q-Pochhammer symbol

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21: 34.5 Basic Properties: $\mathit{6}j$ Symbol

§34.5 Basic Properties: $\mathit{6}j$ Symbol

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§34.5(ii) Symmetry

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§34.5(iv) Orthogonality

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§34.5(vi) Sums

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22: 5.2 Definitions

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§5.2(iii) Pochhammer’s Symbol

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5.2.5 $${\left(a\right)}_n=\mathrm{\Gamma }\left(a+n\right)/\mathrm{\Gamma }\left(a\right),$$ $a\ne 0,-1,-2,\mathrm{\dots }$.

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $n$: nonnegative integer and $a$: real or complex variable

A&S Ref:

6.1.22

Referenced by:

(25.11.28), (25.5.7), (25.8.2)

Permalink:

http://dlmf.nist.gov/5.2.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §5.2(iii), §5.2 and Ch.5

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5.2.6 $${\left(-a\right)}_n={(-1)}^n{\left(a-n+1\right)}_n,$$

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Symbols:

${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $n$: nonnegative integer and $a$: real or complex variable

Referenced by:

§5.2(iii), Erratum (V1.0.17) for Subsection 5.2(iii)

Permalink:

http://dlmf.nist.gov/5.2.E6

Encodings:

pMML, png, TeX

Addition (effective with 1.0.17):

This equation was added.

See also:

Annotations for §5.2(iii), §5.2 and Ch.5

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5.2.7

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Symbols:

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

Permalink:

http://dlmf.nist.gov/5.2.E7

Encodings:

pMML, png, TeX

Addition (effective with 1.0.17):

This equation was added.

See also:

Annotations for §5.2(iii), §5.2 and Ch.5

… ►Pochhammer symbols (rising factorials) ${\left(x\right)}_n=x(x+1)\mathrm{\cdots }(x+n-1)$ and falling factorials ${(-1)}^n{\left(-x\right)}_n=x(x-1)\mathrm{\cdots }(x-n+1)$ can be expressed in terms of each other via …

23: 16.7 Relations to Other Functions

… ►For $\mathit{3}j$, $\mathit{6}j$, $\mathit{9}j$ symbols see Chapter 34. …

24: 18.42 Software

… ►A more complete list of available software for computing these functions, and for generating formulas symbolically, is found in the Software Index. …

25: 17.2 Calculus

… ►For $n=0,1,2,\mathrm{\dots }$, … … ►

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

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

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

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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, $q$: 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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26: About MathML

… ►Since the display of mathematics involves many special symbols not often seen in plain text, a MathML renderer generally needs special fonts. … ►For other browsers, you may see a ? or a box like indicating missing symbols, and thus insufficient fonts. …

27: Guide to Searching the DLMF

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a textual word, a number, or a math symbol.

… ►To recognize the math symbols and structures, and to accommodate equivalence between various notations and various forms of expression, the search system maps the math part of your queries into a different form. …Be careful, however, because if you put quotes around math expressions involving math symbols, you may not get the matches you’d expect. ►

Math Symbols

►You can use in math queries all the symbols and commands defined in LaTeX (you can omit the $\backslash$), and some additional convenient ones, as well as the special functions’ names: …

28: 15.15 Sums

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

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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 Pochhammersymbol now links to its definition.

See also:

Annotations for §15.15 and Ch.15

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29: 10 Bessel Functions

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30: 18 Orthogonal Polynomials

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