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21: 17.14 Constant Term Identities

§17.14 Constant Term Identities

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Rogers–Ramanujan Constant Term Identities

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22: 27.13 Functions

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27.13.5 $${(\vartheta \left(x\right))}^2=1+\sum _{n=1}^{\mathrm{\infty }}{r}_2\left(n\right)x^n.$$

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Symbols:

$\vartheta \left(x\right)={\theta }_3(0,x)$: alternative notation, $r_k\left(n\right)$: number of squares, $n$: positive integer and $x$: real number

Referenced by:

§27.13(iv)

Permalink:

http://dlmf.nist.gov/27.13.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §27.13(iv), §27.13 and Ch.27

►One of Jacobi’s identities implies that … ►Also, Milne (1996, 2002) announce new infinite families of explicit formulas extending Jacobi’s identities. For more than 8 squares, Milne’s identities are not the same as those obtained earlier by Mordell and others.

23: 27.14 Unrestricted Partitions

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§27.14(v) Divisibility Properties

►Ramanujan (1921) gives identities that imply divisibility properties of the partition function. For example, the Ramanujan identity …implies $p\left(5n+4\right)\equiv 0(mod5)$. …For example, $p\left(\mathrm{1575\hspace{0.33em}25693}n+\mathrm{1\hspace{0.33em}11247}\right)\equiv 0(mod13)$. …

24: 26.3 Lattice Paths: Binomial Coefficients

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§26.3(iv) Identities

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25: 27.19 Methods of Computation: Factorization

… ►Type II probabilistic algorithms for factoring $n$ rely on finding a pseudo-random pair of integers $(x,y)$ that satisfy $x^2\equiv y^2(modn)$. …

26: 20.7 Identities

§20.7 Identities

… ►Also, in further development along the lines of the notations of Neville (§20.1) and of Glaisher (§22.2), the identities (20.7.6)–(20.7.9) have been recast in a more symmetric manner with respect to suffices $2,3,4$. … ►

§20.7(v) Watson’s Identities

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20.7.15 $$A\equiv A(\tau )=1/{\theta }_4\left(0\right|2\tau ),$$

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Defines:

$A$ (locally)

Symbols:

${\theta }_j\left(z\right|\tau )$: theta function, $\equiv$: equals by definition and $\tau$: lattice parameter

Permalink:

http://dlmf.nist.gov/20.7.E15

Encodings:

pMML, png, TeX

See also:

Annotations for §20.7(vi), §20.7 and Ch.20

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

27: 24.19 Methods of Computation

… ►Another method is based on the identities … ►For number-theoretic applications it is important to compute $B_{2n}(modp)$ for $2n\le p-3$; in particular to find the irregular pairs $(2n,p)$ for which $B_{2n}\equiv 0(modp)$. …

28: 25.9 Asymptotic Approximations

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25.9.2 $$\chi (s)\equiv {\pi }^{s-\frac{1}{2}}\mathrm{\Gamma }\left(\frac{1}{2}-\frac{1}{2}s\right)/\mathrm{\Gamma }\left(\frac{1}{2}s\right).$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $\equiv$: equals by definition and $s$: complex variable

Keywords:

definition

Source:

Titchmarsh (1986b, p. 78); with (5.5.5), (5.5.3)

Referenced by:

(25.10.2)

Permalink:

http://dlmf.nist.gov/25.9.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §25.9 and Ch.25

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29: 25.13 Periodic Zeta Function

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25.13.1 $$F(x,s)\equiv \sum _{n=1}^{\mathrm{\infty }}\frac{{\mathrm{e}}^{2\pi \mathrm{i}nx}}{n^s},$$

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Defines:

$F(x,s)$: periodic zeta function

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\equiv$: equals by definition, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $n$: nonnegative integer, $x$: real variable and $s$: complex variable

Keywords:

definition, infinite series, series representation

Source:

Apostol (1976, (9), p. 257)

Permalink:

http://dlmf.nist.gov/25.13.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §25.13 and Ch.25

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30: 26.6 Other Lattice Path Numbers

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§26.6(iv) Identities

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26.6.12 $$C\left(n\right)=\sum _{k=1}^{n}N(n,k),$$

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Symbols:

$C\left(n\right)$: Catalan number, $k$: nonnegative integer, $n$: nonnegative integer and $N(n,k)$: Narayana number

Permalink:

http://dlmf.nist.gov/26.6.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §26.6(iv), §26.6 and Ch.26

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