Gauss sum

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21—30 of 68 matching pages

21: 16.12 Products

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16.12.3

\left({{}_{2}F_{1}}\left({a,b\atop c};z\right)\right)^{2}=\sum_{k=0}^{\infty}% \frac{{\left(2a\right)_{k}}{\left(2b\right)_{k}}{\left(c-\frac{1}{2}\right)_{k% }}}{{\left(c\right)_{k}}{\left(2c-1\right)_{k}}k!}{{}_{4}F_{3}}\left({-\frac{1% }{2}k,\frac{1}{2}(1-k),a+b-c+\frac{1}{2},\frac{1}{2}\atop a+\frac{1}{2},b+% \frac{1}{2},\frac{3}{2}-k-c};1\right)z^{k},

|z|<1

ⓘ

Symbols:: ${{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ : $=F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ notation for Gauss’ hypergeometric function, ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{a_{1},\dots,a_{p}};\NVar{b_{1},\dots,b_% {q}};\NVar{z}\right)$ 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 $n!$ ), $z$ : complex variable, $a,a_{1},\ldots,a_{p}$ : real or complex parameters and $b,b_{1},\ldots,b_{q}$ : real or complex parameters
Referenced by:: §16.12
Permalink:: http://dlmf.nist.gov/16.12.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §16.12 and Ch.16

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22: 16.17 Definition

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16.17.2

{G^{m,n}_{p,q}}\left(z;{a_{1},\dots,a_{p}\atop b_{1},\dots,b_{q}}\right)=\sum_% {k=1}^{m}A_{p,q,k}^{m,n}(z){{}_{p}F_{q-1}}\left({1+b_{k}-a_{1},\dots,1+b_{k}-a% _{p}\atop 1+b_{k}-b_{1},\ldots*\dots,1+b_{k}-b_{q}};(-1)^{p-m-n}z\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, ${G^{\NVar{m},\NVar{n}}_{\NVar{p},\NVar{q}}}\left(\NVar{z};{\NVar{a_{1},\dots,a% _{p}}\atop\NVar{b_{1},\dots,b_{q}}}\right)$ or ${G^{\NVar{m},\NVar{n}}_{\NVar{p},\NVar{q}}}\left(\NVar{z};\NVar{\mathbf{a}};% \NVar{\mathbf{b}}\right)$ : Meijer $G$ -function, $p$ : nonnegative integer, $q$ : nonnegative integer, $z$ : complex variable, $a,a_{1},\ldots,a_{p}$ : real or complex parameters, $b,b_{1},\ldots,b_{q}$ : real or complex parameters and $A_{p,q}^{m,n}(z)$ : coefficient
Permalink:: http://dlmf.nist.gov/16.17.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §16.17 and Ch.16

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23: 2.10 Sums and Sequences

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2.10.19

{{}_{0}F_{2}}\left(-;1,1;x\right)=\sum_{j=0}^{\infty}\frac{x^{j}}{(j!)^{3}}.

ⓘ

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 and $!$ : factorial (as in $n!$ )
Permalink:: http://dlmf.nist.gov/2.10.E19
Encodings:: pMML, png, TeX
See also:: Annotations for §2.10(iii), §2.10(iii), §2.10 and Ch.2

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24: 35.6 Confluent Hypergeometric Functions of Matrix Argument

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35.6.1

{{}_{1}F_{1}}\left({a\atop b};\mathbf{T}\right)=\sum_{k=0}^{\infty}\frac{1}{k!% }\sum_{|\kappa|=k}\frac{{\left[a\right]_{\kappa}}}{{\left[b\right]_{\kappa}}}Z% _{\kappa}\left(\mathbf{T}\right).

