.$$

ⓘ

Symbols:

${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), ${(a_1,a_2,\mathrm{\dots },a_r;q)}_n$: multiple $q$-Pochhammer symbol, $\delta$: arbitrary small positive constant, $q$: real variable, $n$: nonnegative integer and $h_n$

Permalink:

http://dlmf.nist.gov/18.28.E22

Encodings:

pMML, png, TeX

See also:

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

…

14: 27.4 Euler Products and Dirichlet Series

… ►Every multiplicative $f$ satisfies the identity …If $f(n)$ is completely multiplicative, then each factor in the product is a geometric series and the Euler product becomes … ►Euler products are used to find series that generate many functions of multiplicative number theory. The completely multiplicative function $f(n)=n^{-s}$ gives the Euler product representation of the Riemann zeta function $\zeta \left(s\right)$ (§25.2(i)): … ►

27.4.4 $$F(s)=\sum _{n=1}^{\mathrm{\infty }}f(n)n^{-s},$$

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Symbols:

$n$: positive integer, $f$: multiplicative function and $F(s)$: generating function

Permalink:

http://dlmf.nist.gov/27.4.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §27.4 and Ch.27

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15: 17.2 Calculus

… ► … ►

17.2.5 $${(a_1,a_2,\mathrm{\dots },a_r;q)}_n=\prod _{j=1}^r{(a_j;q)}_n,$$

ⓘ

Symbols:

${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), ${(a_1,a_2,\mathrm{\dots },a_r;q)}_n$: multiple $q$-Pochhammer symbol, $q$: complex base, $j$: nonnegative integer, $n$: nonnegative integer and $r$: nonnegative integer

Permalink:

http://dlmf.nist.gov/17.2.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §17.2(i), §17.2 and Ch.17

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17.2.6 $${(a_1,a_2,\mathrm{\dots },a_r;q)}_{\mathrm{\infty }}=\prod _{j=1}^r{(a_j;q)}_{\mathrm{\infty }}.$$

ⓘ

Symbols:

${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), ${(a_1,a_2,\mathrm{\dots },a_r;q)}_n$: multiple $q$-Pochhammer symbol, $q$: complex base, $j$: nonnegative integer and $r$: nonnegative integer

Referenced by:

§17.2(i)

Permalink:

http://dlmf.nist.gov/17.2.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §17.2(i), §17.2 and Ch.17

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17.2.19 $${(a;q)}_{2n}={(a,aq;q^2)}_n,$$

ⓘ

Symbols:

${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), ${(a_1,a_2,\mathrm{\dots },a_r;q)}_n$: multiple $q$-Pochhammer symbol, $q$: complex base and $n$: nonnegative integer

Permalink:

http://dlmf.nist.gov/17.2.E19

Encodings:

pMML, png, TeX

See also:

Annotations for §17.2(i), §17.2 and Ch.17

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17.2.22 $$\frac{{(q{a}^{\frac{1}{2}},-q{a}^{\frac{1}{2}};q)}_n}{{(a^{\frac{1}{2}},-a^{\frac{1}{2}};q)}_n}=\frac{{(a{q}^2;q^2)}_n}{{(a;q^2)}_n}=\frac{1-a{q}^{2n}}{1-a},$$

ⓘ

Symbols:

${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), ${(a_1,a_2,\mathrm{\dots },a_r;q)}_n$: multiple $q$-Pochhammer symbol, $q$: complex base and $n$: nonnegative integer

Referenced by:

Erratum (V1.0.16) for Equations (17.2.22) and (17.2.23)

Permalink:

http://dlmf.nist.gov/17.2.E22

Encodings:

pMML, png, TeX

Errata (effective with 1.0.16):

The numerator of the leftmost fraction was corrected to read ${(q{a}^{\frac{1}{2}},-q{a}^{\frac{1}{2}};q)}_n$ instead of ${(q{a}^{\frac{1}{2}},-a{q}^{\frac{1}{2}};q)}_n$.

Reported 2017-06-26 by Jason Zhao

See also:

Annotations for §17.2(i), §17.2 and Ch.17

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16: 18.27 $q$-Hahn Class

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18.27.9 $$v_x=\frac{{(a^{-1}x,c^{-1}x;q)}_{\mathrm{\infty }}}{{(x,b{c}^{-1}x;q)}_{\mathrm{\infty }}},$$ , , ,

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Symbols:

${(a_1,a_2,\mathrm{\dots },a_r;q)}_n$: multiple $q$-Pochhammer symbol, $q$: real variable, $x$: real variable and $v_x$

Permalink:

http://dlmf.nist.gov/18.27.E9

Encodings:

pMML, png, TeX

Correction (effective with 1.1.2):

$q$-Pochhammer symbols now link to their definition.

