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1: 20.11 Generalizations and Analogs

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§20.11(v) Permutation Symmetry

… ►For $m=1,2,3,4$, $n=1,2,3,4$, and $m\ne n$, define twelve combined theta functions ${\phi }_{m,n}(z,q)$ by ►

20.11.6 $${\phi }_{m,1}(z,q)=\frac{{\theta }_1^{\prime }(0,q){\theta }_m(z,q)}{{\theta }_m(0,q){\theta }_1(z,q)},$$ $m=2,3,4$,

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Symbols:

${\theta }_j(z,q)$: theta function, ${\phi }_{n,m}(z,q)$: combined theta function, $m$: integer, $z$: complex and $q$: nome

Permalink:

http://dlmf.nist.gov/20.11.E6

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Annotations for §20.11(v), §20.11 and Ch.20

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20.11.9 $${\phi }_{m,n}(z,q)={\phi }_{m,1}(z,q){\phi }_{1,n}(z,q)=\frac{1}{{\phi }_{n,m}(z,q)}=\frac{{\phi }_{m,1}(z,q)}{{\phi }_{n,1}(z,q)}=\frac{{\phi }_{1,n}(z,q)}{{\phi }_{1,m}(z,q)}.$$

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Symbols:

${\phi }_{n,m}(z,q)$: combined theta function, $m$: integer, $n$: integer, $z$: complex and $q$: nome

Permalink:

http://dlmf.nist.gov/20.11.E9

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Annotations for §20.11(v), §20.11 and Ch.20

►The importance of these combined theta functions is that sets of twelve equations for the theta functions often can be replaced by corresponding sets of three equations of the combined theta functions, plus permutation symmetry. …

2: 27.22 Software

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Maple. isprime combines a strong pseudoprime test and a Lucas pseudoprime test. ifactor uses cfrac (§27.19) after exhausting trial division. Brent–Pollard rho, Square Forms Factorization, and ecm are available also; see §27.19.

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Mathematica. PrimeQ combines strong pseudoprime tests for the bases 2 and 3 and a Lucas pseudoprime test. No known composite numbers pass these three tests, and Bleichenbacher (1996) has shown that this combination of tests proves primality for integers below ${10}^{16}$. Provable PrimeQ uses the Atkin–Goldwasser–Kilian–Morain Elliptic Curve Method to prove primality. FactorInteger tries Brent–Pollard rho, Pollard $p-1$, and then cfrac after trial division. See §27.19. ecm is available also, and the Multiple Polynomial Quadratic sieve is expected in a future release.

For additional Mathematica routines for factorization and primality testing, including several different pseudoprime tests, see Bressoud and Wagon (2000).

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3: 21.8 Abelian Functions

… ►For every Abelian function, there is a positive integer $n$, such that the Abelian function can be expressed as a ratio of linear combinations of products with $n$ factors of Riemann theta functions with characteristics that share a common period lattice. …

4: 35.11 Tables

… ►Each table expresses the zonal polynomials as linear combinations of monomial symmetric functions.

5: 4.44 Other Applications

… ►The Einstein functions and Planck’s radiation function are elementary combinations of exponentials, or exponentials and logarithms. …

6: 34.10 Zeros

… ►However, the $\mathit{3}j$ and $\mathit{6}j$ symbols may vanish for certain combinations of the angular momenta and projective quantum numbers even when the triangle conditions are fulfilled. …

7: 33.18 Limiting Forms for Large $\mathrm{\ell }$

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8: 10.53 Power Series

… ►For ${𝗁}_n^{(1)}\left(z\right)$ and ${𝗁}_n^{(2)}\left(z\right)$ combine (10.47.10), (10.53.1), and (10.53.2). For ${𝗄}_n\left(z\right)$ combine (10.47.11), (10.53.3), and (10.53.4).

9: 33.21 Asymptotic Approximations for Large $|r|$

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When $r\to \pm \mathrm{\infty }$ with $ϵ>0$, Equations (33.16.4)–(33.16.7) are combined with (33.10.1).

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When $r\to \pm \mathrm{\infty }$ with , Equations (33.16.10)–(33.16.13) are combined with

33.21.1

${\zeta }_{\mathrm{\ell }}(\nu ,r)\sim {\mathrm{e}}^{-r/\nu }{(2r/\nu )}^{\nu },$

${\xi }_{\mathrm{\ell }}(\nu ,r)\sim {\mathrm{e}}^{r/\nu }{(2r/\nu )}^{-\nu }$ , $r\to \mathrm{\infty }$,

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Symbols:

$\sim$: asymptotic equality, $\mathrm{e}$: base of natural logarithm, $\mathrm{\ell }$: nonnegative integer, $r$: real variable, $\nu$: parameter, ${\zeta }_{\mathrm{\ell }}(\nu ,r)$: function and ${\xi }_{\mathrm{\ell }}(\nu ,r)$: function

Permalink:

http://dlmf.nist.gov/33.21.E1

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Annotations for §33.21(i), §33.21 and Ch.33

33.21.2

${\zeta }_{\mathrm{\ell }}(-\nu ,r)\sim {\mathrm{e}}^{r/\nu }{(-2r/\nu )}^{-\nu },$

${\xi }_{\mathrm{\ell }}(-\nu ,r)\sim {\mathrm{e}}^{-r/\nu }{(-2r/\nu )}^{\nu },$ $r\to -\mathrm{\infty }$.

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Symbols:

$\sim$: asymptotic equality, $\mathrm{e}$: base of natural logarithm, $\mathrm{\ell }$: nonnegative integer, $r$: real variable, $\nu$: parameter, ${\zeta }_{\mathrm{\ell }}(\nu ,r)$: function and ${\xi }_{\mathrm{\ell }}(\nu ,r)$: function

Permalink:

http://dlmf.nist.gov/33.21.E2

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Annotations for §33.21(i), §33.21 and Ch.33

Corresponding approximations for $s(ϵ,\mathrm{\ell };r)$ and $c(ϵ,\mathrm{\ell };r)$ as $r\to \mathrm{\infty }$ can be obtained via (33.16.17), and as $r\to -\mathrm{\infty }$ via (33.16.18).

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When $r\to \pm \mathrm{\infty }$ with $ϵ=0$, combine (33.20.1), (33.20.2) with §§10.7(ii), 10.30(ii).

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10: 33.23 Methods of Computation

… ►Combined with the Wronskians (33.2.12), the values of $F_{\mathrm{\ell }}$, $G_{\mathrm{\ell }}$, and their derivatives can be extracted. … ►Thompson and Barnett (1985, 1986) and Thompson (2004) use combinations of series, continued fractions, and Padé-accelerated asymptotic expansions (§3.11(iv)) for the analytic continuations of Coulomb functions. ►Noble (2004) obtains double-precision accuracy for $W_{-\eta ,\mu }\left(2\rho \right)$ for a wide range of parameters using a combination of recurrence techniques, power-series expansions, and numerical quadrature; compare (33.2.7). …