Chu–Vandermonde sums (first and second)

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21: 17.5 ${}_0{}^{}\varphi _0^{},{}_1{}^{}\varphi _0^{},{}_1{}^{}\varphi _1^{}$ Functions

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Euler’s Second Sum

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17.5.1 $${}_0{}^{}\varphi _0^{}(-;-;q,z)=\sum _{n=0}^{\mathrm{\infty }}\frac{{(-1)}^n{q}^{\left(\genfrac{}{}{0.0pt}{}{n}{2}\right)}{z}^n}{{(q;q)}_n}={(z;q)}_{\mathrm{\infty }};$$

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Symbols:

$\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, ${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), ${}_{r+1}{}^{}\varphi _s^{}(a_0,\mathrm{\dots },a_r;b_1,\mathrm{\dots },b_s;q,z)$ or ${}_{r+1}{}^{}\varphi _s^{}(\genfrac{}{}{0pt}{}{a_0,\mathrm{\dots },a_r}{b_1,\mathrm{\dots },b_s};q,z)$: basic hypergeometric (or $q$-hypergeometric) function, $q$: complex base, $n$: nonnegative integer and $z$: complex variable

Referenced by:

§17.2(iii), Erratum (V1.1.11) for Equation (17.5.1)

Permalink:

http://dlmf.nist.gov/17.5.E1

Encodings:

pMML, png, TeX

Correction (effective with 1.1.11):

The constraint is not necessary and was removed.

See also:

Annotations for §17.5, §17.5 and Ch.17

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Euler’s First Sum

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17.5.4 $${}_1{}^{}\varphi _0^{}(0;-;q,z)=\sum _{n=0}^{\mathrm{\infty }}\frac{z^n}{{(q;q)}_n}=\frac{1}{{(z;q)}_{\mathrm{\infty }}},$$ ;

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Symbols:

${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), ${}_{r+1}{}^{}\varphi _s^{}(a_0,\mathrm{\dots },a_r;b_1,\mathrm{\dots },b_s;q,z)$ or ${}_{r+1}{}^{}\varphi _s^{}(\genfrac{}{}{0pt}{}{a_0,\mathrm{\dots },a_r}{b_1,\mathrm{\dots },b_s};q,z)$: basic hypergeometric (or $q$-hypergeometric) function, $q$: complex base, $n$: nonnegative integer and $z$: complex variable

Referenced by:

§17.2(iii)

Permalink:

http://dlmf.nist.gov/17.5.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §17.5, §17.5 and Ch.17

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Cauchy’s Sum

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22: 10.25 Definitions

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10.25.2 $$I_{\nu }\left(z\right)={(\frac{1}{2}z)}^{\nu }\sum _{k=0}^{\mathrm{\infty }}\frac{{(\frac{1}{4}{z}^2)}^k}{k!\mathrm{\Gamma }\left(\nu +k+1\right)}.$$

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Defines:

$I_{\nu }\left(z\right)$: modified Bessel function of the first kind

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $!$: factorial (as in $n!$), $k$: nonnegative integer, $z$: complex variable and $\nu$: complex parameter

A&S Ref:

9.6.10

Referenced by:

§10.30(i), §10.31, §10.38, §10.45, §10.46, §10.53, §8.6(i)

Permalink:

http://dlmf.nist.gov/10.25.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §10.25(ii), §10.25 and Ch.10

… ►Both $I_{\nu }\left(z\right)$ and $K_{\nu }\left(z\right)$ are real when $\nu$ is real and $\mathrm{ph}z=0$. ►For fixed $z$ $(\ne 0)$ each branch of $I_{\nu }\left(z\right)$ and $K_{\nu }\left(z\right)$ is entire in $\nu$. … ►Except where indicated otherwise it is assumed throughout the DLMF that the symbols $I_{\nu }\left(z\right)$ and $K_{\nu }\left(z\right)$ denote the principal values of these functions. … ►Corresponding to the symbol ${𝒞}_{\nu }$ introduced in §10.2(ii), we sometimes use ${𝒵}_{\nu }\left(z\right)$ to denote $I_{\nu }\left(z\right)$, ${\mathrm{e}}^{\nu \pi \mathrm{i}}{K}_{\nu }\left(z\right)$, or any nontrivial linear combination of these functions, the coefficients in which are independent of $z$ and $\nu$. …