See also:

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

►

18.27.9_5 $$h_n=\frac{{\left(-c\right)}^n{a}^{n+1}}{1-ab{q}^{2n+1}}\frac{{(q;q)}_n{q}^{\left(\genfrac{}{}{0.0pt}{}{n+2}{2}\right)}}{{(aq,cq;q)}_n}\frac{{(q,c^{-1}aq,a^{-1}c,ab{q}^{n+1};q)}_{\mathrm{\infty }}}{{(aq,cq,b{q}^{n+1},c^{-1}ab{q}^{n+1};q)}_{\mathrm{\infty }}},$$

ⓘ

Symbols:

$\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, ${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), ${(a_1,a_2,\mathrm{\dots },a_r;q)}_n$: multiple $q$-Pochhammer symbol, $q$: real variable, $n$: nonnegative integer and $h_n$

Referenced by:

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

Permalink:

http://dlmf.nist.gov/18.27.E9_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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18.27.12 $$v_x=\frac{{(qx/c,-qx/d;q)}_{\mathrm{\infty }}}{{(q^{\alpha +1}x/c,-q^{\beta +1}x/d;q)}_{\mathrm{\infty }}},$$ $\alpha ,\beta >-1$, $c,d>0$.

ⓘ

Symbols:

${(a_1,a_2,\mathrm{\dots },a_r;q)}_n$: multiple $q$-Pochhammer symbol, $q$: real variable, $x$: real variable and $v_x$

Permalink:

http://dlmf.nist.gov/18.27.E12

Encodings:

pMML, png, TeX

Correction (effective with 1.1.2):

$q$-Pochhammer symbols now link to their definition.

See also:

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

… ►

18.27.14_1 $$h_n=\frac{{\left(aq\right)}^n}{1-ab{q}^{2n+1}}\frac{{(q,bq;q)}_n}{{(aq;q)}_n}\frac{{(ab{q}^{n+1};q)}_{\mathrm{\infty }}}{{(aq;q)}_{\mathrm{\infty }}}.$$

ⓘ

Symbols:

${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), ${(a_1,a_2,\mathrm{\dots },a_r;q)}_n$: multiple $q$-Pochhammer symbol, $q$: real variable, $n$: nonnegative integer and $h_n$

Source:

Koekoek et al. (2010, (14.12.2))

Referenced by:

§18.27(iv), (18.28.27), Erratum (V1.2.0) Section 18.27

Permalink:

http://dlmf.nist.gov/18.27.E14_1

Encodings:

pMML, png, TeX

Addition (effective with 1.2.0):

This equation was added.

See also:

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

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18.27.17_2 $$h_0^{(2)}=\frac{{(q,-c{q}^{\alpha +1},-c^{-1}{q}^{-\alpha };q)}_{\mathrm{\infty }}}{{(q^{\alpha +1},-c,-c^{-1}q;q)}_{\mathrm{\infty }}}.$$

ⓘ

Symbols:

${(a_1,a_2,\mathrm{\dots },a_r;q)}_n$: multiple $q$-Pochhammer symbol, $q$: real variable and $h_n$

Source:

Koekoek et al. (2010, (14.21.3))

Permalink:

http://dlmf.nist.gov/18.27.E17_2

Encodings:

pMML, png, TeX

Modification (effective with 1.2.0):

This equation was added.

See also:

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

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17: 5.5 Functional Relations

… ►

§5.5(iii) Multiplication

… ►

Gauss’s Multiplication Formula

… ►

5.5.7 $$\prod _{k=1}^{n-1}\mathrm{\Gamma }\left(\frac{k}{n}\right)={(2\pi )}^{(n-1)/2}{n}^{-1/2}.$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $n$: nonnegative integer and $k$: nonnegative integer

Permalink:

http://dlmf.nist.gov/5.5.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §5.5(iii), §5.5(iii), §5.5 and Ch.5

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18: 27.1 Special Notation

§27.1 Special Notation

… ► ►►

$d,k,m,n$	positive integers (unless otherwise indicated).
…

19: 24.12 Zeros

… ►

§24.12(iv) Multiple Zeros

► $B_n\left(x\right)$, $n=1,2,\mathrm{\dots }$, has no multiple zeros. The only polynomial $E_n\left(x\right)$ with multiple zeros is $E_5\left(x\right)=(x-\frac{1}{2}){(x^2-x-1)}^2$.

20: 35.10 Methods of Computation

… ►Other methods include numerical quadrature applied to double and multiple integral representations. …