Gauss 2F1(-1) sum

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1—10 of 818 matching pages

1: 32.10 Special Function Solutions

… ►For example, if

\alpha=\tfrac{1}{2}\varepsilon

, with

\varepsilon=\pm 1

, then the Riccati equation is … ►with

n\in\mathbb{Z}

, and

\varepsilon_{1}=\pm 1

\varepsilon_{2}=\pm 1

, independently. … ►with

n\in\mathbb{Z}

and

\varepsilon=\pm 1

. … ►where

n\in\mathbb{Z}

a=\varepsilon_{1}\sqrt{2\alpha}

, and

b=\varepsilon_{2}\sqrt{-2\beta}

, with

\varepsilon_{j}=\pm 1

j=1,2,3

, independently. … ►where

n\in\mathbb{Z}

a=\varepsilon_{1}\sqrt{2\alpha}

b=\varepsilon_{2}\sqrt{-2\beta}

c=\varepsilon_{3}\sqrt{2\gamma}

, and

d=\varepsilon_{4}\sqrt{1-2\delta}

, with

\varepsilon_{j}=\pm 1

j=1,2,3,4

, independently. …

2: 16.10 Expansions in Series of ${{}_{p}F_{q}}$ Functions

§16.10 Expansions in Series of ${{}_{p}F_{q}}$ Functions

… ►

16.10.1

{{}_{p+r}F_{q+s}}\left({a_{1},\dots,a_{p},c_{1},\dots,c_{r}\atop b_{1},\dots,b% _{q},d_{1},\dots,d_{s}};z\zeta\right)=\sum_{k=0}^{\infty}\frac{{\left(\mathbf{% a}\right)_{k}}{\left(\alpha\right)_{k}}{\left(\beta\right)_{k}}(-z)^{k}}{{% \left(\mathbf{b}\right)_{k}}{\left(\gamma+k\right)_{k}}k!}{{}_{p+2}F_{q+1}}% \left({\alpha+k,\beta+k,a_{1}+k,\dots,a_{p}+k\atop\gamma+2k+1,b_{1}+k,\dots,b_% {q}+k};z\right){{}_{r+2}F_{s+2}}\left({-k,\gamma+k,c_{1},\dots,c_{r}\atop% \alpha,\beta,d_{1},\dots,d_{s}};\zeta\right).

ⓘ

Symbols:: ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{a_{1},\dots,a_{p}};\NVar{b_{1},\dots,b_% {q}};\NVar{z}\right)$ or ${{}_{\NVar{p}}F_{\NVar{q}}}\left({\NVar{a_{1},\dots,a_{p}}\atop\NVar{b_{1},% \dots,b_{q}}};\NVar{z}\right)$ : alternatively ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{\mathbf{a}};\NVar{\mathbf{b}};\NVar{z}\right)$ or ${{}_{\NVar{p}}F_{\NVar{q}}}\left({\NVar{\mathbf{a}}\atop\NVar{\mathbf{b}}};% \NVar{z}\right)$
generalized hypergeometric function, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $!$ : factorial (as in $n!$ ), $p$ : nonnegative integer, $q$ : nonnegative integer, $z$ : complex variable, $a,a_{1},\ldots,a_{p}$ : real or complex parameters and $b,b_{1},\ldots,b_{q}$ : real or complex parameters
Permalink:: http://dlmf.nist.gov/16.10.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §16.10 and Ch.16

… ►

16.10.2

{{}_{p+1}F_{p}}\left({a_{1},\dots,a_{p+1}\atop b_{1},\dots,b_{p}};z\zeta\right% )=(1-z)^{-a_{1}}\sum_{k=0}^{\infty}\frac{{\left(a_{1}\right)_{k}}}{k!}{{}_{p+1% }F_{p}}\left({-k,a_{2},\dots,a_{p+1}\atop b_{1},\dots,b_{p}};\zeta\right)\left% (\frac{z}{z-1}\right)^{k}.