23: 14.11 Derivatives with Respect to Degree or Order

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14.11.2 $$\frac{\partial }{\partial \nu }{𝖰}_{\nu }^{\mu }\left(x\right)=-\frac{1}{2}{\pi }^2{𝖯}_{\nu }^{\mu }\left(x\right)+\frac{\pi \sin \left(\mu \pi \right)}{\sin \left(\nu \pi \right)\sin \left((\nu +\mu )\pi \right)}{𝖰}_{\nu }^{\mu }\left(x\right)-\frac{1}{2}\cot \left((\nu +\mu )\pi \right){𝖠}_{\nu }^{\mu }(x)+\frac{1}{2}\csc \left((\nu +\mu )\pi \right){𝖠}_{\nu }^{\mu }(-x),$$

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Symbols:

: Ferrers function of the first kind, : Ferrers function of the second kind, $\pi$: the ratio of the circumference of a circle to its diameter, $\csc z$: cosecant function, $\cot z$: cotangent function, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $\sin z$: sine function, $x$: real variable, $\mu$: general order, $\nu$: general degree and ${𝖠}_{\nu }^{\mu }(x)$

Referenced by:

§14.11

Permalink:

http://dlmf.nist.gov/14.11.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §14.11 and Ch.14

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14.11.3 $${𝖠}_{\nu }^{\mu }(x)=\sin \left(\nu \pi \right){\left(\frac{1+x}{1-x}\right)}^{\mu /2}\sum _{k=0}^{\mathrm{\infty }}\frac{{\left(\frac{1}{2}-\frac{1}{2}x\right)}^k\mathrm{\Gamma }\left(k-\nu \right)\mathrm{\Gamma }\left(k+\nu +1\right)}{k!\mathrm{\Gamma }\left(k-\mu +1\right)}\left(\psi \left(k+\nu +1\right)-\psi \left(k-\nu \right)\right).$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $\psi \left(z\right)$: psi (or digamma) function, $!$: factorial (as in $n!$), $\sin z$: sine function, $x$: real variable, $\mu$: general order, $\nu$: general degree and ${𝖠}_{\nu }^{\mu }(x)$

Referenced by:

§14.11

Permalink:

http://dlmf.nist.gov/14.11.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §14.11 and Ch.14

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14.11.4 $${\frac{\partial }{\partial \mu }{𝖯}_{\nu }^{\mu }\left(x\right)|}_{\mu =0}=\left(\psi \left(-\nu \right)-\pi \cot \left(\nu \pi \right)\right){𝖯}_{\nu }\left(x\right)+{𝖰}_{\nu }\left(x\right),$$

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Symbols:

: Ferrers function of the first kind, $\pi$: the ratio of the circumference of a circle to its diameter, $\cot z$: cotangent function, $\psi \left(z\right)$: psi (or digamma) function, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , ${𝖯}_{\nu }\left(x\right)={𝖯}_{\nu }^0\left(x\right)$: Ferrers function of the first kind, ${𝖰}_{\nu }\left(x\right)={𝖰}_{\nu }^0\left(x\right)$: Ferrers function of the second kind, $x$: real variable, $\mu$: general order and $\nu$: general degree

Referenced by:

§14.11, §14.11

Permalink:

http://dlmf.nist.gov/14.11.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §14.11 and Ch.14

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14.11.5 $${\frac{\partial }{\partial \mu }{𝖰}_{\nu }^{\mu }\left(x\right)|}_{\mu =0}=-\frac{1}{4}{\pi }^2{𝖯}_{\nu }\left(x\right)+\left(\psi \left(-\nu \right)-\pi \cot \left(\nu \pi \right)\right){𝖰}_{\nu }\left(x\right).$$

