Whipple 3F2 sum

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21: 36.6 Scaling Relations

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$\text{umbilics: }{𝐲}^{(\mathrm{U})}(k)=(x{k}^{2/3},y{k}^{2/3},z{k}^{1/3}).$

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$\text{umbilics: }{\beta }^{(\mathrm{U})}={\displaystyle \frac{1}{3}}.$

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$\text{umbilics: }{\gamma }_x^{(\mathrm{U})}=\frac{2}{3},$

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$\text{cuspoids: }{\gamma }_K={\displaystyle \sum _{m=1}^K}{\gamma }_{mK}={\displaystyle \frac{K(K+3)}{2(K+2)}},$

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$\text{umbilics: }{\gamma }^{(\mathrm{U})}={\displaystyle \sum _{m=1}^3}{\gamma }_m^{(\mathrm{U})}=\frac{5}{3}.$

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22: 26.10 Integer Partitions: Other Restrictions

… ► $p({𝒟}^{\prime }3,n)$ denotes the number of partitions of $n$ into parts with difference at least 3, except that multiples of 3 must differ by at least 6. …The set $\{2,3,4,\mathrm{\dots }\}$ is denoted by $T$. … ►where the inner sum is the sum of all positive odd divisors of $t$. … ►where the sum is over nonnegative integer values of $k$ for which $n-\frac{1}{2}(3k^2\pm k)\ge 0$. … ►where the sum is over nonnegative integer values of $k$ for which $n-(3k^2\pm k)\ge 0$. …

23: 25.16 Mathematical Applications

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§25.16(ii) Euler Sums

►Euler sums have the form … ► $H\left(s\right)$ has a simple pole with residue $\zeta \left(1-2r\right)$ ($=-B_{2r}/(2r)$) at each odd negative integer $s=1-2r$, $r=1,2,3,\mathrm{\dots }$. ► $H\left(s\right)$ is the special case $H(s,1)$ of the function …which satisfies the reciprocity law …

24: 9.12 Scorer Functions

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9.12.15 $$\mathrm{Gi}\left(z\right)=\frac{3^{-2/3}}{\pi }\sum _{k=0}^{\mathrm{\infty }}\cos \left(\frac{2k-1}{3}\pi \right)\mathrm{\Gamma }\left(\frac{k+1}{3}\right)\frac{{(3^{1/3}z)}^k}{k!},$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\mathrm{Gi}\left(z\right)$: Scorer function (inhomogeneous Airy function), $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $!$: factorial (as in $n!$), $k$: nonnegative integer and $z$: complex variable

Proof sketch:

Substitute into (9.12.11) by means of (9.12.17), (9.4.3), (9.2.5), (9.2.6), and use (5.5.3).

Referenced by:

(9.12.16)

Permalink:

http://dlmf.nist.gov/9.12.E15

Encodings:

pMML, png, TeX

See also:

Annotations for §9.12(vi), §9.12 and Ch.9

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9.12.16 $${\mathrm{Gi}}^{\prime }\left(z\right)=\frac{3^{-1/3}}{\pi }\sum _{k=0}^{\mathrm{\infty }}\cos \left(\frac{2k+1}{3}\pi \right)\mathrm{\Gamma }\left(\frac{k+2}{3}\right)\frac{{(3^{1/3}z)}^k}{k!}.$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\mathrm{Gi}\left(z\right)$: Scorer function (inhomogeneous Airy function), $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $!$: factorial (as in $n!$), $k$: nonnegative integer and $z$: complex variable

Proof sketch:

Differentiate (9.12.15).

Permalink:

http://dlmf.nist.gov/9.12.E16

Encodings:

pMML, png, TeX

See also:

Annotations for §9.12(vi), §9.12 and Ch.9

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9.12.17 $$\mathrm{Hi}\left(z\right)=\frac{3^{-2/3}}{\pi }\sum _{k=0}^{\mathrm{\infty }}\mathrm{\Gamma }\left(\frac{k+1}{3}\right)\frac{{(3^{1/3}z)}^k}{k!},$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\mathrm{Hi}\left(z\right)$: Scorer function (inhomogeneous Airy function), $\pi$: the ratio of the circumference of a circle to its diameter, $!$: factorial (as in $n!$), $k$: nonnegative integer and $z$: complex variable

Proof sketch:

Expand the integral in (9.12.20) in powers of $zt$ and integrate term-by-term by means of (5.2.1).

