Euler sums (first, second, third)

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11: 16.16 Transformations of Variables

… ►

16.16.2

{F_{2}}\left(\alpha;\beta,\beta^{\prime};\gamma,\beta^{\prime};x,y\right)=(1-y% )^{-\alpha}{{}_{2}F_{1}}\left({\alpha,\beta\atop\gamma};\frac{x}{1-y}\right),

ⓘ

Symbols:: ${F_{2}}\left(\NVar{\alpha};\NVar{\beta},\NVar{\beta^{\prime}};\NVar{\gamma},% \NVar{\gamma^{\prime}};\NVar{x},\NVar{y}\right)$ : second Appell function and ${{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ : $=F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ notation for Gauss’ hypergeometric function
Permalink:: http://dlmf.nist.gov/16.16.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §16.16(i), §16.16 and Ch.16

►

16.16.3

{F_{2}}\left(\alpha;\beta,\beta^{\prime};\gamma,\alpha;x,y\right)=(1-y)^{-% \beta^{\prime}}{F_{1}}\left(\beta;\alpha-\beta^{\prime},\beta^{\prime};\gamma;% x,\frac{x}{1-y}\right),

ⓘ

Symbols:: ${F_{1}}\left(\NVar{\alpha};\NVar{\beta},\NVar{\beta^{\prime}};\NVar{\gamma};% \NVar{x},\NVar{y}\right)$ : first Appell function and ${F_{2}}\left(\NVar{\alpha};\NVar{\beta},\NVar{\beta^{\prime}};\NVar{\gamma},% \NVar{\gamma^{\prime}};\NVar{x},\NVar{y}\right)$ : second Appell function
Permalink:: http://dlmf.nist.gov/16.16.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §16.16(i), §16.16 and Ch.16

►

16.16.4

{F_{3}}\left(\alpha,\gamma-\alpha;\beta,\beta^{\prime};\gamma;x,y\right)=(1-y)% ^{-\beta^{\prime}}{F_{1}}\left(\alpha;\beta,\beta^{\prime};\gamma;x,\frac{y}{y% -1}\right),

ⓘ

Symbols:: ${F_{1}}\left(\NVar{\alpha};\NVar{\beta},\NVar{\beta^{\prime}};\NVar{\gamma};% \NVar{x},\NVar{y}\right)$ : first Appell function and ${F_{3}}\left(\NVar{\alpha},\NVar{\alpha^{\prime}};\NVar{\beta},\NVar{\beta^{% \prime}};\NVar{\gamma};\NVar{x},\NVar{y}\right)$ : third Appell function
Permalink:: http://dlmf.nist.gov/16.16.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §16.16(i), §16.16 and Ch.16

►

16.16.5

{F_{3}}\left(\alpha,\gamma-\alpha;\beta,\gamma-\beta;\gamma;x,y\right)=(1-y)^{% \alpha+\beta-\gamma}{{}_{2}F_{1}}\left({\alpha,\beta\atop\gamma};x+y-xy\right),

ⓘ

Symbols:: ${F_{3}}\left(\NVar{\alpha},\NVar{\alpha^{\prime}};\NVar{\beta},\NVar{\beta^{% \prime}};\NVar{\gamma};\NVar{x},\NVar{y}\right)$ : third Appell function and ${{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ : $=F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ notation for Gauss’ hypergeometric function
Permalink:: http://dlmf.nist.gov/16.16.E5
Encodings:: pMML, png, TeX
See also:: Annotations for §16.16(i), §16.16 and Ch.16

… ►

16.16.9

{F_{2}}\left(\alpha;\beta,\beta^{\prime};\gamma,\gamma^{\prime};x,y\right)=(1-% x)^{-\alpha}{F_{2}}\left(\alpha;\gamma-\beta,\beta^{\prime};\gamma,\gamma^{% \prime};\frac{x}{x-1},\frac{y}{1-x}\right),

