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21: 12.9 Asymptotic Expansions for Large Variable

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12.9.1 $$U(a,z)\sim {\mathrm{e}}^{-\frac{1}{4}{z}^2}{z}^{-a-\frac{1}{2}}\sum _{s=0}^{\mathrm{\infty }}{(-1)}^s\frac{{\left(\frac{1}{2}+a\right)}_{2s}}{s!{(2z^2)}^s},$$ ,

ⓘ

Symbols:

${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $U(a,z)$: parabolic cylinder function, $\mathrm{ph}$: phase, $z$: complex variable, $s$: nonnegative integer, $a$: real or complex parameter and $\delta$: arbitrary small positive constant

A&S Ref:

19.8.1 (modification of)

Referenced by:

§12.9(i), §12.9(ii)

Permalink:

http://dlmf.nist.gov/12.9.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §12.9(i), §12.9 and Ch.12

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12.9.2 $$V(a,z)\sim \sqrt{\frac{2}{\pi }}{\mathrm{e}}^{\frac{1}{4}{z}^2}{z}^{a-\frac{1}{2}}\sum _{s=0}^{\mathrm{\infty }}\frac{{\left(\frac{1}{2}-a\right)}_{2s}}{s!{(2z^2)}^s},$$ .

ⓘ

Symbols:

${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $V(a,z)$: parabolic cylinder function, $\mathrm{ph}$: phase, $z$: complex variable, $s$: nonnegative integer, $a$: real or complex parameter and $\delta$: arbitrary small positive constant

A&S Ref:

19.8.2 (modification of)

Referenced by:

§12.9(i), §12.9(ii)

Permalink:

http://dlmf.nist.gov/12.9.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §12.9(i), §12.9 and Ch.12

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12.9.3 $$U(a,z)\sim {\mathrm{e}}^{-\frac{1}{4}{z}^2}{z}^{-a-\frac{1}{2}}\sum _{s=0}^{\mathrm{\infty }}{(-1)}^s\frac{{\left(\frac{1}{2}+a\right)}_{2s}}{s!{(2z^2)}^s}\pm \mathrm{i}\frac{\sqrt{2\pi }}{\mathrm{\Gamma }\left(\frac{1}{2}+a\right)}{\mathrm{e}}^{\mp \mathrm{i}\pi a}{\mathrm{e}}^{\frac{1}{4}{z}^2}{z}^{a-\frac{1}{2}}\sum _{s=0}^{\mathrm{\infty }}\frac{{\left(\frac{1}{2}-a\right)}_{2s}}{s!{(2z^2)}^s},$$ $\frac{1}{4}\pi +\delta \le \pm \mathrm{ph}z\le \frac{5}{4}\pi -\delta$ ,

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $\mathrm{i}$: imaginary unit, $U(a,z)$: parabolic cylinder function, $\mathrm{ph}$: phase, $z$: complex variable, $s$: nonnegative integer, $a$: real or complex parameter and $\delta$: arbitrary small positive constant

Permalink:

http://dlmf.nist.gov/12.9.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §12.9(i), §12.9 and Ch.12

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12.9.4 $$V(a,z)\sim \sqrt{\frac{2}{\pi }}{\mathrm{e}}^{\frac{1}{4}{z}^2}{z}^{a-\frac{1}{2}}\sum _{s=0}^{\mathrm{\infty }}\frac{{\left(\frac{1}{2}-a\right)}_{2s}}{s!{(2z^2)}^s}\pm \frac{\mathrm{i}}{\mathrm{\Gamma }\left(\frac{1}{2}-a\right)}{\mathrm{e}}^{-\frac{1}{4}{z}^2}{z}^{-a-\frac{1}{2}}\sum _{s=0}^{\mathrm{\infty }}{(-1)}^s\frac{{\left(\frac{1}{2}+a\right)}_{2s}}{s!{(2z^2)}^s},$$ $-\frac{1}{4}\pi +\delta \le \pm \mathrm{ph}z\le \frac{3}{4}\pi -\delta$.