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Symbols:

: Ferrers function of the second kind, $\pi$: the ratio of the circumference of a circle to its diameter, $\cot z$: cotangent function, $\psi \left(z\right)$: psi (or digamma) function, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , ${𝖯}_{\nu }\left(x\right)={𝖯}_{\nu }^0\left(x\right)$: Ferrers function of the first kind, ${𝖰}_{\nu }\left(x\right)={𝖰}_{\nu }^0\left(x\right)$: Ferrers function of the second kind, $x$: real variable, $\mu$: general order and $\nu$: general degree

Referenced by:

§14.11

Permalink:

http://dlmf.nist.gov/14.11.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §14.11 and Ch.14

… ►(14.11.4) holds if ${𝖯}_{\nu }^{\mu }\left(x\right)$, ${𝖯}_{\nu }\left(x\right)$, and ${𝖰}_{\nu }\left(x\right)$ are replaced by $P_{\nu }^{\mu }\left(x\right)$, $P_{\nu }\left(x\right)$, and $Q_{\nu }\left(x\right)$, respectively. …

24: 33.20 Expansions for Small $|ϵ|$

… ►For the functions $J$, $Y$, $I$, and $K$ see §§10.2(ii), 10.25(ii). … ►

33.20.3 $$f(ϵ,\mathrm{\ell };r)=\sum _{k=0}^{\mathrm{\infty }}{ϵ}^k{𝖥}_k(\mathrm{\ell };r),$$

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Symbols:

$f(ϵ,\mathrm{\ell };r)$: regular Coulomb function, $k$: nonnegative integer, $\mathrm{\ell }$: nonnegative integer, $r$: real variable, $ϵ$: real parameter and ${𝖥}_k(\mathrm{\ell };r)$: coefficient

Referenced by:

§33.20(ii)

Permalink:

http://dlmf.nist.gov/33.20.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §33.20(ii), §33.20 and Ch.33

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33.20.4 $${𝖥}_k(\mathrm{\ell };r)=\sum _{p=2k}^{3k}{(2r)}^{(p+1)/2}{C}_{k,p}{J}_{2\mathrm{\ell }+1+p}\left(\sqrt{8r}\right),$$ $r>0$,

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Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $k$: nonnegative integer, $\mathrm{\ell }$: nonnegative integer, $r$: real variable, ${𝖥}_k(\mathrm{\ell };r)$: coefficient and $C_{k,p}$: coefficients

Permalink:

http://dlmf.nist.gov/33.20.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §33.20(ii), §33.20 and Ch.33

►

33.20.5 $${𝖥}_k(\mathrm{\ell };r)=\sum _{p=2k}^{3k}{(-1)}^{\mathrm{\ell }+1+p}{(2|r|)}^{(p+1)/2}{C}_{k,p}{I}_{2\mathrm{\ell }+1+p}\left(\sqrt{8|r|}\right),$$ .

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Symbols:

$I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $k$: nonnegative integer, $\mathrm{\ell }$: nonnegative integer, $r$: real variable, ${𝖥}_k(\mathrm{\ell };r)$: coefficient and $C_{k,p}$: coefficients

Permalink:

http://dlmf.nist.gov/33.20.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §33.20(ii), §33.20 and Ch.33

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33.20.7 $$h(ϵ,\mathrm{\ell };r)\sim -A(ϵ,\mathrm{\ell })\sum _{k=0}^{\mathrm{\infty }}{ϵ}^k{𝖧}_k(\mathrm{\ell };r),$$

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Symbols:

$h(ϵ,\mathrm{\ell };r)$: irregular Coulomb function, $\sim$: Poincaré asymptotic expansion, $k$: nonnegative integer, $\mathrm{\ell }$: nonnegative integer, $r$: real variable, $ϵ$: real parameter, $A(ϵ,\mathrm{\ell })$: function and ${𝖧}_k(\mathrm{\ell };r)$: coefficient

