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11: 1.12 Continued Fractions

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1.12.9 $$C_n=b_0+\frac{a_1}{B_0{B}_1}-\mathrm{\cdots }+{(-1)}^{n-1}\frac{{\prod }_{k=1}^n{a}_k}{B_{n-1}{B}_n}.$$

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Symbols:

$k$: integer, $n$: nonnegative integer, $B_n$: $n$th denominator and $C_n(w)$: continued fraction

Permalink:

http://dlmf.nist.gov/1.12.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §1.12(ii), §1.12(ii), §1.12 and Ch.1

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12: 18.1 Notation

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${(z;q)}_n=(1-z)(1-zq)\mathrm{\cdots }(1-z{q}^{n-1}),$

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$${(z_1,\mathrm{\dots },z_k;q)}_n={(z_1;q)}_n\mathrm{\cdots }{(z_k;q)}_n.$$

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Infinite $q$-Product

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$${(z;q)}_{\mathrm{\infty }}=\prod _{j=0}^{\mathrm{\infty }}(1-z{q}^j),$$

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$${(z_1,\mathrm{\dots },z_k;q)}_{\mathrm{\infty }}={(z_1;q)}_{\mathrm{\infty }}\mathrm{\cdots }{(z_k;q)}_{\mathrm{\infty }}.$$

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13: 5.14 Multidimensional Integrals

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5.14.2 $${\int }_{V_n}{\left(1-\sum _{k=1}^n{t}_k\right)}^{z_{n+1}-1}\prod _{k=1}^n{t}_k^{z_k-1}d{t}_k=\frac{\mathrm{\Gamma }\left(z_1\right)\mathrm{\Gamma }\left(z_2\right)\mathrm{\cdots }\mathrm{\Gamma }\left(z_{n+1}\right)}{\mathrm{\Gamma }\left(z_1+z_2+\mathrm{\cdots }+z_{n+1}\right)}.$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $dx$: differential of $x$ , $\int$: integral, $n$: nonnegative integer, $k$: nonnegative integer, $z$: complex variable and $V_n$: simplex

Referenced by:

§5.19(iii)

Permalink:

http://dlmf.nist.gov/5.14.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §5.14 and Ch.5

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5.14.4 $${\int }_{{[0,1]}^n}{t}_1{t}_2\mathrm{\cdots }{t}_m{|\mathrm{\Delta }(t_1,\mathrm{\dots },t_n)|}^{2c}\prod _{k=1}^n{t}_k^{a-1}{(1-t_k)}^{b-1}d{t}_k=\frac{1}{{(\mathrm{\Gamma }\left(1+c\right))}^n}\prod _{k=1}^m\frac{a+(n-k)c}{a+b+(2n-k-1)c}\prod _{k=1}^n\frac{\mathrm{\Gamma }\left(a+(n-k)c\right)\mathrm{\Gamma }\left(b+(n-k)c\right)\mathrm{\Gamma }\left(1+kc\right)}{\mathrm{\Gamma }\left(a+b+(2n-k-1)c\right)},$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $[a,b]$: closed interval, $dx$: differential of $x$ , $\int$: integral, $m$: nonnegative integer, $n$: nonnegative integer, $k$: nonnegative integer, $a$: real or complex variable, $b$: real or complex variable and $\mathrm{\Delta }$: product

Permalink:

http://dlmf.nist.gov/5.14.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §5.14, §5.14 and Ch.5

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5.14.5 $${\int }_{{[0,\mathrm{\infty })}^n}{t}_1{t}_2\mathrm{\cdots }{t}_m{|\mathrm{\Delta }(t_1,\mathrm{\dots },t_n)|}^{2c}\prod _{k=1}^n{t}_k^{a-1}{\mathrm{e}}^{-t_k}d{t}_k=\prod _{k=1}^m(a+(n-k)c)\frac{{\prod }_{k=1}^n\mathrm{\Gamma }\left(a+(n-k)c\right)\mathrm{\Gamma }\left(1+kc\right)}{{(\mathrm{\Gamma }\left(1+c\right))}^n},$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $[a,b)$: half-closed interval, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral, $m$: nonnegative integer, $n$: nonnegative integer, $k$: nonnegative integer, $a$: real or complex variable and $\mathrm{\Delta }$: product

Permalink:

http://dlmf.nist.gov/5.14.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §5.14, §5.14 and Ch.5

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5.14.7 $\mathrm{\Re }b>-1/n$.