ⓘ

Symbols:: ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{a_{1},\dots,a_{p}};\NVar{b_{1},\dots,b_% {q}};\NVar{\mathbf{T}}\right)$ or ${{}_{\NVar{p}}F_{\NVar{q}}}\left({\NVar{a_{1},\dots,a_{p}}\atop\NVar{b_{1},% \dots,b_{q}}};\NVar{\mathbf{T}}\right)$ : generalized hypergeometric function of matrix argument, $!$ : factorial (as in $n!$ ), ${\left[\NVar{a}\right]_{\NVar{\kappa}}}$ : partitional shifted factorial, $Z_{\NVar{\kappa}}\left(\NVar{\mathbf{T}}\right)$ : zonal polynomial, $a$ : complex variable, $\mathbf{T}$ : real symmetric $m\times m$ matrix, $b$ : complex variable, $k$ : nonnegative integer and $\kappa$ : vector of nonnegative integers
Permalink:: http://dlmf.nist.gov/35.6.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §35.6(i), §35.6 and Ch.35

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25: 35.7 Gaussian Hypergeometric Function of Matrix Argument

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35.7.1

{{{}_{2}F_{1}}\left({a,b\atop c};\mathbf{T}\right)=\sum_{k=0}^{\infty}\frac{1}% {k!}\sum_{|\kappa|=k}\frac{{\left[a\right]_{\kappa}}{\left[b\right]_{\kappa}}}% {{\left[c\right]_{\kappa}}}Z_{\kappa}\left(\mathbf{T}\right)},

-c+\frac{1}{2}(j+1)\notin\mathbb{N}

1\leq j\leq m

;

\|\mathbf{T}\|<1

ⓘ

Symbols:: ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{a_{1},\dots,a_{p}};\NVar{b_{1},\dots,b_% {q}};\NVar{\mathbf{T}}\right)$ or ${{}_{\NVar{p}}F_{\NVar{q}}}\left({\NVar{a_{1},\dots,a_{p}}\atop\NVar{b_{1},% \dots,b_{q}}};\NVar{\mathbf{T}}\right)$ : generalized hypergeometric function of matrix argument, $!$ : factorial (as in $n!$ ), $\notin$ : not an element of, $\mathbb{N}$ : set of all positive integers, ${\left[\NVar{a}\right]_{\NVar{\kappa}}}$ : partitional shifted factorial, $Z_{\NVar{\kappa}}\left(\NVar{\mathbf{T}}\right)$ : zonal polynomial, $a$ : complex variable, $\mathbf{T}$ : real symmetric $m\times m$ matrix, $b$ : complex variable, $\|\mathbf{T}\|$ : spectral norm of $\mathbf{T}$ , $c$ : complex variable, $j$ : nonnegative integer, $k$ : nonnegative integer, $m$ : positive integer and $\kappa$ : vector of nonnegative integers
Permalink:: http://dlmf.nist.gov/35.7.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §35.7(i), §35.7 and Ch.35

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35.7.3

{{}_{2}F_{1}}\left({a,b\atop c};\begin{bmatrix}t_{1}&0\\ 0&t_{2}\end{bmatrix}\right)=\sum_{k=0}^{\infty}\frac{{\left(a\right)_{k}}{% \left(c-a\right)_{k}}{\left(b\right)_{k}}{\left(c-b\right)_{k}}}{k!\,{\left(c% \right)_{2k}}{\left(c-\tfrac{1}{2}\right)_{k}}}\*(t_{1}t_{2})^{k}{{}_{2}F_{1}}% \left({a+k,b+k\atop c+2k};t_{1}+t_{2}-t_{1}t_{2}\right).

ⓘ

Symbols:: ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{a_{1},\dots,a_{p}};\NVar{b_{1},\dots,b_% {q}};\NVar{\mathbf{T}}\right)$ or ${{}_{\NVar{p}}F_{\NVar{q}}}\left({\NVar{a_{1},\dots,a_{p}}\atop\NVar{b_{1},% \dots,b_{q}}};\NVar{\mathbf{T}}\right)$ : generalized hypergeometric function of matrix argument, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $!$ : factorial (as in $n!$ ), $a$ : complex variable, $t_{j}$ : eigenvalues of $\mathbf{T}$ , $b$ : complex variable, $c$ : complex variable and $k$ : nonnegative integer
Referenced by:: Erratum (V1.0.25) for Equation (35.7.3)
Permalink:: http://dlmf.nist.gov/35.7.E3
Encodings:: pMML, png, TeX
Notational change (effective with 1.0.25):: Originally the matrix in the argument of the Gaussian hypergeometric function of matrix argument ${{}_{2}F_{1}}$ was written with round brackets. This matrix has been rewritten with square brackets to be consistent with the rest of the DLMF.
See also:: Annotations for §35.7(ii), §35.7(ii), §35.7 and Ch.35

