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1: 20 Theta Functions

Chapter 20 Theta Functions

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2: 26.21 Tables

… ►Abramowitz and Stegun (1964, Chapter 24) tabulates binomial coefficients $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$ for $m$ up to 50 and $n$ up to 25; extends Table 26.4.1 to $n=10$; tabulates Stirling numbers of the first and second kinds, $s(n,k)$ and $S(n,k)$, for $n$ up to 25 and $k$ up to $n$; tabulates partitions $p\left(n\right)$ and partitions into distinct parts $p(𝒟,n)$ for $n$ up to 500. ►Andrews (1976) contains tables of the number of unrestricted partitions, partitions into odd parts, partitions into parts $\cancel{\equiv }\pm 2(mod5)$, partitions into parts $\cancel{\equiv }\pm 1(mod5)$, and unrestricted plane partitions up to 100. …

3: 26.10 Integer Partitions: Other Restrictions

… ► $p(𝒟,n)$ denotes the number of partitions of $n$ into distinct parts. $p_m(𝒟,n)$ denotes the number of partitions of $n$ into at most $m$ distinct parts. …$p(𝒪,n)$ denotes the number of partitions of $n$ into odd parts. … ►where the last right-hand side is the sum over $m\ge 0$ of the generating functions for partitions into distinct parts with largest part equal to $m$. … ►where the inner sum is the sum of all positive odd divisors of $t$. …

4: 1.12 Continued Fractions

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§1.12(iv) Contraction and Extension

… ►The even part of $C$ exists iff $b_{2k}\ne 0$, $k=1,2,\mathrm{\dots }$, and up to equivalence is given by …If $C_n^{\prime }=C_{2n+1}$, $n=0,1,2,\mathrm{\dots }$, then $C^{\prime }$ is called the odd part of $C$. The odd part of $C$ exists iff $b_{2k+1}\ne 0$, $k=0,1,2,\mathrm{\dots }$, and up to equivalence is given by … ►and the even and odd parts of the continued fraction converge to finite values. …

5: 26.13 Permutations: Cycle Notation

… ►For the example (26.13.2), this decomposition is given by $\left(1,3,2,5,7\right)\left(6,8\right)=\left(1,3\right)\left(2,3\right)\left(2,5\right)\left(5,7\right)\left(6,8\right).$ ►A permutation is even or odd according to the parity of the number of transpositions. The sign of a permutation is $+$ if the permutation is even, $-$ if it is odd. …

6: 8 Incomplete Gamma and Related
Functions

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7: 28 Mathieu Functions and Hill’s Equation

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8: 27.9 Quadratic Characters

… ►For an odd prime $p$, the Legendre symbol $(n|p)$ is defined as follows. … ►If $p,q$ are distinct odd primes, then the quadratic reciprocity law states that ►

27.9.3 $$(p|q)(q|p)={(-1)}^{(p-1)(q-1)/4}.$$

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Symbols:

$(n|p)$: Legendre symbol, $p$: odd prime and $q$: odd prime

Referenced by:

§27.9

Permalink:

http://dlmf.nist.gov/27.9.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §27.9 and Ch.27

►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 $(n|1)=1$. …Both (27.9.1) and (27.9.2) are valid with $p$ replaced by $P$; the reciprocity law (27.9.3) holds if $p,q$ are replaced by any two relatively prime odd integers $P,Q$.

9: 23 Weierstrass Elliptic and Modular
Functions

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10: 6.19 Tables

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Zhang and Jin (1996, pp. 652, 689) includes $\mathrm{Si}\left(x\right)$, $\mathrm{Ci}\left(x\right)$, $x=0(.5)20(2)30$, 8D; $\mathrm{Ei}\left(x\right)$, $E_1\left(x\right)$, $x=[0,100]$, 8S.

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Abramowitz and Stegun (1964, Chapter 5) includes the real and imaginary parts of $z{\mathrm{e}}^z{E}_1\left(z\right)$, $x=-19(1)20$, $y=0(1)20$, 6D; ${\mathrm{e}}^z{E}_1\left(z\right)$, $x=-4(.5)-2$, $y=0(.2)1$, 6D; $E_1\left(z\right)+\ln z$, $x=-2(.5)2.5$, $y=0(.2)1$, 6D.

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Zhang and Jin (1996, pp. 690–692) includes the real and imaginary parts of $E_1\left(z\right)$, $\pm x=0.5,1,3,5,10,15,20,50,100$, $y=0(.5)1(1)5(5)30,50,100$, 8S.