Referenced by:

(9.12.15), (9.12.18)

Permalink:

http://dlmf.nist.gov/9.12.E17

Encodings:

pMML, png, TeX

See also:

Annotations for §9.12(vi), §9.12 and Ch.9

… ►If $\zeta =\frac{2}{3}{z}^{3/2}$ or $\frac{2}{3}{x}^{3/2}$, and $K_{1/3}$ is the modified Bessel function (§10.25(ii)), then … ►where the integration contour separates the poles of $\mathrm{\Gamma }\left(\frac{1}{3}+\frac{1}{3}t\right)$ from those of $\mathrm{\Gamma }\left(-t\right)$. …

25: 10.20 Uniform Asymptotic Expansions for Large Order

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10.20.4 $$J_{\nu }\left(\nu z\right)\sim {\left(\frac{4\zeta }{1-z^2}\right)}^{\frac{1}{4}}\left(\frac{\mathrm{Ai}\left({\nu }^{\frac{2}{3}}\zeta \right)}{{\nu }^{\frac{1}{3}}}\sum _{k=0}^{\mathrm{\infty }}\frac{A_k(\zeta )}{{\nu }^{2k}}+\frac{{\mathrm{Ai}}^{\prime }\left({\nu }^{\frac{2}{3}}\zeta \right)}{{\nu }^{\frac{5}{3}}}\sum _{k=0}^{\mathrm{\infty }}\frac{B_k(\zeta )}{{\nu }^{2k}}\right),$$

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Symbols:

$\mathrm{Ai}\left(z\right)$: Airy function, $J_{\nu }\left(z\right)$: Bessel function of the first kind, $\sim$: Poincaré asymptotic expansion, $k$: nonnegative integer, $z$: complex variable, $\nu$: complex parameter, $\zeta (z)$: solution, $A_k(\zeta )$: coefficients and $B_k(\zeta )$: coefficients

A&S Ref:

9.3.35

Referenced by:

§10.20(ii), §10.20(iii)

Permalink:

http://dlmf.nist.gov/10.20.E4

Encodings:

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See also:

Annotations for §10.20(i), §10.20 and Ch.10

… ►In the following formulas for the coefficients $A_k(\zeta )$, $B_k(\zeta )$, $C_k(\zeta )$, and $D_k(\zeta )$, $u_k$, $v_k$ are the constants defined in §9.7(i), and $U_k(p)$, $V_k(p)$ are the polynomials in $p$ of degree $3k$ defined in §10.41(ii). … ►

10.20.10 $$A_k(\zeta )=\sum _{j=0}^{2k}{(\frac{3}{2})}^j{v}_j{\zeta }^{-3j/2}{U}_{2k-j}\left({(1-z^2)}^{-\frac{1}{2}}\right),$$

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Defines:

$A_k(\zeta )$: coefficients (locally)

Symbols:

$k$: nonnegative integer, $z$: complex variable, $\zeta (z)$: solution, $v_k$: constants and $U_k(p)$: polynomial coefficient

A&S Ref:

9.3.40

Referenced by:

§10.20(i), §10.41(v), §10.74(i)

Permalink:

http://dlmf.nist.gov/10.20.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §10.20(i), §10.20(i), §10.20 and Ch.10

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10.20.13 $$D_k(\zeta )=\sum _{j=0}^{2k}{(\frac{3}{2})}^j{u}_j{\zeta }^{-3j/2}{V}_{2k-j}\left({(1-z^2)}^{-\frac{1}{2}}\right).$$

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Defines:

$D_k(\zeta )$: coefficients (locally)

Symbols:

$k$: nonnegative integer, $z$: complex variable, $\zeta (z)$: solution, $u_k$: constants and $V_k(p)$: polynomial coefficient

A&S Ref:

9.3.46

Referenced by:

§10.20(i), §10.74(i)

Permalink:

http://dlmf.nist.gov/10.20.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §10.20(i), §10.20(i), §10.20 and Ch.10

… ►In formulas (10.20.10)–(10.20.13) replace ${\zeta }^{\frac{1}{2}}$, ${\zeta }^{-\frac{1}{2}}$, ${\zeta }^{-3j/2}$, and ${(1-z^2)}^{-\frac{1}{2}}$ by $-\mathrm{i}{(-\zeta )}^{\frac{1}{2}}$, $\mathrm{i}{(-\zeta )}^{-\frac{1}{2}}$, ${\mathrm{i}}^{3j}{(-\zeta )}^{-3j/2}$, and $\mathrm{i}{(z^2-1)}^{-\frac{1}{2}}$, respectively. …

26: 34.11 Higher-Order $3nj$ Symbols

§34.11 Higher-Order $3nj$ Symbols

…

27: 6.15 Sums

§6.15 Sums

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6.15.1 $$\sum _{n=1}^{\mathrm{\infty }}\mathrm{Ci}\left(\pi n\right)=\frac{1}{2}(\ln 2-\gamma ),$$

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Symbols:

$\gamma$: Euler’s constant, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{Ci}\left(z\right)$: cosine integral, $\ln z$: principal branch of logarithm function and $n$: nonnegative integer