ⓘ

Symbols:: ${F_{2}}\left(\NVar{\alpha};\NVar{\beta},\NVar{\beta^{\prime}};\NVar{\gamma},% \NVar{\gamma^{\prime}};\NVar{x},\NVar{y}\right)$ : second Appell function
Permalink:: http://dlmf.nist.gov/16.16.E9
Encodings:: pMML, png, TeX
See also:: Annotations for §16.16(ii), §16.16 and Ch.16

…

12: 19.29 Reduction of General Elliptic Integrals

… ►The only cases of

I(\mathbf{m})

that are integrals of the first kind are the two (

h=3

or 4) with

\mathbf{m}=\boldsymbol{{0}}

. The only cases that are integrals of the third kind are those in which at least one

m_{j}

with

j>h

is a negative integer and those in which

h=4

and

\sum_{j=1}^{n}m_{j}

is a positive integer. All other cases are integrals of the second kind. … ►The first choice gives a formula that includes the 18+9+18 = 45 formulas in Gradshteyn and Ryzhik (2000, 3.133, 3.156, 3.158), and the second choice includes the 8+8+8+12 = 36 formulas in Gradshteyn and Ryzhik (2000, 3.151, 3.149, 3.137, 3.157) (after setting

x^{2}=t

in some cases). … ►The first formula replaces (19.14.4)–(19.14.10). …

13: 10.40 Asymptotic Expansions for Large Argument

… ►Corresponding expansions for

I_{\nu}\left(z\right)

K_{\nu}\left(z\right)

I_{\nu}'\left(z\right)

, and

K_{\nu}'\left(z\right)

for other ranges of

\operatorname{ph}z

are obtainable by combining (10.34.3), (10.34.4), (10.34.6), and their differentiated forms, with (10.40.2) and (10.40.4). … ►

10.40.6

I_{\nu}\left(z\right)K_{\nu}\left(z\right)\sim\frac{1}{2z}\left(1-\frac{1}{2}% \frac{\mu-1}{(2z)^{2}}+\frac{1\cdot 3}{2\cdot 4}\frac{(\mu-1)(\mu-9)}{(2z)^{4}% }-\dotsb\right),

ⓘ

Symbols:: $\sim$ : Poincaré asymptotic expansion, $I_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Bessel function of the first kind, $K_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Bessel function of the second kind, $z$ : complex variable and $\nu$ : complex parameter
A&S Ref:: 9.7.5
Referenced by:: §10.40(i), §10.40(i)
Permalink:: http://dlmf.nist.gov/10.40.E6
Encodings:: pMML, png, TeX
See also:: Annotations for §10.40(i), §10.40(i), §10.40 and Ch.10

… ►In the expansion (10.40.2) assume that

z>0

and the sum is truncated when

k=\ell-1

. Then the remainder term does not exceed the first neglected term in absolute value and has the same sign provided that

\ell\geq\max(|\nu|-\tfrac{1}{2},1)

. … ►where

\chi(\ell)=\pi^{\frac{1}{2}}\Gamma\left(\tfrac{1}{2}\ell+1\right)/\Gamma\left(% \tfrac{1}{2}\ell+\tfrac{1}{2}\right)

; see §9.7(i). …

14: Errata

… ►

Equations (18.5.1), (18.5.2), (18.5.3), (18.5.4)

18.5.1

T_{n}\left(x\right)=\cos\left(n\theta\right)=\tfrac{1}{2}\left(z^{n}+z^{-n}\right)

18.5.2

U_{n}\left(x\right)=\frac{\sin\left((n+1)\theta\right)}{\sin\theta}=\dfrac{z^{% n+1}-z^{-n-1}}{z-z^{-1}}

18.5.3

V_{n}\left(x\right)=\frac{\cos\left((n+\tfrac{1}{2})\theta\right)}{\cos\left(% \tfrac{1}{2}\theta\right)}=\dfrac{z^{n+1}+z^{-n}}{z+1}

18.5.4

W_{n}\left(x\right)=\frac{\sin\left((n+\tfrac{1}{2})\theta\right)}{\sin\left(% \tfrac{1}{2}\theta\right)}=\dfrac{z^{n+1}-z^{-n}}{z-1}

These equations were updated to include the definition in terms of $z$ where $x=\cos\theta=\tfrac{1}{2}(z+z^{-1})$ .