Permalink:

http://dlmf.nist.gov/33.20.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §33.20(iii), §33.20 and Ch.33

…

25: 10.43 Integrals

… ►

10.43.22

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, : Ferrers function of the first kind, $P_{\nu }^{\mu }\left(z\right)$: associated Legendre function of the first kind, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $\mathrm{\Re }$: real part and $\nu$: complex parameter

Keywords:

Laplace transform, Mellin transform

Referenced by:

§10.43(iv)

Permalink:

http://dlmf.nist.gov/10.43.E22

Encodings:

pMML, png, TeX

See also:

Annotations for §10.43(iv), §10.43 and Ch.10

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10.43.26 $${\int }_0^{\mathrm{\infty }}\frac{K_{\mu }\left(at\right)J_{\nu }\left(bt\right)}{t^{\lambda }}dt=\frac{b^{\nu }\mathrm{\Gamma }\left(\frac{1}{2}\nu -\frac{1}{2}\lambda +\frac{1}{2}\mu +\frac{1}{2}\right)\mathrm{\Gamma }\left(\frac{1}{2}\nu -\frac{1}{2}\lambda -\frac{1}{2}\mu +\frac{1}{2}\right)}{2^{\lambda +1}{a}^{\nu -\lambda +1}}𝐅(\frac{\nu -\lambda +\mu +1}{2},\frac{\nu -\lambda -\mu +1}{2};\nu +1;-\frac{b^2}{a^2}),$$ $\mathrm{\Re }\left(\nu +1-\lambda \right)>|\mathrm{\Re }\mu |,\mathrm{\Re }a>|\mathrm{\Im }b|$.

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Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $\mathrm{\Gamma }\left(z\right)$: gamma function, $dx$: differential of $x$ , $\mathrm{\Im }$: imaginary part, $\int$: integral, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $\mathrm{\Re }$: real part, $𝐅(a,b;c;z)$ or $𝐅(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}𝐅_1^{}(a,b;c;z)$ Olver’s hypergeometric function and $\nu$: complex parameter

Keywords:

Mellin transform

Permalink:

http://dlmf.nist.gov/10.43.E26

Encodings:

pMML, png, TeX

See also:

Annotations for §10.43(iv), §10.43 and Ch.10

… ►

10.43.27 $${\int }_0^{\mathrm{\infty }}{t}^{\mu +\nu +1}{K}_{\mu }\left(at\right)J_{\nu }\left(bt\right)dt=\frac{{(2a)}^{\mu }{(2b)}^{\nu }\mathrm{\Gamma }\left(\mu +\nu +1\right)}{{(a^2+b^2)}^{\mu +\nu +1}},$$ $\mathrm{\Re }\left(\nu +1\right)>|\mathrm{\Re }\mu |,\mathrm{\Re }a>|\mathrm{\Im }b|$.

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Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $\mathrm{\Gamma }\left(z\right)$: gamma function, $dx$: differential of $x$ , $\mathrm{\Im }$: imaginary part, $\int$: integral, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $\mathrm{\Re }$: real part and $\nu$: complex parameter

Keywords:

Mellin transform

Permalink:

http://dlmf.nist.gov/10.43.E27

Encodings:

pMML, png, TeX

See also:

Annotations for §10.43(iv), §10.43 and Ch.10

… ►

10.43.29 $${\int }_0^{\mathrm{\infty }}t\exp \left(-p^2{t}^2\right)I_0\left(at\right)K_0\left(at\right)dt=\frac{1}{4p^2}\exp \left(\frac{a^2}{2p^2}\right)K_0\left(\frac{a^2}{2p^2}\right),$$ $\mathrm{\Re }\left(p^2\right)>0$.

