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21: 16.8 Differential Equations

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16.8.8 $${}_{q+1}{}^{}F_q^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_{q+1}}{b_1,\mathrm{\dots },b_q};z)=\sum _{j=1}^{q+1}\left(\prod _{\begin{array}{c}k=1\\ k\ne j\end{array}}^{q+1}\frac{\mathrm{\Gamma }\left(a_k-a_j\right)}{\mathrm{\Gamma }\left(a_k\right)}/\prod _{k=1}^q\frac{\mathrm{\Gamma }\left(b_k-a_j\right)}{\mathrm{\Gamma }\left(b_k\right)}\right){\tilde{w}}_j(z),$$ $|\mathrm{ph}\left(-z\right)|\le \pi$.

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, ${}_p{}^{}F_q^{}(a_1,\mathrm{\dots },a_p;b_1,\mathrm{\dots },b_q;z)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q};z)$: alternatively ${}_p{}^{}F_q^{}(𝐚;𝐛;z)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{𝐚}{𝐛};z)$
generalized hypergeometric function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{ph}$: phase, $q$: nonnegative integer, $z$: complex variable, $a,a_1,\mathrm{\dots },a_p$: real or complex parameters, $b,b_1,\mathrm{\dots },b_q$: real or complex parameters and $w$: solution

Referenced by:

§16.11(ii)

Permalink:

http://dlmf.nist.gov/16.8.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §16.8(ii), §16.8 and Ch.16

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16.8.9 $$\left(\prod _{k=1}^{q+1}\mathrm{\Gamma }\left(a_k\right)/\prod _{k=1}^q\mathrm{\Gamma }\left(b_k\right)\right){}_{q+1}{}^{}F_q^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_{q+1}}{b_1,\mathrm{\dots },b_q};z)=\sum _{j=1}^{q+1}{\left(z_0-z\right)}^{-a_j}\sum _{n=0}^{\mathrm{\infty }}\frac{\mathrm{\Gamma }\left(a_j+n\right)}{n!}\left(\prod _{\begin{array}{c}k=1\\ k\ne j\end{array}}^{q+1}\mathrm{\Gamma }\left(a_k-a_j-n\right)/\prod _{k=1}^q\mathrm{\Gamma }\left(b_k-a_j-n\right)\right){}_{q+1}{}^{}F_q^{}(\genfrac{}{}{0pt}{}{a_1-a_j-n,\mathrm{\dots },a_{q+1}-a_j-n}{b_1-a_j-n,\mathrm{\dots },b_q-a_j-n};z_0){\left(z-z_0\right)}^{-n}.$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, ${}_p{}^{}F_q^{}(a_1,\mathrm{\dots },a_p;b_1,\mathrm{\dots },b_q;z)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q};z)$: alternatively ${}_p{}^{}F_q^{}(𝐚;𝐛;z)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{𝐚}{𝐛};z)$
generalized hypergeometric function, $!$: factorial (as in $n!$), $q$: nonnegative integer, $z$: complex variable, $a,a_1,\mathrm{\dots },a_p$: real or complex parameters and $b,b_1,\mathrm{\dots },b_q$: real or complex parameters

Referenced by:

§16.8(ii)

Permalink:

http://dlmf.nist.gov/16.8.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §16.8(ii), §16.8 and Ch.16

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22: 27.5 Inversion Formulas

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27.5.1 $$h(n)=\sum _{d|n}f(d)g\left(\frac{n}{d}\right),$$

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Symbols:

$d$: positive integer, $n$: positive integer, $f$: function, $g$: function and $h$: function

Permalink:

http://dlmf.nist.gov/27.5.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §27.5 and Ch.27

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23: 26.12 Plane Partitions

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26.12.20 $$\sum _{\pi \subseteq \mathbb{N}\times \mathbb{N}\times \mathbb{N}}{q}^{|\pi |}=\prod _{k=1}^{\mathrm{\infty }}\frac{1}{{(1-q^k)}^k},$$

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Symbols:

$\mathbb{N}$: set of all positive integers, $\subseteq$: is, or is contained in, $k$: nonnegative integer, $\pi$: plane partition and $q$: parameter

Permalink:

http://dlmf.nist.gov/26.12.E20

Encodings:

pMML, png, TeX

See also:

Annotations for §26.12(ii), §26.12 and Ch.26

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26.12.21 $$\sum _{\pi \subseteq B(r,s,t)}{q}^{|\pi |}=\prod _{(h,j,k)\in B(r,s,t)}\frac{1-q^{h+j+k-1}}{1-q^{h+j+k-2}}=\prod _{h=1}^r\prod _{j=1}^s\frac{1-q^{h+j+t-1}}{1-q^{h+j-1}},$$