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $\mathrm{i}$: imaginary unit, $V(a,z)$: parabolic cylinder function, $\mathrm{ph}$: phase, $z$: complex variable, $s$: nonnegative integer, $a$: real or complex parameter and $\delta$: arbitrary small positive constant

Referenced by:

§12.9(i)

Permalink:

http://dlmf.nist.gov/12.9.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §12.9(i), §12.9 and Ch.12

…

22: 13.13 Addition and Multiplication Theorems

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13.13.1 $$\sum _{n=0}^{\mathrm{\infty }}\frac{{\left(a\right)}_n{y}^n}{{\left(b\right)}_{n}n!}M(a+n,b+n,x),$$

ⓘ

Symbols:

$M(a,b,z)$: $={}_1{}^{}F_1^{}(a;b;z)$ Kummer confluent hypergeometric function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $!$: factorial (as in $n!$), $n$: nonnegative integer, $x$: real variable and $y$: real variable

Permalink:

http://dlmf.nist.gov/13.13.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §13.13(i), §13.13 and Ch.13

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13.13.2 $${\left(\frac{x+y}{x}\right)}^{1-b}\sum _{n=0}^{\mathrm{\infty }}\frac{{\left(1-b\right)}_n{(-y/x)}^n}{n!}M(a,b-n,x),$$ ,

ⓘ

Symbols:

$M(a,b,z)$: $={}_1{}^{}F_1^{}(a;b;z)$ Kummer confluent hypergeometric function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $!$: factorial (as in $n!$), $n$: nonnegative integer, $x$: real variable and $y$: real variable

Permalink:

http://dlmf.nist.gov/13.13.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §13.13(i), §13.13 and Ch.13

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13.13.3 $${\left(\frac{x}{x+y}\right)}^a\sum _{n=0}^{\mathrm{\infty }}\frac{{\left(a\right)}_n{y}^n}{n!{(x+y)}^n}M(a+n,b,x),$$ $\mathrm{\Re }\left(y/x\right)>-\frac{1}{2}$,

ⓘ

Symbols:

$M(a,b,z)$: $={}_1{}^{}F_1^{}(a;b;z)$ Kummer confluent hypergeometric function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $!$: factorial (as in $n!$), $\mathrm{\Re }$: real part, $n$: nonnegative integer, $x$: real variable and $y$: real variable

Permalink:

http://dlmf.nist.gov/13.13.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §13.13(i), §13.13 and Ch.13

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13.13.4 $${\mathrm{e}}^y\sum _{n=0}^{\mathrm{\infty }}\frac{{\left(b-a\right)}_n{(-y)}^n}{{\left(b\right)}_{n}n!}M(a,b+n,x),$$

ⓘ

Symbols:

$M(a,b,z)$: $={}_1{}^{}F_1^{}(a;b;z)$ Kummer confluent hypergeometric function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $n$: nonnegative integer, $x$: real variable and $y$: real variable

Permalink:

http://dlmf.nist.gov/13.13.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §13.13(i), §13.13 and Ch.13

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13.13.5 $${\mathrm{e}}^y{\left(\frac{x}{x+y}\right)}^{b-a}\sum _{n=0}^{\mathrm{\infty }}\frac{{\left(b-a\right)}_n{y}^n}{n!{(x+y)}^n}M(a-n,b,x),$$ $\mathrm{\Re }\left((y+x)/x\right)>\frac{1}{2}$,

ⓘ

Symbols:

$M(a,b,z)$: $={}_1{}^{}F_1^{}(a;b;z)$ Kummer confluent hypergeometric function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $\mathrm{\Re }$: real part, $n$: nonnegative integer, $x$: real variable and $y$: real variable

Permalink:

http://dlmf.nist.gov/13.13.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §13.13(i), §13.13 and Ch.13

…

23: 13.26 Addition and Multiplication Theorems

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13.26.1 $${\mathrm{e}}^{-\frac{1}{2}y}{\left(\frac{x}{x+y}\right)}^{\mu -\frac{1}{2}}\sum _{n=0}^{\mathrm{\infty }}\frac{{\left(-2\mu \right)}_n}{n!}{\left(\frac{-y}{\sqrt{x}}\right)}^n{M}_{\kappa -\frac{1}{2}n,\mu -\frac{1}{2}n}\left(x\right),$$ ,

ⓘ

Symbols:

${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $M_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $n$: nonnegative integer, $x$: real variable and $y$: real variable

Permalink:

http://dlmf.nist.gov/13.26.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §13.26(i), §13.26 and Ch.13

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13.26.2 $${\mathrm{e}}^{-\frac{1}{2}y}{\left(\frac{x+y}{x}\right)}^{\mu +\frac{1}{2}}\sum _{n=0}^{\mathrm{\infty }}\frac{{\left(\frac{1}{2}+\mu -\kappa \right)}_n}{{\left(1+2\mu \right)}_{n}n!}{\left(\frac{y}{\sqrt{x}}\right)}^n{M}_{\kappa -\frac{1}{2}n,\mu +\frac{1}{2}n}\left(x\right),$$

ⓘ

Symbols:

${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $M_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $n$: nonnegative integer, $x$: real variable and $y$: real variable

Permalink:

http://dlmf.nist.gov/13.26.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §13.26(i), §13.26 and Ch.13

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13.26.3 $${\mathrm{e}}^{-\frac{1}{2}y}{\left(\frac{x+y}{x}\right)}^{\kappa }\sum _{n=0}^{\mathrm{\infty }}\frac{{\left(\frac{1}{2}+\mu -\kappa \right)}_n{y}^n}{n!{(x+y)}^n}{M}_{\kappa -n,\mu }\left(x\right),$$ $\mathrm{\Re }\left(y/x\right)>-\frac{1}{2}$,

ⓘ

Symbols:

${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $M_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $\mathrm{\Re }$: real part, $n$: nonnegative integer, $x$: real variable and $y$: real variable

Referenced by:

§13.26(i)

Permalink:

http://dlmf.nist.gov/13.26.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §13.26(i), §13.26 and Ch.13

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13.26.5 $${\mathrm{e}}^{\frac{1}{2}y}{\left(\frac{x+y}{x}\right)}^{\mu +\frac{1}{2}}\sum _{n=0}^{\mathrm{\infty }}\frac{{\left(\frac{1}{2}+\mu +\kappa \right)}_n}{{\left(1+2\mu \right)}_{n}n!}{\left(\frac{-y}{\sqrt{x}}\right)}^n{M}_{\kappa +\frac{1}{2}n,\mu +\frac{1}{2}n}\left(x\right),$$

ⓘ

Symbols:

${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $M_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $n$: nonnegative integer, $x$: real variable and $y$: real variable

Permalink:

http://dlmf.nist.gov/13.26.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §13.26(i), §13.26 and Ch.13

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13.26.6 $${\mathrm{e}}^{\frac{1}{2}y}{\left(\frac{x}{x+y}\right)}^{\kappa }\sum _{n=0}^{\mathrm{\infty }}\frac{{\left(\frac{1}{2}+\mu +\kappa \right)}_n{y}^n}{n!{(x+y)}^n}{M}_{\kappa +n,\mu }\left(x\right),$$ $\mathrm{\Re }\left((y+x)/x\right)>\frac{1}{2}$.

ⓘ

Symbols:

${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $M_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $\mathrm{\Re }$: real part, $n$: nonnegative integer, $x$: real variable and $y$: real variable

Referenced by:

§13.26(i)

Permalink:

http://dlmf.nist.gov/13.26.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §13.26(i), §13.26 and Ch.13

…

24: 15.15 Sums

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15.15.1 $$𝐅(\genfrac{}{}{0pt}{}{a,b}{c};\frac{1}{z})={\left(1-\frac{z_0}{z}\right)}^{-a}\sum _{s=0}^{\mathrm{\infty }}\frac{{\left(a\right)}_s}{s!}𝐅(\genfrac{}{}{0pt}{}{-s,b}{c};\frac{1}{z_0}){\left(1-\frac{z}{z_0}\right)}^{-s}.$$

ⓘ

Symbols:

${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $!$: factorial (as in $n!$), $𝐅(a,b;c;z)$ or $𝐅(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}𝐅_1^{}(a,b;c;z)$ Olver’s hypergeometric function, $s$: nonnegative integer, $z$: complex variable, $a$: real or complex parameter, $b$: real or complex parameter and $c$: real or complex parameter

Referenced by:

§15.19(i)

Permalink:

http://dlmf.nist.gov/15.15.E1

Encodings:

pMML, png, TeX

Correction (effective with 1.1.2):

The Pochhammer symbol now links to its definition.