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Symbols:

$dx$: differential of $x$ , $\exp z$: exponential function, $\int$: integral, $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind and $\mathrm{\Re }$: real part

Referenced by:

§10.43(iv)

Permalink:

http://dlmf.nist.gov/10.43.E29

Encodings:

pMML, png, TeX

See also:

Annotations for §10.43(iv), §10.43 and Ch.10

… ►For collections of integrals of the functions $I_{\nu }\left(z\right)$ and $K_{\nu }\left(z\right)$, including integrals with respect to the order, see 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, 6.5–6.7), Gröbner and Hofreiter (1950, pp. 197–203), Luke (1962), Magnus et al. (1966, §3.8), Marichev (1983, pp. 191–216), Oberhettinger (1972), Oberhettinger (1974, §§1.11 and 2.7), Oberhettinger (1990, §§1.17–1.20 and 2.17–2.20), Oberhettinger and Badii (1973, §§1.15 and 2.13), Okui (1974, 1975), Prudnikov et al. (1986b, §§1.11–1.12, 2.15–2.16, 3.2.8–3.2.10, and 3.4.1), Prudnikov et al. (1992a, §§3.15, 3.16), Prudnikov et al. (1992b, §§3.15, 3.16), Watson (1944, Chapter 13), and Wheelon (1968).

26: 10.40 Asymptotic Expansions for Large Argument

… ►Corresponding expansions for $I_{\nu }\left(z\right)$, $K_{\nu }\left(z\right)$, $I_{\nu }^{\prime }\left(z\right)$, and $K_{\nu }^{\prime }\left(z\right)$ for other ranges of $\mathrm{ph}z$ 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). … ►

10.40.5 $$I_{\nu }\left(z\right)\sim \frac{{\mathrm{e}}^z}{{(2\pi z)}^{\frac{1}{2}}}\sum _{k=0}^{\mathrm{\infty }}{(-1)}^k\frac{a_k(\nu )}{z^k}\pm \mathrm{i}{\mathrm{e}}^{\pm \nu \pi \mathrm{i}}\frac{{\mathrm{e}}^{-z}}{{(2\pi z)}^{\frac{1}{2}}}\sum _{k=0}^{\mathrm{\infty }}\frac{a_k(\nu )}{z^k},$$ $-\frac{1}{2}\pi +\delta \le \pm \mathrm{ph}z\le \frac{3}{2}\pi -\delta$.

ⓘ

Symbols:

$\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $\mathrm{ph}$: phase, $k$: nonnegative integer, $z$: complex variable, $\nu$: complex parameter, $\delta$: small positive constant and $a_k(\nu )$: polynomial coefficient

Referenced by:

§10.40(i)

Permalink:

http://dlmf.nist.gov/10.40.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §10.40(i), §10.40 and Ch.10

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10.40.6 $$I_{\nu }\left(z\right)K_{\nu }\left(z\right)\sim \frac{1}{2z}\left(1-\frac{1}{2}\frac{\mu -1}{{(2z)}^2}+\frac{1\cdot 3}{2\cdot 4}\frac{(\mu -1)(\mu -9)}{{(2z)}^4}-\mathrm{\cdots }\right),$$

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Symbols:

$\sim$: Poincaré asymptotic expansion, $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $K_{\nu }\left(z\right)$: 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

►

10.40.7 $$I_{\nu }^{\prime }\left(z\right)K_{\nu }^{\prime }\left(z\right)\sim -\frac{1}{2z}\left(1+\frac{1}{2}\frac{\mu -3}{{(2z)}^2}-\frac{1}{2\cdot 4}\frac{(\mu -1)(\mu -45)}{{(2z)}^4}+\mathrm{\cdots }\right),$$

ⓘ

Symbols:

$\sim$: Poincaré asymptotic expansion, $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $z$: complex variable and $\nu$: complex parameter

A&S Ref:

9.7.6

Referenced by:

§10.40(i), §10.40(i)