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Symbols:

$\in$: element of, $\subseteq$: is, or is contained in, $h$: nonnegative integer, $j$: nonnegative integer, $k$: nonnegative integer, $\pi$: plane partition, $r$: positive integer, $s$: positive integer, $t$: positive integer, $B(r,s,t)$: Box and $q$: parameter

Permalink:

http://dlmf.nist.gov/26.12.E21

Encodings:

pMML, png, TeX

See also:

Annotations for §26.12(ii), §26.12 and Ch.26

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26.12.22

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Symbols:

$\subseteq$: is, or is contained in, $h$: nonnegative integer, $j$: nonnegative integer, $\pi$: plane partition, $r$: positive integer, $t$: positive integer, $B(r,s,t)$: Box and $q$: parameter

Referenced by:

§26.12(ii)

Permalink:

http://dlmf.nist.gov/26.12.E22

Encodings:

pMML, png, TeX

See also:

Annotations for §26.12(ii), §26.12 and Ch.26

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26.12.23

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Symbols:

$\subseteq$: is, or is contained in, $h$: nonnegative integer, $j$: nonnegative integer, $\pi$: plane partition, $r$: positive integer, $B(r,s,t)$: Box and $q$: parameter

Permalink:

http://dlmf.nist.gov/26.12.E23

Encodings:

pMML, png, TeX

See also:

Annotations for §26.12(ii), §26.12 and Ch.26

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26.12.24

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Symbols:

$\subseteq$: is, or is contained in, $h$: nonnegative integer, $j$: nonnegative integer, $\pi$: plane partition, $r$: positive integer, $B(r,s,t)$: Box and $q$: parameter

Permalink:

http://dlmf.nist.gov/26.12.E24

Encodings:

pMML, png, TeX

See also:

Annotations for §26.12(ii), §26.12 and Ch.26

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24: 26.10 Integer Partitions: Other Restrictions

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26.10.2 $$\sum _{n=0}^{\mathrm{\infty }}p(𝒟,n)q^n=\prod _{j=1}^{\mathrm{\infty }}(1+q^j)=\prod _{j=1}^{\mathrm{\infty }}\frac{1}{1-q^{2j-1}}=1+\sum _{m=1}^{\mathrm{\infty }}\frac{q^{m(m+1)/2}}{(1-q)(1-q^2)\mathrm{\cdots }(1-q^m)}=1+\sum _{m=1}^{\mathrm{\infty }}{q}^m(1+q)(1+q^2)\mathrm{\cdots }(1+q^{m-1}),$$

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Symbols:

$p(\mathrm{condition},n)$: restricted number of partitions of $n$ , $j$: nonnegative integer, $m$: nonnegative integer, $n$: nonnegative integer and $q$: parameter

A&S Ref:

24.2.2

Permalink:

http://dlmf.nist.gov/26.10.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §26.10(ii), §26.10 and Ch.26

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26.10.3 $$(1-x)\sum _{m,n=0}^{\mathrm{\infty }}{p}_m(\le k,𝒟,n)x^m{q}^n=\sum _{m=0}^k{\left[\genfrac{}{}{0.0pt}{}{k}{m}\right]}_q{q}^{m(m+1)/2}{x}^m=\prod _{j=1}^k(1+x{q}^j),$$ ,

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Symbols:

${\left[\genfrac{}{}{0.0pt}{}{n}{m}\right]}_q$: $q$-binomial coefficient (or Gaussian polynomial), $p_k(\le m,n)$: number of partitions of $n$ into at most $k$ parts, each less than or equal to $m$ , $x$: real variable, $j$: nonnegative integer, $k$: nonnegative integer, $m$: nonnegative integer, $n$: nonnegative integer and $q$: parameter

Referenced by:

§26.10(ii)

Permalink:

http://dlmf.nist.gov/26.10.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §26.10(ii), §26.10 and Ch.26

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26.10.5 $$\sum _{n=0}^{\mathrm{\infty }}p(\in S,n)q^n=\prod _{j\in S}\frac{1}{1-q^j}.$$

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Symbols:

$\in$: element of, $p(\mathrm{condition},n)$: restricted number of partitions of $n$ , $j$: nonnegative integer, $n$: nonnegative integer, $S$: set and $q$: parameter

Permalink:

http://dlmf.nist.gov/26.10.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §26.10(ii), §26.10 and Ch.26