… ►

Subsection 19.2(ii) and Equation (19.2.9)

The material surrounding (19.2.8), (19.2.9) has been updated so that the complementary complete elliptic integrals of the first and second kind are defined with consistent multivalued properties and correct analytic continuation. In particular, (19.2.9) has been corrected to read

19.2.9

{K^{\prime}}\left(k\right)=\begin{cases}K\left(k^{\prime}\right),&|% \operatorname{ph}k|\leq\tfrac{1}{2}\pi,\\ K\left(k^{\prime}\right)\mp 2\mathrm{i}K\left(-k\right),&\tfrac{1}{2}\pi<\pm% \operatorname{ph}k<\pi,\end{cases}

{E^{\prime}}\left(k\right)=\begin{cases}E\left(k^{\prime}\right),&|% \operatorname{ph}k|\leq\tfrac{1}{2}\pi,\\ E\left(k^{\prime}\right)\mp 2\mathrm{i}(K\left(-k\right)-E\left(-k\right)),&% \tfrac{1}{2}\pi<\pm\operatorname{ph}k<\pi\end{cases}

… ►

Additions

Equations: (5.9.2_5), (5.9.10_1), (5.9.10_2), (5.9.11_1), (5.9.11_2), the definition of the scaled gamma function $\Gamma^{*}\left(z\right)$ was inserted after the first equals sign in (5.11.3), post equality added in (7.17.2) which gives “ $=\sum_{m=0}^{\infty}a_{m}t^{2m+1}$ ”, (7.17.2_5), (31.11.3_1), (31.11.3_2) with some explanatory text.

… ►

Equation (8.12.18)

8.12.18

\rselection{Q\left(a,z\right)\\ P\left(a,z\right)}\sim\frac{z^{a-\frac{1}{2}}{\mathrm{e}}^{-z}}{\Gamma\left(a% \right)}{\left(d(\pm\chi)\sum_{k=0}^{\infty}\frac{A_{k}(\chi)}{z^{k/2}}\mp\sum% _{k=1}^{\infty}\frac{B_{k}(\chi)}{z^{k/2}}\right)}

The original $\pm$ in front of the second summation was replaced by $\mp$ to correct an error in Paris (2002b); for details see https://arxiv.org/abs/1611.00548.

Reported 2017-01-28 by Richard Paris.

… ►

Table 3.5.19

The correct headings for the second and third columns of this table are $J_{0}\left(t\right)$ and $g(t)$ , respectively. Previously these columns were mislabeled as $g(t)$ and $J_{0}\left(t\right)$ .

$t$	$J_{0}(t)$	$g(t)$
0.0	1.00000 00000	1.00000 00000
0.5	0.93846 98072	0.93846 98072
1.0	0.76519 76866	0.76519 76865
2.0	0.22389 07791	0.22389 10326
5.0	$-$ 0.17759 67713	$-$ 0.17902 54097
10.0	$-$ 0.24593 57645	$-$ 0.07540 53543

Reported 2014-01-31 by Masataka Urago.

…

15: 19.16 Definitions

… ►

19.16.2

R_{J}\left(x,y,z,p\right)=\frac{3}{2}\int_{0}^{\infty}\frac{\,\mathrm{d}t}{s(t% )(t+p)},

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Defines:: $R_{J}\left(\NVar{x},\NVar{y},\NVar{z},\NVar{p}\right)$ : symmetric elliptic integral of third kind
Symbols:: $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral and $s(t)$ : scaling factor
Referenced by:: §19.16(i), §19.19, §19.20(iii), §19.28
Permalink:: http://dlmf.nist.gov/19.16.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §19.16(i), §19.16 and Ch.19