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26: 18.26 Wilson Class: Continued

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18.26.18

{{}_{2}F_{1}}\left({a+\mathrm{i}y,d+\mathrm{i}y\atop a+d};z\right){{}_{2}F_{1}% }\left({b-\mathrm{i}y,c-\mathrm{i}y\atop b+c};z\right)=\sum_{n=0}^{\infty}% \frac{W_{n}\left(y^{2};a,b,c,d\right)}{{\left(a+d\right)_{n}}{\left(b+c\right)% _{n}}n!}z^{n},

|z|<1

ⓘ

Symbols:: ${{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ : $=F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ notation for Gauss’ hypergeometric function, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $W_{\NVar{n}}\left(\NVar{x};\NVar{a},\NVar{b},\NVar{c},\NVar{d}\right)$ : Wilson polynomial, $!$ : factorial (as in $n!$ ), $\mathrm{i}$ : imaginary unit, $y$ : real variable, $z$ : complex variable and $n$ : nonnegative integer
Referenced by:: §18.23, §18.26(iv)
Permalink:: http://dlmf.nist.gov/18.26.E18
Encodings:: pMML, png, TeX
See also:: Annotations for §18.26(iv), §18.26(iv), §18.26 and Ch.18

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18.26.19

(1-z)^{-c+\mathrm{i}y}{{}_{2}F_{1}}\left({a+\mathrm{i}y,b+\mathrm{i}y\atop a+b% };z\right)=\sum_{n=0}^{\infty}\frac{S_{n}\left(y^{2};a,b,c\right)}{{\left(a+b% \right)_{n}}n!}z^{n},

|z|<1

ⓘ

Symbols:: ${{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ : $=F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ notation for Gauss’ hypergeometric function, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $S_{\NVar{n}}\left(\NVar{x};\NVar{a},\NVar{b},\NVar{c}\right)$ : continuous dual Hahn polynomial, $!$ : factorial (as in $n!$ ), $\mathrm{i}$ : imaginary unit, $y$ : real variable, $z$ : complex variable and $n$ : nonnegative integer
Referenced by:: §18.26(iv)
Permalink:: http://dlmf.nist.gov/18.26.E19
Encodings:: pMML, png, TeX
See also:: Annotations for §18.26(iv), §18.26(iv), §18.26 and Ch.18

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18.26.20

{{}_{2}F_{1}}\left({-y,-y+\beta-\gamma\atop\beta+\delta+1};z\right){{}_{2}F_{1% }}\left({y-N,y+\gamma+1\atop-\delta-N};z\right)=\sum_{n=0}^{N}\frac{{\left(-N% \right)_{n}}{\left(\gamma+1\right)_{n}}}{{\left(-\delta-N\right)_{n}}n!}R_{n}% \left(y(y+\gamma+\delta+1);-N-1,\beta,\gamma,\delta\right)z^{n}.

ⓘ

Symbols:: ${{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ : $=F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ notation for Gauss’ hypergeometric function, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $R_{\NVar{n}}\left(\NVar{x};\NVar{\alpha},\NVar{\beta},\NVar{\gamma},\NVar{% \delta}\right)$ : Racah polynomial, $!$ : factorial (as in $n!$ ), $y$ : real variable, $N$ : positive integer, $\delta$ : arbitrary small positive constant, $z$ : complex variable and $n$ : nonnegative integer
Referenced by:: §18.26(iv)
Permalink:: http://dlmf.nist.gov/18.26.E20
Encodings:: pMML, png, TeX
See also:: Annotations for §18.26(iv), §18.26(iv), §18.26 and Ch.18

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18.26.21

(1-z)^{y}{{}_{2}F_{1}}\left({y-N,y+\gamma+1\atop-\delta-N};z\right)=\sum_{n=0}% ^{N}\frac{{\left(\gamma+1\right)_{n}}{\left(-N\right)_{n}}}{{\left(-\delta-N% \right)_{n}}n!}\*R_{n}\left(y(y+\gamma+\delta+1);\gamma,\delta,N\right)z^{n}.