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25: 26.9 Integer Partitions: Restricted Number and Part Size

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26.9.5 $$\sum _{n=0}^{\mathrm{\infty }}{p}_k\left(n\right)q^n=\prod _{j=1}^k\frac{1}{1-q^j}=1+\sum _{m=1}^{\mathrm{\infty }}{\left[\genfrac{}{}{0.0pt}{}{k+m-1}{m}\right]}_q{q}^m,$$

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Symbols:

$p_k\left(n\right)$: total number of partitions of $n$ into at most $k$ parts, ${\left[\genfrac{}{}{0.0pt}{}{n}{m}\right]}_q$: $q$-binomial coefficient (or Gaussian polynomial), $j$: nonnegative integer, $k$: nonnegative integer, $m$: nonnegative integer, $n$: nonnegative integer and $q$: parameter

Permalink:

http://dlmf.nist.gov/26.9.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §26.9(ii), §26.9 and Ch.26

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26.9.7 $$\sum _{m,n=0}^{\mathrm{\infty }}{p}_k(\le m,n)x^k{q}^n=1+\sum _{k=1}^{\mathrm{\infty }}{\left[\genfrac{}{}{0.0pt}{}{m+k}{k}\right]}_q{x}^k=\prod _{j=0}^m\frac{1}{1-x{q}^j}.$$

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Symbols:

${\left[\genfrac{}{}{0.0pt}{}{n}{m}\right]}_q$: $q$-binomial coefficient (or Gaussian polynomial), $p_k(\le m,n)$: number of partitions of $n$ into at most $k$ parts, each less than or equal to $m$ , $x$: real variable, $j$: nonnegative integer, $k$: nonnegative integer, $m$: nonnegative integer, $n$: nonnegative integer and $q$: parameter

Permalink:

http://dlmf.nist.gov/26.9.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §26.9(ii), §26.9 and Ch.26

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26: 26.14 Permutations: Order Notation

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26.14.3 $$\sum _{\sigma \in {𝔖}_n}{q}^{inv(\sigma )}=\sum _{\sigma \in {𝔖}_n}{q}^{maj(\sigma )}=\prod _{j=1}^n\frac{1-q^j}{1-q}.$$

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Symbols:

$\in$: element of, ${𝔖}_n$: set of permutations of $\{1,2,\mathrm{\dots },n\}$ , $j$: nonnegative integer, $n$: nonnegative integer and $q$: parameter

Permalink:

http://dlmf.nist.gov/26.14.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §26.14(ii), §26.14 and Ch.26

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27: 19.19 Taylor and Related Series

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19.19.4 $$\prod _{j=1}^n(1+t{z}_j)=\sum _{s=0}^n{t}^s{E}_s(𝐳),$$

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Defines:

$E_s(𝐳)$: symmetric function (locally)

Symbols:

$n$: nonnegative integer

Referenced by:

§19.36(i)

Permalink:

http://dlmf.nist.gov/19.19.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §19.19 and Ch.19

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28: 1.3 Determinants, Linear Operators, and Spectral Expansions

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1.3.20 $$𝐮=\sum _{i=1}^n{c}_i{𝐚}_i,$$ $c_i=⟨𝐮,{𝐚}_i⟩$.

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Symbols:

$⟨𝐮,𝐯⟩$: inner product over vectors and $n$: nonnegative integer

Referenced by:

Erratum (V1.2.0) Section 1.3

Permalink:

http://dlmf.nist.gov/1.3.E20

Encodings:

pMML, png, TeX

See also:

Annotations for §1.3(iv), §1.3(iv), §1.3 and Ch.1

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29: 10.23 Sums

§10.23 Sums

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§10.23(i) Multiplication Theorem

… ►For expansions of products of Bessel functions of the first kind in partial fractions see Rogers (2005). … ► …

30: 27.3 Multiplicative Properties

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27.3.6 $${\sigma }_{\alpha }\left(n\right)=\prod _{r=1}^{\nu \left(n\right)}\frac{p_r^{\alpha (1+a_r)}-1}{p_r^{\alpha }-1},$$ $\alpha \ne 0$.

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Symbols:

${\sigma }_{\alpha }\left(n\right)$: sum of powers of divisors of a number, $\nu \left(n\right)$: number of distinct primes dividing a number, $n$: positive integer, $p,p_1,\mathrm{\dots }$: prime numbers and $a_r$: positive exponent

A&S Ref:

24.3.3 I.C

Permalink:

http://dlmf.nist.gov/27.3.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §27.3 and Ch.27

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