… ►where

\mathrm{B}\left(x,y\right)

is the beta function (§5.12) and … ►The only cases that are integrals of the first kind are the four in which each of

a

and

a^{\prime}

is either

\frac{1}{2}

or 1 and each

b_{j}

\frac{1}{2}

. The only cases that are integrals of the third kind are those in which at least one

b_{j}

is a positive integer. All other elliptic cases are integrals of the second kind. …

16: 10.22 Integrals

… ►where

\gamma

is Euler’s constant (§5.2(ii)). … ►For

I

and

K

see §10.25(ii). … ►For

I

and

K

see §10.25(ii). … ►For the Ferrers function

\mathsf{P}

and the associated Legendre function

Q

, see §§14.3(i) and 14.3(ii), respectively. … ►For collections of integrals of the functions

J_{\nu}\left(z\right)

Y_{\nu}\left(z\right)

{H^{(1)}_{\nu}}\left(z\right)

, and

{H^{(2)}_{\nu}}\left(z\right)

, including integrals with respect to the order, see Andrews et al. (1999, pp. 216–225), Apelblat (1983, §12), Erdélyi et al. (1953b, §§7.7.1–7.7.7 and 7.14–7.14.2), Erdélyi et al. (1954a, b), Gradshteyn and Ryzhik (2000, §§5.5 and 6.5–6.7), Gröbner and Hofreiter (1950, pp. 196–204), Luke (1962), Magnus et al. (1966, §3.8), Marichev (1983, pp. 191–216), Oberhettinger (1974, §§1.10 and 2.7), Oberhettinger (1990, §§1.13–1.16 and 2.13–2.16), Oberhettinger and Badii (1973, §§1.14 and 2.12), Okui (1974, 1975), Prudnikov et al. (1986b, §§1.8–1.10, 2.12–2.14, 3.2.4–3.2.7, 3.3.2, and 3.4.1), Prudnikov et al. (1992a, §§3.12–3.14), Prudnikov et al. (1992b, §§3.12–3.14), Watson (1944, Chapters 5, 12, 13, and 14), and Wheelon (1968).

17: 32.10 Special Function Solutions

… ►All solutions of

\mbox{P}_{\mbox{\scriptsize II}}

–

\mbox{P}_{\mbox{\scriptsize VI}}

that are expressible in terms of special functions satisfy a first-order equation of the form … ►

§32.10(ii) Second Painlevé Equation

… ►

§32.10(iii) Third Painlevé Equation

►If

\gamma\delta\neq 0

, then as in §32.2(ii) we may set

\gamma=1

and

\delta=-1

. … ►The solution (32.10.34) is an essentially transcendental function of both constants of integration since

\mbox{P}_{\mbox{\scriptsize VI}}

with

\alpha=\beta=\gamma=0

and

\delta=\tfrac{1}{2}

does not admit an algebraic first integral of the form

P(z,w,w^{\prime},C)=0

, with

C

a constant. …

18: 16.14 Partial Differential Equations

… ►

x(1-x)\frac{{\partial}^{2}{F_{1}}}{{\partial x}^{2}}+y(1-x)\frac{\,{\partial}^% {2}{F_{1}}}{\,\partial x\,\partial y}+\left(\gamma-(\alpha+\beta+1)x\right)% \frac{\partial{F_{1}}}{\partial x}-\beta y\frac{\partial{F_{1}}}{\partial y}-% \alpha\beta{F_{1}}=0,

►

y(1-y)\frac{{\partial}^{2}{F_{1}}}{{\partial y}^{2}}+x(1-y)\frac{\,{\partial}^% {2}{F_{1}}}{\,\partial x\,\partial y}+\left(\gamma-(\alpha+\beta^{\prime}+1)y% \right)\frac{\partial{F_{1}}}{\partial y}-\beta^{\prime}x\frac{\partial{F_{1}}% }{\partial x}-\alpha\beta^{\prime}{F_{1}}=0,

… ►In addition to the four Appell functions there are

24

other sums of double series that cannot be expressed as a product of two

{{}_{2}F_{1}}

functions, and which satisfy pairs of linear partial differential equations of the second order. … ►