ⓘ

Symbols:: ${{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ : $=F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ notation for Gauss’ hypergeometric function, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $R_{\NVar{n}}\left(\NVar{x};\NVar{\gamma},\NVar{\delta},\NVar{N}\right)$ : dual Hahn polynomial, $!$ : factorial (as in $n!$ ), $y$ : real variable, $N$ : positive integer, $\delta$ : arbitrary small positive constant, $z$ : complex variable and $n$ : nonnegative integer
Referenced by:: §18.26(iv)
Permalink:: http://dlmf.nist.gov/18.26.E21
Encodings:: pMML, png, TeX
See also:: Annotations for §18.26(iv), §18.26(iv), §18.26 and Ch.18

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27: 18.23 Hahn Class: Generating Functions

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18.23.2

{{}_{2}F_{0}}\left({-x,-x+\beta+N+1\atop-};-z\right)\*{{}_{2}F_{0}}\left({x-N,% x+\alpha+1\atop-};z\right)=\sum_{n=0}^{N}\frac{{\left(-N\right)_{n}}{\left(% \alpha+1\right)_{n}}}{n!}Q_{n}\left(x;\alpha,\beta,N\right)z^{n},

x=0,1,\dots,N

ⓘ

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, $Q_{\NVar{n}}\left(\NVar{x};\NVar{\alpha},\NVar{\beta},\NVar{N}\right)$ : Hahn polynomial, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $!$ : factorial (as in $n!$ ), $N$ : positive integer, $z$ : complex variable, $n$ : nonnegative integer and $x$ : real variable
Referenced by:: §18.23
Permalink:: http://dlmf.nist.gov/18.23.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §18.23, §18.23 and Ch.18

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28: 19.5 Maclaurin and Related Expansions

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19.5.1

K\left(k\right)=\frac{\pi}{2}\sum_{m=0}^{\infty}\frac{{\left(\tfrac{1}{2}% \right)_{m}}{\left(\tfrac{1}{2}\right)_{m}}}{m!\;m!}k^{2m}=\frac{\pi}{2}{{}_{2% }F_{1}}\left({\tfrac{1}{2},\tfrac{1}{2}\atop 1};k^{2}\right),

ⓘ

Symbols:: ${{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ : $=F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ notation for Gauss’ hypergeometric function, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $\pi$ : the ratio of the circumference of a circle to its diameter, $K\left(\NVar{k}\right)$ : Legendre’s complete elliptic integral of the first kind, $!$ : factorial (as in $n!$ ), $m$ : nonnegative integer and $k$ : real or complex modulus
Source:: Carlson (1961b, (2.4))
Referenced by:: §19.36(iv), §19.5
Permalink:: http://dlmf.nist.gov/19.5.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §19.5 and Ch.19

… ►

19.5.2

E\left(k\right)=\frac{\pi}{2}\sum_{m=0}^{\infty}\frac{{\left(-\tfrac{1}{2}% \right)_{m}}{\left(\tfrac{1}{2}\right)_{m}}}{m!\;m!}k^{2m}=\frac{\pi}{2}{{}_{2% }F_{1}}\left({-\tfrac{1}{2},\tfrac{1}{2}\atop 1};k^{2}\right),

ⓘ

Symbols:: ${{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ : $=F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ notation for Gauss’ hypergeometric function, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $\pi$ : the ratio of the circumference of a circle to its diameter, $E\left(\NVar{k}\right)$ : Legendre’s complete elliptic integral of the second kind, $!$ : factorial (as in $n!$ ), $m$ : nonnegative integer and $k$ : real or complex modulus
Source:: Carlson (1961b, (2.5))
Referenced by:: §19.36(iv)
Permalink:: http://dlmf.nist.gov/19.5.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §19.5 and Ch.19