ⓘ

Symbols:: ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{a_{1},\dots,a_{p}};\NVar{b_{1},\dots,b_% {q}};\NVar{z}\right)$ or ${{}_{\NVar{p}}F_{\NVar{q}}}\left({\NVar{a_{1},\dots,a_{p}}\atop\NVar{b_{1},% \dots,b_{q}}};\NVar{z}\right)$ : alternatively ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{\mathbf{a}};\NVar{\mathbf{b}};\NVar{z}\right)$ or ${{}_{\NVar{p}}F_{\NVar{q}}}\left({\NVar{\mathbf{a}}\atop\NVar{\mathbf{b}}};% \NVar{z}\right)$
generalized hypergeometric function, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $!$ : factorial (as in $n!$ ), $p$ : nonnegative integer, $z$ : complex variable, $a,a_{1},\ldots,a_{p}$ : real or complex parameters and $b,b_{1},\ldots,b_{q}$ : real or complex parameters
Permalink:: http://dlmf.nist.gov/16.10.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §16.10 and Ch.16

►When

|\zeta-1|<1

the series on the right-hand side converges in the half-plane

\Re z<\frac{1}{2}

. ►Expansions of the form

\sum_{n=1}^{\infty}(\pm 1)^{n}{{}_{p}F_{p+1}}\left(\mathbf{a};\mathbf{b};-n^{2% }z^{2}\right)

are discussed in Miller (1997), and further series of generalized hypergeometric functions are given in Luke (1969b, Chapter 9), Luke (1975, §§5.10.2 and 5.11), and Prudnikov et al. (1990, §§5.3, 6.8–6.9).

3: 5.5 Functional Relations

… ►

5.5.3

\Gamma\left(z\right)\Gamma\left(1-z\right)=\pi/\sin\left(\pi z\right),

z\neq 0,\pm 1,\dots

ⓘ

Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\sin\NVar{z}$ : sine function and $z$ : complex variable
A&S Ref:: 6.1.17 (without the condition on $z$ .)
Referenced by:: §10.19(i), §15.4(ii), §15.8(ii), (25.9.2), §5.21, (9.10.18), (9.12.15)
Permalink:: http://dlmf.nist.gov/5.5.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §5.5(ii), §5.5 and Ch.5

►

5.5.4

\psi\left(z\right)-\psi\left(1-z\right)=-\pi/\tan\left(\pi z\right),

z\neq 0,\pm 1,\dots

ⓘ

Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\psi\left(\NVar{z}\right)$ : psi (or digamma) function, $\tan\NVar{z}$ : tangent function and $z$ : complex variable
A&S Ref:: 6.3.7 (without the condition on $z$ .)
Permalink:: http://dlmf.nist.gov/5.5.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §5.5(ii), §5.5 and Ch.5

… ►For

2z\neq 0,-1,-2,\dots

, … ►

Gauss’s Multiplication Formula

►For

nz\neq 0,-1,-2,\dots

, …

4: 15.4 Special Cases

… ►Exceptions are (15.4.8) and (15.4.10), that hold for

|z|<\ifrac{\pi}{4}

, and (15.4.12), (15.4.14), and (15.4.16), that hold for

|z|<\ifrac{\pi}{2}

. … ►

§15.4(ii) Argument Unity

… ►

Dougall’s Bilateral Sum

… ►

§15.4(iii) Other Arguments

… ►where the limit interpretation (15.2.6), rather than (15.2.5), has to be taken when in (15.4.33)

a=-\frac{1}{3},-\frac{4}{3},-\frac{7}{3},\dots

, and in (15.4.34)

a=0,-1,-2,\dots

. …

5: 15.5 Derivatives and Contiguous Functions

… ►The six functions

F\left(a\pm 1,b;c;z\right)

F\left(a,b\pm 1;c;z\right)

F\left(a,b;c\pm 1;z\right)

are said to be contiguous to

F\left(a,b;c;z\right)

. … ►

15.5.12

(b-a)F\left(a,b;c;z\right)+aF\left(a+1,b;c;z\right)-bF\left(a,b+1;c;z\right)=0,

ⓘ

Symbols:: $F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ or $F\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)$ : $={{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ Gauss’ hypergeometric function, $z$ : complex variable, $a$ : real or complex parameter, $b$ : real or complex parameter and $c$ : real or complex parameter
Permalink:: http://dlmf.nist.gov/15.5.E12
Encodings:: pMML, png, TeX
See also:: Annotations for §15.5(ii), §15.5 and Ch.15