See also:

Annotations for §15.15 and Ch.15

…

25: 16.13 Appell Functions

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16.13.1 $$F_1(\alpha ;\beta ,{\beta }^{\prime };\gamma ;x,y)=\sum _{m,n=0}^{\mathrm{\infty }}\frac{{\left(\alpha \right)}_{m+n}{\left(\beta \right)}_m{\left({\beta }^{\prime }\right)}_n}{{\left(\gamma \right)}_{m+n}m!n!}{x}^m{y}^n,$$ ,

ⓘ

Defines:

$F_1(\alpha ;\beta ,{\beta }^{\prime };\gamma ;x,y)$: first Appell function

Symbols:

${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial) and $!$: factorial (as in $n!$)

Permalink:

http://dlmf.nist.gov/16.13.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §16.13 and Ch.16

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16.13.2 $$F_2(\alpha ;\beta ,{\beta }^{\prime };\gamma ,{\gamma }^{\prime };x,y)=\sum _{m,n=0}^{\mathrm{\infty }}\frac{{\left(\alpha \right)}_{m+n}{\left(\beta \right)}_m{\left({\beta }^{\prime }\right)}_n}{{\left(\gamma \right)}_m{\left({\gamma }^{\prime }\right)}_{n}m!n!}{x}^m{y}^n,$$ ,

ⓘ

Defines:

$F_2(\alpha ;\beta ,{\beta }^{\prime };\gamma ,{\gamma }^{\prime };x,y)$: second Appell function

Symbols:

${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial) and $!$: factorial (as in $n!$)

Permalink:

http://dlmf.nist.gov/16.13.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §16.13 and Ch.16

►

16.13.3 $$F_3(\alpha ,{\alpha }^{\prime };\beta ,{\beta }^{\prime };\gamma ;x,y)=\sum _{m,n=0}^{\mathrm{\infty }}\frac{{\left(\alpha \right)}_m{\left({\alpha }^{\prime }\right)}_n{\left(\beta \right)}_m{\left({\beta }^{\prime }\right)}_n}{{\left(\gamma \right)}_{m+n}m!n!}{x}^m{y}^n,$$ ,

ⓘ

Defines:

$F_3(\alpha ,{\alpha }^{\prime };\beta ,{\beta }^{\prime };\gamma ;x,y)$: third Appell function

Symbols:

${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial) and $!$: factorial (as in $n!$)

Permalink:

http://dlmf.nist.gov/16.13.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §16.13 and Ch.16

►

16.13.4 $$F_4(\alpha ,\beta ;\gamma ,{\gamma }^{\prime };x,y)=\sum _{m,n=0}^{\mathrm{\infty }}\frac{{\left(\alpha \right)}_{m+n}{\left(\beta \right)}_{m+n}}{{\left(\gamma \right)}_m{\left({\gamma }^{\prime }\right)}_{n}m!n!}{x}^m{y}^n,$$ .

ⓘ

Defines:

$F_4(\alpha ,\beta ;\gamma ,{\gamma }^{\prime };x,y)$: fourth Appell function

Symbols:

${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial) and $!$: factorial (as in $n!$)

Permalink:

http://dlmf.nist.gov/16.13.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §16.13 and Ch.16

…

26: 17.12 Bailey Pairs

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17.12.3 $${\beta }_n=\sum _{j=0}^n\frac{{\alpha }_j}{{(q;q)}_{n-j}{(aq;q)}_{n+j}}.$$

ⓘ

Symbols:

${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), $q$: complex base, $j$: nonnegative integer, $n$: nonnegative integer, ${\alpha }_n$: part of Bailey pair and ${\beta }_n$: part of Bailey pair

Permalink:

http://dlmf.nist.gov/17.12.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §17.12, §17.12 and Ch.17

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17.12.4 $$\sum _{n=0}^{\mathrm{\infty }}{q}^{n^2}{a}^n{\beta }_n=\frac{1}{{(aq;q)}_{\mathrm{\infty }}}\sum _{n=0}^{\mathrm{\infty }}{q}^{n^2}{a}^n{\alpha }_n.$$