►

19.5.3

D\left(k\right)=\frac{\pi}{4}\sum_{m=0}^{\infty}\frac{{\left(\tfrac{3}{2}% \right)_{m}}{\left(\tfrac{1}{2}\right)_{m}}}{(m+1)!\;m!}k^{2m}=\frac{\pi}{4}{{% }_{2}F_{1}}\left({\tfrac{3}{2},\tfrac{1}{2}\atop 2};k^{2}\right),

ⓘ

Symbols:: ${{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ : $=F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ notation for Gauss’ hypergeometric function, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $\pi$ : the ratio of the circumference of a circle to its diameter, $D\left(\NVar{k}\right)$ : complete elliptic integral of Legendre’s type, $!$ : factorial (as in $n!$ ), $m$ : nonnegative integer and $k$ : real or complex modulus
Permalink:: http://dlmf.nist.gov/19.5.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §19.5 and Ch.19

… ►

19.5.4_1

F\left(\phi,k\right)=\sum_{m=0}^{\infty}\frac{{\left(\tfrac{1}{2}\right)_{m}}{% \sin}^{2m+1}\phi}{(2m+1)m!}{{}_{2}F_{1}}\left({m+\tfrac{1}{2},\tfrac{1}{2}% \atop m+\tfrac{3}{2}};{\sin}^{2}{\phi}\right)k^{2m}=\sin\phi\,{F_{1}}\left(% \tfrac{1}{2};\tfrac{1}{2},\tfrac{1}{2};\tfrac{3}{2};{\sin}^{2}\phi,k^{2}{\sin}% ^{2}\phi\right),

ⓘ

Symbols:: ${F_{1}}\left(\NVar{\alpha};\NVar{\beta},\NVar{\beta^{\prime}};\NVar{\gamma};% \NVar{x},\NVar{y}\right)$ : first Appell function, ${{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ : $=F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ notation for Gauss’ hypergeometric function, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $F\left(\NVar{\phi},\NVar{k}\right)$ : Legendre’s incomplete elliptic integral of the first kind, $!$ : factorial (as in $n!$ ), $\sin\NVar{z}$ : sine function, $m$ : nonnegative integer, $\phi$ : real or complex argument and $k$ : real or complex modulus
Source:: Carlson (1961b, (2.8))
Referenced by:: §19.5, Erratum (V1.1.2) for Additions
Permalink:: http://dlmf.nist.gov/19.5.E4_1
Encodings:: pMML, png, TeX
Addition (effective with 1.1.2):: This equation was added.
See also:: Annotations for §19.5 and Ch.19

►

19.5.4_2

E\left(\phi,k\right)=\sum_{m=0}^{\infty}\frac{{\left(-\tfrac{1}{2}\right)_{m}}% {\sin}^{2m+1}\phi}{(2m+1)m!}{{}_{2}F_{1}}\left({m+\tfrac{1}{2},\tfrac{1}{2}% \atop m+\tfrac{3}{2}};{\sin}^{2}{\phi}\right)k^{2m}=\sin\phi\,{F_{1}}\left(% \tfrac{1}{2};\tfrac{1}{2},-\tfrac{1}{2};\tfrac{3}{2};{\sin}^{2}\phi,k^{2}{\sin% }^{2}\phi\right),

ⓘ

Symbols:: ${F_{1}}\left(\NVar{\alpha};\NVar{\beta},\NVar{\beta^{\prime}};\NVar{\gamma};% \NVar{x},\NVar{y}\right)$ : first Appell function, ${{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ : $=F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ notation for Gauss’ hypergeometric function, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $E\left(\NVar{\phi},\NVar{k}\right)$ : Legendre’s incomplete elliptic integral of the second kind, $!$ : factorial (as in $n!$ ), $\sin\NVar{z}$ : sine function, $m$ : nonnegative integer, $\phi$ : real or complex argument and $k$ : real or complex modulus
Source:: Carlson (1961b, (2.9))
Referenced by:: §19.5, Erratum (V1.1.2) for Additions
Permalink:: http://dlmf.nist.gov/19.5.E4_2
Encodings:: pMML, png, TeX
Addition (effective with 1.1.2):: This equation was added.
See also:: Annotations for §19.5 and Ch.19