Permalink:

http://dlmf.nist.gov/10.40.E7

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 $k=\mathrm{\ell }-1$. …

27: 18.3 Definitions

… ►

Table 18.3.1: Orthogonality properties for classical OP’s: intervals, weight functions, standardizations, leading coefficients, and parameter constraints. …

► ►►

Name	$p_n(x)$	$(a,b)$	$w(x)$	$h_n$	$k_n$	${\tilde{k}}_n/k_n$	Constraints
…

►

… ►In this chapter, formulas for the Chebyshev polynomials of the second, third, and fourth kinds will not be given as extensively as those of the first kind. … ►

18.3.1 $$\sum _{n=1}^{N+1}{T}_j\left(x_{N+1,n}\right)T_k\left(x_{N+1,n}\right)=0,$$ $0\le j\le N$, $0\le k\le N$, $j\ne k$,

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Symbols:

$T_n\left(x\right)$: Chebyshev polynomial of the first kind, $N$: positive integer, $n$: nonnegative integer and $x$: real variable

Referenced by:

§18.3, §18.3

Permalink:

http://dlmf.nist.gov/18.3.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §18.3, §18.3 and Ch.18

… ►When $j=k\ne 0$ the sum in (18.3.1) is $\frac{1}{2}(N+1)$. When $j=k=0$ the sum in (18.3.1) is $N+1$. …

28: 29.15 Fourier Series and Chebyshev Series

… ►

29.15.10 $$\sum _{p=0}^n{A}_{2p+1}^2=1,$$

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Symbols:

$n$: nonnegative integer, $p$: nonnegative integer and $A_{2p}$: coefficients

Permalink:

http://dlmf.nist.gov/29.15.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §29.15(i), §29.15(i), §29.15 and Ch.29

►

29.15.11 $$\sum _{p=0}^n{A}_{2p+1}>0.$$

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Symbols:

$n$: nonnegative integer, $p$: nonnegative integer and $A_{2p}$: coefficients

Permalink:

http://dlmf.nist.gov/29.15.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §29.15(i), §29.15(i), §29.15 and Ch.29

… ►

29.15.15 $$\sum _{p=0}^n{B}_{2p+1}^2=1,$$

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Symbols:

$n$: nonnegative integer, $p$: nonnegative integer and $B_{2p+1}$: coefficents

Permalink:

http://dlmf.nist.gov/29.15.E15

Encodings:

pMML, png, TeX

See also:

Annotations for §29.15(i), §29.15(i), §29.15 and Ch.29

… ►

29.15.25 $$\sum _{p=0}^n{B}_{2p+2}^2=1,$$

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Symbols:

$n$: nonnegative integer, $p$: nonnegative integer and $B_{2p+1}$: coefficents

Permalink:

http://dlmf.nist.gov/29.15.E25

Encodings:

pMML, png, TeX

See also:

Annotations for §29.15(i), §29.15(i), §29.15 and Ch.29

… ►

29.15.31 $$\sum _{p=0}^n{C}_{2p+1}>0.$$

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Symbols:

$n$: nonnegative integer, $p$: nonnegative integer and $C_{2p+1}$: coefficents

Permalink:

http://dlmf.nist.gov/29.15.E31

Encodings:

pMML, png, TeX

See also:

Annotations for §29.15(i), §29.15(i), §29.15 and Ch.29

…

29: 19.8 Quadratic Transformations

… ►

19.8.6 $$E\left(k\right)=\frac{\pi }{2M(1,k^{\prime })}\left(a_0^2-\sum _{n=0}^{\mathrm{\infty }}{2}^{n-1}{c}_n^2\right)=K\left(k\right)\left(a_1^2-\sum _{n=2}^{\mathrm{\infty }}{2}^{n-1}{c}_n^2\right),$$ , $a_0=1$, $g_0=k^{\prime }$,

ⓘ

Symbols:

$M(a,g)$: arithmetic-geometric mean, $\pi$: the ratio of the circumference of a circle to its diameter, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $E\left(k\right)$: Legendre’s complete elliptic integral of the second kind, $n$: nonnegative integer, $k$: real or complex modulus, $k^{\prime }$: complementary modulus, $a_n$: iterate and $g_n$: iterate

Permalink:

http://dlmf.nist.gov/19.8.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §19.8(i), §19.8 and Ch.19

… ►

$$E\left(k\right)=(1+k^{\prime })E\left(k_1\right)-k^{\prime }K\left(k\right).$$

… ►

$$E(\varphi ,k)=\frac{1}{2}(1+k^{\prime })E({\varphi }_1,k_1)-k^{\prime }F(\varphi ,k)+\frac{1}{2}(1-k^{\prime })\sin {\varphi }_1.$$

… ►

$$E(\varphi ,k)=(1+k)E({\varphi }_2,k_2)+(1-k)F({\varphi }_2,k_2)-k\sin \varphi .$$

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$$E(\varphi ,k)=(1+k^{\prime })E({\psi }_1,k_1)-k^{\prime }F(\varphi ,k)+(1-\mathrm{\Delta })\cot \varphi ,$$

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30: 22.16 Related Functions

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22.16.28 $$ℰ(x+K,k)=ℰ(x,k)+E\left(k\right)-k^2\mathrm{sn}(x,k)\mathrm{cd}(x,k),$$

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Symbols:

$ℰ(x,k)$: Jacobi’s epsilon function, $\mathrm{cd}(z,k)$: Jacobian elliptic function, $\mathrm{sn}(z,k)$: Jacobian elliptic function, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $E\left(k\right)$: Legendre’s complete elliptic integral of the second kind, $x$: real and $k$: modulus

Permalink:

http://dlmf.nist.gov/22.16.E28

Encodings:

pMML, png, TeX

See also:

Annotations for §22.16(ii), §22.16(ii), §22.16 and Ch.22

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22.16.29 $$ℰ(x+2K,k)=ℰ(x,k)+2E\left(k\right).$$

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Symbols:

$ℰ(x,k)$: Jacobi’s epsilon function, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $E\left(k\right)$: Legendre’s complete elliptic integral of the second kind, $x$: real and $k$: modulus

Permalink:

http://dlmf.nist.gov/22.16.E29

Encodings:

pMML, png, TeX

See also:

Annotations for §22.16(ii), §22.16(ii), §22.16 and Ch.22

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22.16.30 $$ℰ(x,k)=\frac{1}{{\theta }_3^2(0,q){\theta }_4(\xi ,q)}\frac{d}{d\xi }{\theta }_4(\xi ,q)+\frac{E\left(k\right)}{K\left(k\right)}x,$$

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Symbols:

$ℰ(x,k)$: Jacobi’s epsilon function, ${\theta }_j(z,q)$: theta function, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $E\left(k\right)$: Legendre’s complete elliptic integral of the second kind, $\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $q$: nome, $x$: real and $k$: modulus

Referenced by:

§22.20(vi)

Permalink:

http://dlmf.nist.gov/22.16.E30

Encodings:

pMML, png, TeX

See also:

Annotations for §22.16(ii), §22.16(ii), §22.16 and Ch.22

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22.16.31 $$E(\mathrm{am}(x,k),k)=ℰ(x,k),$$ $-K\le x\le K$.

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Symbols:

$ℰ(x,k)$: Jacobi’s epsilon function, $\mathrm{am}(x,k)$: Jacobi’s amplitude function, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $E(\varphi ,k)$: Legendre’s incomplete elliptic integral of the second kind, $x$: real and $k$: modulus

Permalink:

http://dlmf.nist.gov/22.16.E31

Encodings:

pMML, png, TeX

See also:

Annotations for §22.16(ii), §22.16(ii), §22.16 and Ch.22

… ►With $E\left(k\right)$ and $K\left(k\right)$ as in §19.2(ii) and $x\in ℝ$, …