ⓘ

Symbols:

${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), $q$: complex base, $n$: nonnegative integer, ${\alpha }_n$: part of Bailey pair and ${\beta }_n$: part of Bailey pair

Permalink:

http://dlmf.nist.gov/17.12.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §17.12, §17.12 and Ch.17

…

27: 17.13 Integrals

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17.13.1 $${\int }_{-c}^d\frac{{(-qx/c;q)}_{\mathrm{\infty }}{(qx/d;q)}_{\mathrm{\infty }}}{{(-ax/c;q)}_{\mathrm{\infty }}{(bx/d;q)}_{\mathrm{\infty }}}{d}_{q}x=\frac{(1-q){(q;q)}_{\mathrm{\infty }}{(ab;q)}_{\mathrm{\infty }}cd{(-c/d;q)}_{\mathrm{\infty }}{(-d/c;q)}_{\mathrm{\infty }}}{{(a;q)}_{\mathrm{\infty }}{(b;q)}_{\mathrm{\infty }}(c+d){(-bc/d;q)}_{\mathrm{\infty }}{(-ad/c;q)}_{\mathrm{\infty }}},$$

ⓘ

Symbols:

$\int$: integral, ${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), $d_{q}x$: $q$-differential, $q$: complex base and $x$: real variable

Permalink:

http://dlmf.nist.gov/17.13.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §17.13 and Ch.17

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17.13.2 $${\int }_{-c}^d\frac{{(-qx/c;q)}_{\mathrm{\infty }}{(qx/d;q)}_{\mathrm{\infty }}}{{(-x{q}^{\alpha }/c;q)}_{\mathrm{\infty }}{(x{q}^{\beta }/d;q)}_{\mathrm{\infty }}}{d}_{q}x=\frac{{\mathrm{\Gamma }}_q\left(\alpha \right){\mathrm{\Gamma }}_q\left(\beta \right)}{{\mathrm{\Gamma }}_q\left(\alpha +\beta \right)}\frac{cd}{c+d}\frac{{(-c/d;q)}_{\mathrm{\infty }}{(-d/c;q)}_{\mathrm{\infty }}}{{(-q^{\beta }c/d;q)}_{\mathrm{\infty }}{(-q^{\alpha }d/c;q)}_{\mathrm{\infty }}}.$$

ⓘ

Symbols:

$\int$: integral, ${\mathrm{\Gamma }}_q\left(z\right)$: $q$-gamma function, ${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), $d_{q}x$: $q$-differential, $q$: complex base and $x$: real variable

Permalink:

http://dlmf.nist.gov/17.13.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §17.13 and Ch.17

… ►

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

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $dx$: differential of $x$ , $\int$: integral, ${\mathrm{\Gamma }}_q\left(z\right)$: $q$-gamma function, ${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial) and $q$: complex base

Referenced by:

Erratum (V1.0.1) for Equation (17.13.3)

Permalink:

http://dlmf.nist.gov/17.13.E3

Encodings:

pMML, png, TeX

Errata (effective with 1.0.1):

Originally the differential was identified incorrectly as a $q$-differential; the correct differential is $dt$.

See also:

Annotations for §17.13, §17.13 and Ch.17

►

17.13.4 $${\int }_0^{\mathrm{\infty }}{t}^{\alpha -1}\frac{{(-ct{q}^{\alpha +\beta };q)}_{\mathrm{\infty }}}{{(-ct;q)}_{\mathrm{\infty }}}{d}_{q}t=\frac{{\mathrm{\Gamma }}_q\left(\alpha \right){\mathrm{\Gamma }}_q\left(\beta \right){(-c{q}^{\alpha };q)}_{\mathrm{\infty }}{(-q^{1-\alpha }/c;q)}_{\mathrm{\infty }}}{{\mathrm{\Gamma }}_q\left(\alpha +\beta \right){(-c;q)}_{\mathrm{\infty }}{(-q/c;q)}_{\mathrm{\infty }}}.$$

ⓘ

Symbols:

$\int$: integral, ${\mathrm{\Gamma }}_q\left(z\right)$: $q$-gamma function, ${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), $d_{q}x$: $q$-differential and $q$: complex base

Permalink:

http://dlmf.nist.gov/17.13.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §17.13, §17.13 and Ch.17

…

28: 5.18 $q$-Gamma and $q$-Beta Functions

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5.18.1 $${(a;q)}_n=\prod _{k=0}^{n-1}(1-a{q}^k),$$ $n=0,1,2,\mathrm{\dots }$,

ⓘ

Symbols:

${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), $q$: real or complex variable, $n$: nonnegative integer, $k$: nonnegative integer and $a$: real or complex variable

Permalink:

http://dlmf.nist.gov/5.18.E1

Encodings:

pMML, png, TeX

See also:

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

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5.18.2 $$n{!}_q=1(1+q)\mathrm{\cdots }(1+q+\mathrm{\cdots }+q^{n-1})={(q;q)}_n{(1-q)}^{-n}.$$

ⓘ

Defines:

${!}_q$: $q$-factorial (as in $n{!}_q$)

Symbols:

${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), $q$: real or complex variable and $n$: nonnegative integer

Permalink:

http://dlmf.nist.gov/5.18.E2

Encodings:

pMML, png, TeX

See also:

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

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5.18.3 $${(a;q)}_{\mathrm{\infty }}=\prod _{k=0}^{\mathrm{\infty }}(1-a{q}^k).$$

ⓘ

Symbols:

${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), $q$: real or complex variable, $k$: nonnegative integer and $a$: real or complex variable

Permalink:

http://dlmf.nist.gov/5.18.E3

Encodings:

pMML, png, TeX

See also:

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

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5.18.4 $${\mathrm{\Gamma }}_q\left(z\right)={(q;q)}_{\mathrm{\infty }}{(1-q)}^{1-z}/{(q^z;q)}_{\mathrm{\infty }},$$

ⓘ

Defines:

${\mathrm{\Gamma }}_q\left(z\right)$: $q$-gamma function

Symbols:

${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), $q$: real or complex variable and $z$: complex variable

Permalink:

http://dlmf.nist.gov/5.18.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §5.18(ii), §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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29: 17.14 Constant Term Identities

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17.14.1

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Symbols:

${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), $k$: nonnegative integer, $q$: complex base, $j$: nonnegative integer, $n$: nonnegative integer and $x$: real variable

Referenced by:

§17.14

Permalink:

http://dlmf.nist.gov/17.14.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §17.14, §17.14 and Ch.17

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17.14.2 $$\sum _{n=0}^{\mathrm{\infty }}\frac{q^{n(n+1)}}{{(q^2;q^2)}_n{(-q;q^2)}_{n+1}}=\text{ coeff. of }{z}^0\text{ in }\frac{{(-zq;q^2)}_{\mathrm{\infty }}{(-z^{-1}q;q^2)}_{\mathrm{\infty }}{(q^2;q^2)}_{\mathrm{\infty }}}{{(z^{-1}{q}^2;q^2)}_{\mathrm{\infty }}{(-q;q^2)}_{\mathrm{\infty }}{(z^{-1}q;q^2)}_{\mathrm{\infty }}}=\frac{1}{{(-q;q^2)}_{\mathrm{\infty }}}\text{ coeff. of }{z}^0\text{ in }\frac{{(-zq;q^2)}_{\mathrm{\infty }}{(-z^{-1}q;q^2)}_{\mathrm{\infty }}{(q^2;q^2)}_{\mathrm{\infty }}}{{(z^{-1}q;q)}_{\mathrm{\infty }}}=\frac{H(q)}{{(-q;q^2)}_{\mathrm{\infty }}},$$

ⓘ

Symbols:

${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), $q$: complex base, $n$: nonnegative integer, $z$: complex variable and $H(q)$: LHS of (17.2.50)

Referenced by:

§17.14

Permalink:

http://dlmf.nist.gov/17.14.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §17.14, §17.14 and Ch.17