…

29: 19.19 Taylor and Related Series

… ►where the summation extends over all nonnegative integers

m_{1},\dots,m_{n}

whose sum is

N

. … ►If

n=2

, then (19.19.3) is a Gauss hypergeometric series (see (19.25.43) and (15.2.1)). … ►

19.19.4

\prod_{j=1}^{n}(1+tz_{j})=\sum_{s=0}^{n}t^{s}E_{s}(\mathbf{z}),

ⓘ

Defines:: $E_{s}(\mathbf{z})$ : symmetric function (locally)
Symbols:: $n$ : nonnegative integer
Referenced by:: §19.36(i)
Permalink:: http://dlmf.nist.gov/19.19.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §19.19 and Ch.19

… ►where

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

and the summation extends over all nonnegative integers

m_{1},\dots,m_{n}

such that

\sum_{j=1}^{n}jm_{j}=N

. … ►

A=\frac{1}{n}\sum_{j=1}^{n}z_{j},

…

30: 18.5 Explicit Representations

… ►

18.5.10

C^{(\lambda)}_{n}\left(x\right)=\sum_{\ell=0}^{\left\lfloor n/2\right\rfloor}% \frac{(-1)^{\ell}{\left(\lambda\right)_{n-\ell}}}{\ell!\;(n-2\ell)!}(2x)^{n-2% \ell}=(2x)^{n}\frac{{\left(\lambda\right)_{n}}}{n!}{{}_{2}F_{1}}\left({-\tfrac% {1}{2}n,-\tfrac{1}{2}n+\tfrac{1}{2}\atop 1-\lambda-n};\frac{1}{x^{2}}\right),

ⓘ

Symbols:: ${{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ : $=F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ notation for Gauss’ hypergeometric function, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $!$ : factorial (as in $n!$ ), $\left\lfloor\NVar{x}\right\rfloor$ : floor of $x$ , $C^{(\NVar{\lambda})}_{\NVar{n}}\left(\NVar{x}\right)$ : ultraspherical (or Gegenbauer) polynomial, $\ell$ : nonnegative integer, $n$ : nonnegative integer and $x$ : real variable
A&S Ref:: 22.3.4
Referenced by:: §18.17(vii), §18.5(iii), §18.6(ii)
Permalink:: http://dlmf.nist.gov/18.5.E10
Encodings:: pMML, png, TeX
See also:: Annotations for §18.5(iii), §18.5(iii), §18.5 and Ch.18

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18.5.11

C^{(\lambda)}_{n}\left(\cos\theta\right)=\sum_{\ell=0}^{n}\frac{{\left(\lambda% \right)_{\ell}}{\left(\lambda\right)_{n-\ell}}}{\ell!\;(n-\ell)!}\cos\left((n-% 2\ell)\theta\right)={\mathrm{e}}^{\mathrm{i}n\theta}\frac{{\left(\lambda\right% )_{n}}}{n!}{{}_{2}F_{1}}\left({-n,\lambda\atop 1-\lambda-n};{\mathrm{e}}^{-2% \mathrm{i}\theta}\right).

