quintuple product identity

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21: 16.12 Products

§16.12 Products

… ►For further identities see Goursat (1883) and Erdélyi et al. (1953a, §4.3).

22: 26.10 Integer Partitions: Other Restrictions

… ►The set $\{n\ge 1|n\equiv \pm j(modk)\}$ is denoted by $A_{j,k}$. … ►

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:

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See also:

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

… ►

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:

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See also:

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

… ►

§26.10(iv) Identities

►Equations (26.10.13) and (26.10.14) are the Rogers–Ramanujan identities. …

23: 17.14 Constant Term Identities

§17.14 Constant Term Identities

… ►

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:

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See also:

Annotations for §17.14, §17.14 and Ch.17

►

Rogers–Ramanujan Constant Term Identities

…

24: 20.7 Identities

§20.7 Identities

… ►

§20.7(iv) Reduction Formulas for Products

… ►

§20.7(v) Watson’s Identities

… ►This reference also gives the eleven additional identities for the permutations of the four theta functions. … ►

§20.7(ix) Addendum to 20.7(iv) Reduction Formulas for Products

…

25: 26.13 Permutations: Cycle Notation

… ►See §26.8 for generating functions, recurrence relations, identities, and asymptotic approximations. … ►Every permutation is a product of transpositions. … ►Every transposition is the product of adjacent transpositions. If , then $\left(j,k\right)$ is a product of $2k-2j-1$ adjacent transpositions: …Every permutation is a product of adjacent transpositions. …

26: 1.3 Determinants, Linear Operators, and Spectral Expansions

… ►If two rows (columns) of a determinant are identical, then the determinant is zero. … ►The determinant of an upper or lower triangular, or diagonal, square matrix $𝐀$ is the product of the diagonal elements $det(𝐀)={\prod }_{i=1}^n{a}_{ii}$. … ►

1.3.14

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Symbols:

$det$: determinant, $j$: integer, $k$: integer and $n$: nonnegative integer

Permalink:

http://dlmf.nist.gov/1.3.E14

Encodings:

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

… ►The adjoint of a matrix $𝐀$ is the matrix ${𝐀}^{\ast }$ such that $⟨𝐀𝐚,𝐛⟩=⟨𝐚,{𝐀}^{\ast }𝐛⟩$ for all $𝐚,𝐛\in {𝐄}_n$. … ►

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:

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See also:

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

…

27: 24.14 Sums

… ►In the following two identities, valid for $n\ge 2$, the sums are taken over all nonnegative integers $j,k,\mathrm{\ell }$ with $j+k+\mathrm{\ell }=n$. … ►In the next identity, valid for $n\ge 4$, the sum is taken over all positive integers $j,k,\mathrm{\ell },m$ with $j+k+\mathrm{\ell }+m=n$. … ►These identities can be regarded as higher-order recurrences. … ►

24.14.11 $$det[B_{r+s}]={(-1)}^{n(n+1)/2}{\left(\prod _{k=1}^{n}k!\right)}^6/\left(\prod _{k=1}^{2n+1}k!\right),$$

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Symbols:

$B_n$: Bernoulli numbers, $det$: determinant, $!$: factorial (as in $n!$), $k$: integer, $n$: integer, $r$: row and $s$: column

Referenced by:

§24.14(ii)

Permalink:

http://dlmf.nist.gov/24.14.E11

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Annotations for §24.14(ii), §24.14 and Ch.24

►

24.14.12 $$det[E_{r+s}]={(-1)}^{n(n+1)/2}{\left(\prod _{k=1}^{n}k!\right)}^2.$$

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Symbols:

$E_n$: Euler numbers, $det$: determinant, $!$: factorial (as in $n!$), $k$: integer, $n$: integer, $r$: row and $s$: column

Referenced by:

§24.14(ii)

Permalink:

http://dlmf.nist.gov/24.14.E12

Encodings:

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See also:

Annotations for §24.14(ii), §24.14 and Ch.24

…

28: 31.17 Physical Applications

… ►The problem of adding three quantum spins $𝐬$, $𝐭$, and $𝐮$ can be solved by the method of separation of variables, and the solution is given in terms of a product of two Heun functions. … ►

$${𝐉}^2\mathrm{\Psi }(𝐱)\equiv {(𝐬+𝐭+𝐮)}^2\mathrm{\Psi }(𝐱)=j(j+1)\mathrm{\Psi }(𝐱),$$

►

$${𝐻}_s\mathrm{\Psi }(𝐱)\equiv (-2𝐬\cdot 𝐭-(2/a)𝐬\cdot 𝐮)\mathrm{\Psi }(𝐱)=h_s\mathrm{\Psi }(𝐱),$$

…

29: 4.21 Identities

§4.21 Identities

… ►

§4.21(ii) Squares and Products

… ►

4.21.35 $$\sin \left(nz\right)=2^{n-1}\prod _{k=0}^{n-1}\sin \left(z+\frac{k\pi }{n}\right),$$ $n=1,2,3,\mathrm{\dots }$.

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\sin z$: sine function, $k$: integer, $n$: integer and $z$: complex variable

Referenced by:

§23.10(iii), §4.21(iii)

Permalink:

http://dlmf.nist.gov/4.21.E35

Encodings:

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See also:

Annotations for §4.21(iii), §4.21(iii), §4.21 and Ch.4

…

30: 27.9 Quadratic Characters

… ►If $p$ does not divide $n$, then $(n|p)$ has the value $1$ when the quadratic congruence $x^2\equiv n(modp)$ has a solution, and the value $-1$ when this congruence has no solution. … ►If an odd integer $P$ has prime factorization $P={\prod }_{r=1}^{\nu \left(n\right)}{p}_r^{a_r}$, then the Jacobi symbol $(n|P)$ is defined by $(n|P)={\prod }_{r=1}^{\nu \left(n\right)}{(n|p_r)}^{a_r}$, with $(n|1)=1$. …