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17.14.3 $$\sum _{n=0}^{\mathrm{\infty }}\frac{q^{n(n+1)}}{{(q^2;q^2)}_n{(-q;q^2)}_{n+1}}=\text{ coeff. of }{z}^0\text{ in }\frac{{(-zq;q^2)}_{\mathrm{\infty }}{(-z^{-1}q;q^2)}_{\mathrm{\infty }}{(q^2;q^2)}_{\mathrm{\infty }}}{{(z^{-1};q^2)}_{\mathrm{\infty }}{(-q;q^2)}_{\mathrm{\infty }}{(z^{-1}q;q^2)}_{\mathrm{\infty }}}=\frac{1}{{(-q;q^2)}_{\mathrm{\infty }}}\text{ coeff. of }{z}^0\text{ in }\frac{{(-zq;q^2)}_{\mathrm{\infty }}{(-z^{-1}q;q^2)}_{\mathrm{\infty }}{(q^2;q^2)}_{\mathrm{\infty }}}{{(z^{-1};q)}_{\mathrm{\infty }}}=\frac{G(q)}{{(-q;q^2)}_{\mathrm{\infty }}},$$

ⓘ

Symbols:

${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), $q$: complex base, $n$: nonnegative integer, $z$: complex variable and $G(q)$: LHS of (17.2.49)

Permalink:

http://dlmf.nist.gov/17.14.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §17.14, §17.14 and Ch.17

►

17.14.4 $$\sum _{n=0}^{\mathrm{\infty }}\frac{q^{n^2}}{{(q^2;q^2)}_n{(q;q^2)}_n}=\text{ coeff. of }{z}^0\text{ in }\frac{{(-zq;q^2)}_{\mathrm{\infty }}{(-z^{-1}q;q^2)}_{\mathrm{\infty }}{(q^2;q^2)}_{\mathrm{\infty }}}{{(-z^{-1};q^2)}_{\mathrm{\infty }}{(q;q^2)}_{\mathrm{\infty }}{(z^{-1};q^2)}_{\mathrm{\infty }}}=\frac{1}{{(q;q^2)}_{\mathrm{\infty }}}\text{ coeff. of }{z}^0\text{ in }\frac{{(-zq;q^2)}_{\mathrm{\infty }}{(-z^{-1}q;q^2)}_{\mathrm{\infty }}{(q^2;q^2)}_{\mathrm{\infty }}}{{(z^{-2};q^4)}_{\mathrm{\infty }}}=\frac{G(q^4)}{{(q;q^2)}_{\mathrm{\infty }}},$$

ⓘ

Symbols:

${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), $q$: complex base, $n$: nonnegative integer, $z$: complex variable and $G(q)$: LHS of (17.2.49)

Permalink:

http://dlmf.nist.gov/17.14.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §17.14, §17.14 and Ch.17

►

17.14.5 $$\sum _{n=0}^{\mathrm{\infty }}\frac{q^{n^2+2n}}{{(q^2;q^2)}_n{(q;q^2)}_{n+1}}=\text{ coeff. of }{z}^0\text{ in }\frac{{(-zq;q^2)}_{\mathrm{\infty }}{(-z^{-1}q;q^2)}_{\mathrm{\infty }}{(q^2;q^2)}_{\mathrm{\infty }}}{{(-q^2{z}^{-1};q^2)}_{\mathrm{\infty }}{(q;q^2)}_{\mathrm{\infty }}{(z^{-1}{q}^2;q^2)}_{\mathrm{\infty }}}=\frac{1}{{(q;q^2)}_{\mathrm{\infty }}}\text{ coeff. of }{z}^0\text{ in }\frac{{(-zq;q^2)}_{\mathrm{\infty }}{(-z^{-1}q;q^2)}_{\mathrm{\infty }}{(q^2;q^2)}_{\mathrm{\infty }}}{{(q^4{z}^{-2};q^4)}_{\mathrm{\infty }}}=\frac{H(q^4)}{{(q;q^2)}_{\mathrm{\infty }}}.$$

ⓘ

Symbols:

${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), $q$: complex base, $n$: nonnegative integer, $z$: complex variable and $H(q)$: LHS of (17.2.50)

Referenced by:

§17.14

Permalink:

http://dlmf.nist.gov/17.14.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §17.14, §17.14 and Ch.17

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30: 19.19 Taylor and Related Series