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Symbols:: ${{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ : $=F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ notation for Gauss’ hypergeometric function, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $\cos\NVar{z}$ : cosine function, $\mathrm{e}$ : base of natural logarithm, $!$ : factorial (as in $n!$ ), $\mathrm{i}$ : imaginary unit, $C^{(\NVar{\lambda})}_{\NVar{n}}\left(\NVar{x}\right)$ : ultraspherical (or Gegenbauer) polynomial, $\ell$ : nonnegative integer and $n$ : nonnegative integer
A&S Ref:: 22.3.12
Referenced by:: §18.18(iv), §18.5(iii)
Permalink:: http://dlmf.nist.gov/18.5.E11
Encodings:: pMML, png, TeX
See also:: Annotations for §18.5(iii), §18.5(iii), §18.5 and Ch.18

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18.5.11_1

T_{n}\left(x\right)=\tfrac{1}{2}n\sum_{\ell=0}^{\left\lfloor n/2\right\rfloor}% \frac{(-1)^{\ell}(n-\ell-1)!}{\ell!\;(n-2\ell)!}(2x)^{n-2\ell}=2^{n-1}x^{n}{{}% _{2}F_{1}}\left({-\tfrac{1}{2}n,-\tfrac{1}{2}n+\tfrac{1}{2}\atop 1-n};\frac{1}% {x^{2}}\right),

n\geq 1

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Symbols:: $T_{\NVar{n}}\left(\NVar{x}\right)$ : Chebyshev polynomial of the first kind, ${{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ : $=F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ notation for Gauss’ hypergeometric function, $!$ : factorial (as in $n!$ ), $\left\lfloor\NVar{x}\right\rfloor$ : floor of $x$ , $\ell$ : nonnegative integer, $n$ : nonnegative integer and $x$ : real variable
Source:: Erdélyi et al. (1953b, 10.11(22))
Referenced by:: §18.5(iii), Erratum (V1.2.0) Section 18.5
Permalink:: http://dlmf.nist.gov/18.5.E11_1
Encodings:: pMML, png, TeX
Addition (effective with 1.2.0):: This equation was added.
See also:: Annotations for §18.5(iii), §18.5(iii), §18.5 and Ch.18

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18.5.11_3

U_{n}\left(x\right)=\sum_{\ell=0}^{\left\lfloor n/2\right\rfloor}\frac{(-1)^{% \ell}(n-\ell)!}{\ell!\;(n-2\ell)!}(2x)^{n-2\ell}=\left(2x\right)^{n}{{}_{2}F_{% 1}}\left({-\tfrac{1}{2}n,-\tfrac{1}{2}n+\tfrac{1}{2}\atop-n};\frac{1}{x^{2}}% \right),

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Symbols:: $U_{\NVar{n}}\left(\NVar{x}\right)$ : Chebyshev polynomial of the second kind, ${{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ : $=F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ notation for Gauss’ hypergeometric function, $!$ : factorial (as in $n!$ ), $\left\lfloor\NVar{x}\right\rfloor$ : floor of $x$ , $\ell$ : nonnegative integer, $n$ : nonnegative integer and $x$ : real variable
Source:: Erdélyi et al. (1953b, 10.11(23))
Permalink:: http://dlmf.nist.gov/18.5.E11_3
Encodings:: pMML, png, TeX
Addition (effective with 1.2.0):: This equation was added.
See also:: Annotations for §18.5(iii), §18.5(iii), §18.5 and Ch.18

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18.5.13

H_{n}\left(x\right)=n!\sum_{\ell=0}^{\left\lfloor n/2\right\rfloor}\frac{(-1)^% {\ell}(2x)^{n-2\ell}}{\ell!\;(n-2\ell)!}=(2x)^{n}{{}_{2}F_{0}}\left({-\tfrac{1% }{2}n,-\tfrac{1}{2}n+\tfrac{1}{2}\atop-};-\frac{1}{x^{2}}\right).

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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, $H_{\NVar{n}}\left(\NVar{x}\right)$ : Hermite polynomial, $!$ : factorial (as in $n!$ ), $\left\lfloor\NVar{x}\right\rfloor$ : floor of $x$ , $\ell$ : nonnegative integer, $n$ : nonnegative integer and $x$ : real variable
A&S Ref:: 22.3.10
Referenced by:: §18.17(v), §18.17(vii), §18.5(iii)
Permalink:: http://dlmf.nist.gov/18.5.E13
Encodings:: pMML, png, TeX
See also:: Annotations for §18.5(iii), §18.5(iii), §18.5 and Ch.18

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