… ►By repeated applications of (15.5.11)–(15.5.18) any function

F\left(a+k,b+\ell;c+m;z\right)

, in which

k,\ell,m

are integers, can be expressed as a linear combination of

F\left(a,b;c;z\right)

and any one of its contiguous functions, with coefficients that are rational functions of

a, b, c

, and

z

. … ►

15.5.19

{z(1-z)(a+1)(b+1)}F\left(a+2,b+2;c+2;z\right)+{(c-(a+b+1)z)(c+1)}F\left(a+1,b+% 1;c+1;z\right)-{c(c+1)}F\left(a,b;c;z\right)=0.

ⓘ

Symbols:: $F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ or $F\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)$ : $={{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ Gauss’ hypergeometric function, $z$ : complex variable, $a$ : real or complex parameter, $b$ : real or complex parameter and $c$ : real or complex parameter
Permalink:: http://dlmf.nist.gov/15.5.E19
Encodings:: pMML, png, TeX
See also:: Annotations for §15.5(ii), §15.5 and Ch.15

… ►

15.5.20

z(1-z)\left(\ifrac{\mathrm{d}F\left(a,b;c;z\right)}{\mathrm{d}z}\right)=(c-a)F% \left(a-1,b;c;z\right)+(a-c+bz)F\left(a,b;c;z\right)=(c-b)F\left(a,b-1;c;z% \right)+(b-c+az)F\left(a,b;c;z\right),

ⓘ

Symbols:: $F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ or $F\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)$ : $={{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ Gauss’ hypergeometric function, $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $z$ : complex variable, $a$ : real or complex parameter, $b$ : real or complex parameter and $c$ : real or complex parameter
Permalink:: http://dlmf.nist.gov/15.5.E20
Encodings:: pMML, png, TeX
See also:: Annotations for §15.5(ii), §15.5 and Ch.15

…

6: 18.33 Polynomials Orthogonal on the Unit Circle

… ►Instead of (18.33.9) one might take monic OP’s

\{q_{n}(x)\}

with weight function

(1+x)w_{1}(x)

, and then express

q_{n}\left(\tfrac{1}{2}(z+z^{-1})\right)

in terms of

\phi_{2n}(z^{\pm 1})

\phi_{2n+1}(z^{\pm 1})

. …See Zhedanov (1998, §2). … ►For the hypergeometric function

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

see §§15.1 and 15.2(i). … ►For the notation, including the basic hypergeometric function

{{}_{2}\phi_{1}}

, see §§17.2 and 17.4(i). … ►See Simon (2005a, p. 2, item (2)). …

7: 10.59 Integrals

… ►

10.59.1

\int_{-\infty}^{\infty}e^{ibt}\mathsf{j}_{n}\left(t\right)\,\mathrm{d}t=\begin% {cases}\pi i^{n}P_{n}\left(b\right),&-1<b<1,\\ \frac{1}{2}\pi(\pm\mathrm{i})^{n},&b=\pm 1,\\ 0,&\pm b>1,\end{cases}

ⓘ

Symbols:: $P_{\NVar{n}}\left(\NVar{x}\right)$ : Legendre polynomial, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $\int$ : integral, $\mathsf{j}_{\NVar{n}}\left(\NVar{z}\right)$ : spherical Bessel function of the first kind and $n$ : integer
A&S Ref:: 11.4.26
Referenced by:: §10.59
Permalink:: http://dlmf.nist.gov/10.59.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §10.59 and Ch.10

…

8: 19.21 Connection Formulas

… ►

19.21.1

R_{F}\left(0,z+1,z\right)R_{D}\left(0,z+1,1\right)+R_{D}\left(0,z+1,z\right)R_% {F}\left(0,z+1,1\right)=3\pi/(2z),

z\in\mathbb{C}\setminus(-\infty,0]