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19.19.1 $$T_N(𝐛,𝐳)=\sum \frac{{\left(b_1\right)}_{m_1}\mathrm{\cdots }{\left(b_n\right)}_{m_n}}{m_1!\mathrm{\cdots }{m}_n!}{z}_1^{m_1}\mathrm{\cdots }{z}_n^{m_n},$$

ⓘ

Symbols:

${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $!$: factorial (as in $n!$), $m$: nonnegative integer, $n$: nonnegative integer, $N$: nonnegative integer and $T_N(𝐛,𝐳)$: polynomial

Referenced by:

§19.20(v)

Permalink:

http://dlmf.nist.gov/19.19.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §19.19 and Ch.19

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19.19.2 $$R_{-a}(𝐛;𝐳)=\sum _{N=0}^{\mathrm{\infty }}\frac{{\left(a\right)}_N}{{\left(c\right)}_N}{T}_N(𝐛,\mathrm{𝟏}-𝐳),$$ $c={\sum }_{j=1}^n{b}_j$, ,

ⓘ

Symbols:

$R_{-a}(b_1,\mathrm{\dots },b_n;z_1,\mathrm{\dots },z_n)$ or $R_{-a}(𝐛;𝐳)$: multivariate hypergeometric function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $n$: nonnegative integer, $N$: nonnegative integer and $T_N(𝐛,𝐳)$: polynomial

Referenced by:

§19.19

Permalink:

http://dlmf.nist.gov/19.19.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §19.19 and Ch.19

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19.19.3 $$R_{-a}(𝐛;𝐳)=z_n^{-a}\sum _{N=0}^{\mathrm{\infty }}\frac{{\left(a\right)}_N}{{\left(c\right)}_N}{T}_N(b_1,\mathrm{\dots },b_{n-1};1-(z_1/z_n),\mathrm{\dots },1-(z_{n-1}/z_n)),$$ $c={\sum }_{j=1}^n{b}_j$, .

ⓘ

Symbols:

$R_{-a}(b_1,\mathrm{\dots },b_n;z_1,\mathrm{\dots },z_n)$ or $R_{-a}(𝐛;𝐳)$: multivariate hypergeometric function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $n$: nonnegative integer, $N$: nonnegative integer and $T_N(𝐛,𝐳)$: polynomial

Referenced by:

§19.19, §19.19, §19.28

Permalink:

http://dlmf.nist.gov/19.19.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §19.19 and Ch.19

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19.19.5 $$T_N(\frac{\mathrm{𝟏}}{\mathrm{𝟐}},𝐳)=\sum {(-1)}^{M+N}{\left(\frac{1}{2}\right)}_M\frac{E_1^{m_1}(𝐳)\mathrm{\cdots }{E}_n^{m_n}(𝐳)}{m_1!\mathrm{\cdots }{m}_n!},$$

ⓘ

Symbols:

${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $!$: factorial (as in $n!$), $m$: nonnegative integer, $n$: nonnegative integer, $N$: nonnegative integer, $T_N(𝐛,𝐳)$: polynomial, $E_s(𝐳)$: symmetric function and $M$: sum

Referenced by:

§19.19, §19.36(i)

Permalink:

http://dlmf.nist.gov/19.19.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §19.19 and Ch.19

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19.19.7 $$R_{-a}(\frac{\mathrm{𝟏}}{\mathrm{𝟐}};𝐳)=A^{-a}\sum _{N=0}^{\mathrm{\infty }}\frac{{\left(a\right)}_N}{{\left(\frac{1}{2}n\right)}_N}{T}_N(\frac{\mathrm{𝟏}}{\mathrm{𝟐}},𝐙),$$

ⓘ

Symbols:

$R_{-a}(b_1,\mathrm{\dots },b_n;z_1,\mathrm{\dots },z_n)$ or $R_{-a}(𝐛;𝐳)$: multivariate hypergeometric function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $n$: nonnegative integer, $N$: nonnegative integer, $T_N(𝐛,𝐳)$: polynomial, $𝐙$: variable and $A$

Referenced by:

§19.15, §19.36(i), §19.36(i), §19.36(i)

Permalink:

http://dlmf.nist.gov/19.19.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §19.19 and Ch.19

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