16.14.5

G_{2}(\alpha,\alpha^{\prime};\beta,\beta^{\prime};x,y)=\sum_{m,n=0}^{\infty}% \frac{\Gamma\left(\alpha+m\right)\Gamma\left(\alpha^{\prime}+n\right)\Gamma% \left(\beta+n-m\right)\Gamma\left(\beta^{\prime}+m-n\right)}{\Gamma\left(% \alpha\right)\Gamma\left(\alpha^{\prime}\right)\Gamma\left(\beta\right)\Gamma% \left(\beta^{\prime}\right)}\frac{x^{m}y^{n}}{m!n!},

|x|<1

|y|<1

ⓘ

Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function and $!$ : factorial (as in $n!$ )
Permalink:: http://dlmf.nist.gov/16.14.E5
Encodings:: pMML, png, TeX
See also:: Annotations for §16.14(ii), §16.14 and Ch.16

►

16.14.6

G_{3}(\alpha,\alpha^{\prime};x,y)=\sum_{m,n=0}^{\infty}\frac{\Gamma\left(% \alpha+2n-m\right)\Gamma\left(\alpha^{\prime}+2m-n\right)}{\Gamma\left(\alpha% \right)\Gamma\left(\alpha^{\prime}\right)}\frac{x^{m}y^{n}}{m!n!},

|x|+|y|<\frac{1}{4}

ⓘ

Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function and $!$ : factorial (as in $n!$ )
Permalink:: http://dlmf.nist.gov/16.14.E6
Encodings:: pMML, png, TeX
See also:: Annotations for §16.14(ii), §16.14 and Ch.16

…

19: 3.5 Quadrature

… ►As in Simpson’s rule, by combining the rule for

h

with that for

h/2

, the first error term

c_{1}h^{2}

in (3.5.9) can be eliminated. … ►For the latter

a=-1

b=1

, and the nodes

x_{k}

are the extrema of the Chebyshev polynomial

T_{n}\left(x\right)

(§3.11(ii) and §18.3). … ►The given quantities

\gamma_{n}

follow from (18.2.5), (18.2.7), Table 18.3.1, and the relation

\gamma_{n}=\ifrac{h_{n}}{k_{n}^{2}}

. … ►Furthermore,

h_{0}=\ifrac{2^{\alpha+\beta+1}\Gamma(\alpha+1)\Gamma(\beta+1)}{\Gamma(\alpha+% \beta+2)}

. … ►A second example is provided in Gil et al. (2001), where the method of contour integration is used to evaluate Scorer functions of complex argument (§9.12). …

20: 19.20 Special Cases

… ►The first lemniscate constant is given by … ►

19.20.9

R_{J}\left(0,y,z,\pm\sqrt{yz}\right)=\pm\frac{3}{2\sqrt{yz}}R_{F}\left(0,y,z% \right).

ⓘ

Symbols:: $R_{F}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : symmetric elliptic integral of first kind and $R_{J}\left(\NVar{x},\NVar{y},\NVar{z},\NVar{p}\right)$ : symmetric elliptic integral of third kind
Referenced by:: §19.20(iii), §19.22(i)
Permalink:: http://dlmf.nist.gov/19.20.E9
Encodings:: pMML, png, TeX
See also:: Annotations for §19.20(iii), §19.20 and Ch.19

… ►

19.20.12

\lim_{p\to\pm\infty}pR_{J}\left(x,y,z,p\right)=3R_{F}\left(x,y,z\right).

ⓘ

Symbols:: $R_{F}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : symmetric elliptic integral of first kind and $R_{J}\left(\NVar{x},\NVar{y},\NVar{z},\NVar{p}\right)$ : symmetric elliptic integral of third kind
Referenced by:: §19.20(iii)
Permalink:: http://dlmf.nist.gov/19.20.E12
Encodings:: pMML, png, TeX
See also:: Annotations for §19.20(iii), §19.20 and Ch.19

… ►The second lemniscate constant is given by … ►Define

c=\sum_{j=1}^{n}b_{j}

. …