ⓘ

Symbols:: $R_{D}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : elliptic integral symmetric in only two variables, $R_{F}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : symmetric elliptic integral of first kind, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathbb{C}$ : complex plane, $\in$ : element of, $(\NVar{a},\NVar{b}]$ : half-closed interval and $\setminus$ : set subtraction
Referenced by:: §19.21(i), §19.21(i), §19.7(i)
Permalink:: http://dlmf.nist.gov/19.21.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §19.21(i), §19.21 and Ch.19

… ►The complete cases of

R_{F}

and

R_{G}

have connection formulas resulting from those for the Gauss hypergeometric function (Erdélyi et al. (1953a, §2.9)). … ►

19.21.4

R_{F}\left(0,z-1,z\right)=R_{F}\left(0,1-z,1\right)\mp\mathrm{i}R_{F}\left(0,z% ,1\right),

ⓘ

Symbols:: $R_{F}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : symmetric elliptic integral of first kind and $\mathrm{i}$ : imaginary unit
Permalink:: http://dlmf.nist.gov/19.21.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §19.21(i), §19.21 and Ch.19

►

19.21.5

2R_{G}\left(0,z-1,z\right)=2R_{G}\left(0,1-z,1\right)\pm\mathrm{i}2R_{G}\left(% 0,z,1\right)+(z-1)R_{F}\left(0,1-z,1\right)\mp\mathrm{i}zR_{F}\left(0,z,1% \right).

ⓘ

Symbols:: $R_{F}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : symmetric elliptic integral of first kind, $R_{G}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : symmetric elliptic integral of second kind and $\mathrm{i}$ : imaginary unit
Permalink:: http://dlmf.nist.gov/19.21.E5
Encodings:: pMML, png, TeX
See also:: Annotations for §19.21(i), §19.21 and Ch.19

… ►

19.21.11

6R_{G}\left(x,y,z\right)=3(x+y+z)R_{F}\left(x,y,z\right)-\sum x^{2}R_{D}\left(% y,z,x\right)=\sum x(y+z)R_{D}\left(y,z,x\right),

ⓘ

Symbols:: $R_{D}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : elliptic integral symmetric in only two variables, $R_{F}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : symmetric elliptic integral of first kind and $R_{G}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : symmetric elliptic integral of second kind
Referenced by:: §19.21(i), §19.21(ii)
Permalink:: http://dlmf.nist.gov/19.21.E11
Encodings:: pMML, png, TeX
See also:: Annotations for §19.21(ii), §19.21 and Ch.19

…

9: 4.24 Inverse Trigonometric Functions: Further Properties

… ►

4.24.13

\operatorname{Arcsin}u\pm\operatorname{Arcsin}v=\operatorname{Arcsin}\left(u(1% -v^{2})^{1/2}\pm v(1-u^{2})^{1/2}\right),

ⓘ

Symbols:: $\operatorname{Arcsin}\NVar{z}$ : general arcsine function
A&S Ref:: 4.4.32
Permalink:: http://dlmf.nist.gov/4.24.E13
Encodings:: pMML, png, TeX
See also:: Annotations for §4.24(iii), §4.24 and Ch.4

►

4.24.14

\operatorname{Arccos}u\pm\operatorname{Arccos}v=\operatorname{Arccos}\left(uv% \mp((1-u^{2})(1-v^{2}))^{1/2}\right),

ⓘ

Symbols:: $\operatorname{Arccos}\NVar{z}$ : general arccosine function
A&S Ref:: 4.4.33
Permalink:: http://dlmf.nist.gov/4.24.E14
Encodings:: pMML, png, TeX
See also:: Annotations for §4.24(iii), §4.24 and Ch.4

►

4.24.15

\operatorname{Arctan}u\pm\operatorname{Arctan}v=\operatorname{Arctan}\left(% \frac{u\pm v}{1\mp uv}\right),

ⓘ

Symbols:: $\operatorname{Arctan}\NVar{z}$ : general arctangent function
A&S Ref:: 4.4.34
Referenced by:: §4.45(i)
Permalink:: http://dlmf.nist.gov/4.24.E15
Encodings:: pMML, png, TeX
See also:: Annotations for §4.24(iii), §4.24 and Ch.4