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $[a,b]$: closed interval, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\int$: integral, $\mathrm{\Re }$: real part, $j$: nonnegative integer, $n$: nonnegative integer, $k$: nonnegative integer and $b$: real or complex variable

Referenced by:

§5.14, §5.20

Permalink:

http://dlmf.nist.gov/5.14.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §5.14, §5.14 and Ch.5

14: 27.12 Asymptotic Formulas: Primes

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27.12.3 $x\ge 1$,

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Symbols:

$\lfloor x\rfloor$: floor of $x$ , $\pi \left(x\right)$: number of primes not exceeding a number, $p,p_1,\mathrm{\dots }$: prime numbers and $x$: real number

Permalink:

http://dlmf.nist.gov/27.12.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §27.12 and Ch.27

►where the series terminates when the product of the first $r$ primes exceeds $x$. …

15: 5.8 Infinite Products

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5.8.5 $$\prod _{k=0}^{\mathrm{\infty }}\frac{(a_1+k)(a_2+k)\mathrm{\cdots }(a_m+k)}{(b_1+k)(b_2+k)\mathrm{\cdots }(b_m+k)}=\frac{\mathrm{\Gamma }\left(b_1\right)\mathrm{\Gamma }\left(b_2\right)\mathrm{\cdots }\mathrm{\Gamma }\left(b_m\right)}{\mathrm{\Gamma }\left(a_1\right)\mathrm{\Gamma }\left(a_2\right)\mathrm{\cdots }\mathrm{\Gamma }\left(a_m\right)},$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $m$: nonnegative integer, $k$: nonnegative integer, $a_k$: coefficient and $b_k$: coefficient

Referenced by:

§5.8

Permalink:

http://dlmf.nist.gov/5.8.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §5.8 and Ch.5

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16: 26.16 Multiset Permutations

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26.16.1 $${\left[\genfrac{}{}{0.0pt}{}{a_1+a_2+\mathrm{\cdots }+a_n}{a_1,a_2,\mathrm{\dots },a_n}\right]}_q=\prod _{k=1}^{n-1}{\left[\genfrac{}{}{0.0pt}{}{a_k+a_{k+1}+\mathrm{\cdots }+a_n}{a_k}\right]}_q,$$

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Symbols:

${\left[\genfrac{}{}{0.0pt}{}{n}{m}\right]}_q$: $q$-binomial coefficient (or Gaussian polynomial), ${\left[\genfrac{}{}{0.0pt}{}{a_1+a_2+\mathrm{\cdots }+a_n}{a_1,a_2,\mathrm{\dots },a_n}\right]}_q$: $q$-multinomial coefficient, $k$: nonnegative integer, $n$: nonnegative integer, $a_j$: numbers and $q$: parameter

Permalink:

http://dlmf.nist.gov/26.16.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §26.16 and Ch.26

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17: 31.15 Stieltjes Polynomials

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31.15.4 $$G({\zeta }_1,{\zeta }_2,\mathrm{\dots },{\zeta }_n)=\prod _{k=1}^n\prod _{\mathrm{\ell }=1}^N{({\zeta }_k-a_{\mathrm{\ell }})}^{{\gamma }_{\mathrm{\ell }}/2}\prod _{j=k+1}^n({\zeta }_k-{\zeta }_j).$$

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Symbols:

$\gamma$: real or complex parameter, $j$: nonnegative integer, $n$: nonnegative integer, $a$: complex parameter and $N+1$: number of singularities

Permalink:

http://dlmf.nist.gov/31.15.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §31.15(ii), §31.15 and Ch.31

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§31.15(iii) Products of Stieltjes Polynomials

… ►The products …with respect to the inner product …The normalized system of products (31.15.8) forms an orthonormal basis in the Hilbert space $L_{\rho }^2(Q)$. …

18: 22.9 Cyclic Identities

… ►These identities are cyclic in the sense that each of the indices $m,n$ in the first product of, for example, the form $s_{m,p}^{(4)}{s}_{n,p}^{(4)}$ are simultaneously permuted in the cyclic order: $m\to m+1\to m+2\to \mathrm{\cdots }p\to 1\to 2\to \mathrm{\cdots }m-1$; $n\to n+1\to n+2\to \mathrm{\cdots }p\to 1\to 2\to \mathrm{\cdots }n-1$. …

19: 27.5 Inversion Formulas

… ►If a Dirichlet series $F(s)$ generates $f(n)$, and $G(s)$ generates $g(n)$, then the product $F(s)G(s)$ generates ►

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

►called the Dirichlet product (or convolution) of $f$ and $g$. …For example, the equation $\zeta \left(s\right)\cdot (1/\zeta \left(s\right))=1$ is equivalent to the identity … ►

27.5.8 $$g(n)=\prod _{d|n}f(d)⟺f(n)=\prod _{d|n}{\left(g\left(\frac{n}{d}\right)\right)}^{\mu \left(d\right)}.$$

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Symbols:

$\mu \left(n\right)$: Möbius function, $d$: positive integer, $n$: positive integer, $f$: function and $g$: function

A&S Ref:

24.3.1 II.C

Permalink:

http://dlmf.nist.gov/27.5.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §27.5 and Ch.27

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20: 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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