Permalink:

http://dlmf.nist.gov/6.15.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §6.15 and Ch.6

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6.15.2 $$\sum _{n=1}^{\mathrm{\infty }}\frac{\mathrm{si}\left(\pi n\right)}{n}=\frac{1}{2}\pi (\ln \pi -1),$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\ln z$: principal branch of logarithm function, $\mathrm{si}\left(z\right)$: sine integral and $n$: nonnegative integer

Permalink:

http://dlmf.nist.gov/6.15.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §6.15 and Ch.6

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6.15.4 $$\sum _{n=1}^{\mathrm{\infty }}{(-1)}^n\frac{\mathrm{si}\left(2\pi n\right)}{n}=\pi (\frac{3}{2}\ln 2-1).$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\ln z$: principal branch of logarithm function, $\mathrm{si}\left(z\right)$: sine integral and $n$: nonnegative integer

Permalink:

http://dlmf.nist.gov/6.15.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §6.15 and Ch.6

►For further sums see Fempl (1960), Hansen (1975, pp. 423–424), Harris (2000), Prudnikov et al. (1986b, vol. 2, pp. 649–650), and Slavić (1974).

28: 5.19 Mathematical Applications

… ►As shown in Temme (1996b, §3.4), the results given in §5.7(ii) can be used to sum infinite series of rational functions. … ►

$S={\displaystyle \sum _{k=0}^{\mathrm{\infty }}}{a}_k,$

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$a_k={\displaystyle \frac{k}{(3k+2)(2k+1)(k+1)}}.$

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5.19.3 $$S=\psi \left(\frac{1}{2}\right)-2\psi \left(\frac{2}{3}\right)-\gamma =3\ln 3-2\ln 2-\frac{1}{3}\pi \sqrt{3}.$$

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Symbols:

$\gamma$: Euler’s constant, $\pi$: the ratio of the circumference of a circle to its diameter, $\psi \left(z\right)$: psi (or digamma) function and $\ln z$: principal branch of logarithm function

Permalink:

http://dlmf.nist.gov/5.19.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §5.19(i), §5.19(i), §5.19 and Ch.5

… ►By translating the contour parallel to itself and summing the residues of the integrand, asymptotic expansions of $f(z)$ for large $|z|$, or small $|z|$, can be obtained complete with an integral representation of the error term. …

29: 23.2 Definitions and Periodic Properties

… ►If ${\omega }_1$ and ${\omega }_3$ are nonzero real or complex numbers such that $\mathrm{\Im }\left({\omega }_3/{\omega }_1\right)>0$, then the set of points $2m{\omega }_1+2n{\omega }_3$, with $m,n\in \mathbb{Z}$, constitutes a lattice $𝕃$ with $2{\omega }_1$ and $2{\omega }_3$ lattice generators. … ►then $2{\omega }_2$, $2{\omega }_3$ are generators, as are $2{\omega }_2$, $2{\omega }_1$. …where $a,b,c,d$ are integers, then $2{\chi }_1$, $2{\chi }_3$ are generators of $𝕃$ iff … ►For $j=1,2,3$, the function $\sigma \left(z\right)$ satisfies …More generally, if $j=1,2,3$, $k=1,2,3$, $j\ne k$, and $m,n\in \mathbb{Z}$, then …

30: 23.20 Mathematical Applications

… ►There is a unique point $z_0\in [{\omega }_1,{\omega }_1+{\omega }_3]\cup [{\omega }_1+{\omega }_3,{\omega }_3]$ such that $\mathrm{\wp }\left(z_0\right)=0$. … ►The interior of the rectangle with vertices $0$, ${\omega }_1$, $2{\omega }_3$, $2{\omega }_3-{\omega }_1$ is mapped two-to-one onto the lower half-plane. The interior of the rectangle with vertices $0$, ${\omega }_1$, $\frac{1}{2}{\omega }_1+{\omega }_3$, $\frac{1}{2}{\omega }_1-{\omega }_3$ is mapped one-to-one onto the lower half-plane with a cut from $e_3$ to $\mathrm{\wp }\left(\frac{1}{2}{\omega }_1+{\omega }_3\right)(=\mathrm{\wp }\left(\frac{1}{2}{\omega }_1-{\omega }_3\right))$. The cut is the image of the edge from $\frac{1}{2}{\omega }_1+{\omega }_3$ to $\frac{1}{2}{\omega }_1-{\omega }_3$ and is not a line segment. … ►It follows from the addition formula (23.10.1) that the points $P_j=P(z_j)$, $j=1,2,3$, have zero sum iff $z_1+z_2+z_3\in 𝕃$, so that addition of points on the curve $C$ corresponds to addition of parameters $z_j$ on the torus $ℂ/𝕃$; see McKean and Moll (1999, §§2.11, 2.14). …