►

4.24.16

\operatorname{Arcsin}u\pm\operatorname{Arccos}v=\operatorname{Arcsin}\left(uv% \pm((1-u^{2})(1-v^{2}))^{1/2}\right)=\operatorname{Arccos}\left(v(1-u^{2})^{1/% 2}\mp u(1-v^{2})^{1/2}\right),

ⓘ

Symbols:: $\operatorname{Arccos}\NVar{z}$ : general arccosine function and $\operatorname{Arcsin}\NVar{z}$ : general arcsine function
A&S Ref:: 4.4.35
Permalink:: http://dlmf.nist.gov/4.24.E16
Encodings:: pMML, png, TeX
See also:: Annotations for §4.24(iii), §4.24 and Ch.4

►

4.24.17

\operatorname{Arctan}u\pm\operatorname{Arccot}v=\operatorname{Arctan}\left(% \frac{uv\pm 1}{v\mp u}\right)=\operatorname{Arccot}\left(\frac{v\mp u}{uv\pm 1% }\right).

ⓘ

Symbols:: $\operatorname{Arccot}\NVar{z}$ : general arccotangent function and $\operatorname{Arctan}\NVar{z}$ : general arctangent function
A&S Ref:: 4.4.36
Permalink:: http://dlmf.nist.gov/4.24.E17
Encodings:: pMML, png, TeX
See also:: Annotations for §4.24(iii), §4.24 and Ch.4

…

10: 15.12 Asymptotic Approximations

… ►

(a)

$a$ and/or $b\in\{0,-1,-2,\dots\}$ .

… ►

(d)

$\Re z>\tfrac{1}{2}$ and $\alpha_{-}-\tfrac{1}{2}\pi+\delta\leq\operatorname{ph}c\leq\alpha_{+}+\tfrac{1% }{2}\pi-\delta$ , where

15.12.1

\alpha_{\pm}=\operatorname{arctan}\left(\frac{\operatorname{ph}z-\operatorname% {ph}\left(1-z\right)\mp\pi}{\ln|1-z^{-1}|}\right),

ⓘ

Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\operatorname{arctan}\NVar{z}$ : arctangent function, $\ln\NVar{z}$ : principal branch of logarithm function, $\operatorname{ph}$ : phase, $z$ : complex variable and $\alpha_{\pm}$
Permalink:: http://dlmf.nist.gov/15.12.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §15.12(ii), §15.12 and Ch.15

with $z$ restricted so that $\pm\alpha_{\pm}\in[0,\tfrac{1}{2}\pi)$ .

►Then for fixed

m\in\{0,1,2,\dots\}

, … ►For the more general case in which

a^{2}=o\left(c\right)

and

b^{2}=o\left(c\right)

see Wagner (1990). … ►By combination of the foregoing results of this subsection with the linear transformations of §15.8(i) and the connection formulas of §15.10(ii), similar asymptotic approximations for

F\left(a+e_{1}\lambda,b+e_{2}\lambda;c+e_{3}\lambda;z\right)

can be obtained with

e_{j}=\pm 1

0

j=1,2,3

. …

Gauss 2F1(-1) sum

1—10 of 818 matching pages

1: 32.10 Special Function Solutions

2: 16.10 Expansions in Series of F q p Functions

§16.10 Expansions in Series of F q p Functions

3: 5.5 Functional Relations

Gauss’s Multiplication Formula

4: 15.4 Special Cases

§15.4(ii) Argument Unity

Dougall’s Bilateral Sum

§15.4(iii) Other Arguments

5: 15.5 Derivatives and Contiguous Functions

6: 18.33 Polynomials Orthogonal on the Unit Circle

7: 10.59 Integrals

8: 19.21 Connection Formulas

9: 4.24 Inverse Trigonometric Functions: Further Properties

10: 15.12 Asymptotic Approximations

2: 16.10 Expansions in Series of ${{}_{p}F_{q}}$ Functions

§16.10 Expansions in Series of ${{}_{p}F_{q}}$ Functions