Gauss%E2%80%93Jacobi%20formula

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►The classical OP’s comprise the Jacobi, Laguerre and Hermite polynomials. … ►However, most of these formulas can be obtained by specialization of formulas for Jacobi polynomials, via (18.7.4)–(18.7.6). … ►Formula (18.3.1) can be understood as a Gauss-Chebyshev quadrature, see (3.5.22), (3.5.23). … ►

Jacobi on Other Intervals

…

3: 15.5 Derivatives and Contiguous Functions

… ►

§15.5(i) Differentiation Formulas

… ►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

. … ►

15.5.20

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

…

4: 16.12 Products

… ►

16.12.1

{{}_{0}F_{1}}\left(-;a;z\right){{}_{0}F_{1}}\left(-;b;z\right)={{}_{2}F_{3}}% \left({\frac{1}{2}(a+b),\frac{1}{2}(a+b-1)\atop a,b,a+b-1};4z\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, $z$ : 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:: (10.8.3), §10.8, §10.8, §16.12, Erratum (V1.0.17) for Section 10.8
Permalink:: http://dlmf.nist.gov/16.12.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §16.12 and Ch.16

►The following formula is often referred to as Clausen’s formula ►

16.12.2

\left({{}_{2}F_{1}}\left({a,b\atop a+b+\frac{1}{2}};z\right)\right)^{2}={{}_{3% }F_{2}}\left({2a,2b,a+b\atop a+b+\frac{1}{2},2a+2b};z\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, ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{a_{1},\dots,a_{p}};\NVar{b_{1},\dots,b_% {q}};\NVar{z}\right)$ 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, $z$ : 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, §17.9(iii), Erratum (V1.1.12) for Subsection 17.9(iii)
Permalink:: http://dlmf.nist.gov/16.12.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §16.12 and Ch.16

… ►

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

…

5: 15.4 Special Cases

… ►

15.4.1

F\left(1,1;2;z\right)=-z^{-1}\ln\left(1-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, $\ln\NVar{z}$ : principal branch of logarithm function and $z$ : complex variable
Referenced by:: §15.4(i)
Permalink:: http://dlmf.nist.gov/15.4.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §15.4(i), §15.4 and Ch.15

►

15.4.2

F\left(\tfrac{1}{2},1;\tfrac{3}{2};z^{2}\right)=\frac{1}{2z}\ln\left(\frac{1+z% }{1-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, $\ln\NVar{z}$ : principal branch of logarithm function and $z$ : complex variable
Permalink:: http://dlmf.nist.gov/15.4.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §15.4(i), §15.4 and Ch.15

►

15.4.3

F\left(\tfrac{1}{2},1;\tfrac{3}{2};-z^{2}\right)=z^{-1}\operatorname{arctan}z,

ⓘ

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, $\operatorname{arctan}\NVar{z}$ : arctangent function and $z$ : complex variable
Permalink:: http://dlmf.nist.gov/15.4.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §15.4(i), §15.4 and Ch.15

… ►

F\left(a,b;a;z\right)=(1-z)^{-b},

►

F\left(a,b;b;z\right)=(1-z)^{-a},

…

6: 15.2 Definitions and Analytical Properties

… ►

§15.2(i) Gauss Series

►The hypergeometric function

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

is defined by the Gauss series … ►On the circle of convergence,

|z|=1

, the Gauss series: … ►The same properties hold for

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

, except that as a function of

c

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

in general has poles at

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

. … ►Formula (15.4.6) reads

F\left(a,b;a;z\right)=(1-z)^{-b}

. …

7: 18.5 Explicit Representations

… ►

§18.5(ii) Rodrigues Formulas

… ►Related formula: … ►For the definitions of

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

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

, and

{{}_{2}F_{0}}

see §16.2. ►

Jacobi

… ►For corresponding formulas for Chebyshev, Legendre, and the Hermite

\mathit{He}_{n}

polynomials apply (18.7.3)–(18.7.6), (18.7.9), and (18.7.11). …

8: 20.11 Generalizations and Analogs

… ►

§20.11(i) Gauss Sum

►For relatively prime integers

m, n

with

n>0

and

m n

even, the Gauss sum

G(m,n)

is defined by … … ► … ►Similar identities can be constructed for

{{}_{2}F_{1}}\left(\tfrac{1}{3},\tfrac{2}{3};1;k^{2}\right)

{{}_{2}F_{1}}\left(\tfrac{1}{4},\tfrac{3}{4};1;k^{2}\right)

, and

{{}_{2}F_{1}}\left(\tfrac{1}{6},\tfrac{5}{6};1;k^{2}\right)

. …

9: 16.3 Derivatives and Contiguous Functions

… ►

§16.3(i) Differentiation Formulas

►

16.3.1

\frac{{\mathrm{d}}^{n}}{{\mathrm{d}z}^{n}}{{}_{p}F_{q}}\left({a_{1},\dots,a_{p% }\atop b_{1},\dots,b_{q}};z\right)=\frac{{\left(\mathbf{a}\right)_{n}}}{{\left% (\mathbf{b}\right)_{n}}}{{}_{p}F_{q}}\left({a_{1}+n,\dots,a_{p}+n\atop b_{1}+n% ,\dots,b_{q}+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, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $p$ : 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/16.3.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §16.3(i), §16.3 and Ch.16

… ►

16.3.3

\left(z\frac{\mathrm{d}}{\mathrm{d}z}z\right)^{n}\left(z^{\gamma-1}{{}_{p+1}F_% {q}}\left({\gamma,a_{1},\dots,a_{p}\atop b_{1},\dots,b_{q}};z\right)\right)={% \left(\gamma\right)_{n}}z^{\gamma+n-1}{{}_{p+1}F_{q}}\left({\gamma+n,a_{1},% \dots,a_{p}\atop b_{1},\dots,b_{q}};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, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $p$ : 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/16.3.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §16.3(i), §16.3 and Ch.16

… ►Two generalized hypergeometric functions

{{}_{p}F_{q}}\left(\mathbf{a};\mathbf{b};z\right)

are (generalized) contiguous if they have the same pair of values of

p

and

q

, and corresponding parameters differ by integers. … ►

16.3.6

z{{}_{0}F_{1}}\left(-;b+1;z\right)+b(b-1){{}_{0}F_{1}}\left(-;b;z\right)-b(b-1% ){{}_{0}F_{1}}\left(-;b-1;z\right)=0,

ⓘ

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, $z$ : complex variable and $b,b_{1},\ldots,b_{q}$ : real or complex parameters
Permalink:: http://dlmf.nist.gov/16.3.E6
Encodings:: pMML, png, TeX
See also:: Annotations for §16.3(ii), §16.3 and Ch.16

…

10: 15.8 Transformations of Variable

… ►The transformation formulas between two hypergeometric functions in Group 2, or two hypergeometric functions in Group 3, are the linear transformations (15.8.1). … ►

15.8.13

F\left({a,b\atop 2b};z\right)=\left(1-\tfrac{1}{2}z\right)^{-a}F\left({\tfrac{% 1}{2}a,\tfrac{1}{2}a+\tfrac{1}{2}\atop b+\tfrac{1}{2}};\left(\frac{z}{2-z}% \right)^{2}\right),

|\operatorname{ph}\left(1-z\right)|<\pi

ⓘ

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, $\pi$ : the ratio of the circumference of a circle to its diameter, $\operatorname{ph}$ : phase, $z$ : complex variable, $a$ : real or complex parameter and $b$ : real or complex parameter
Referenced by:: §15.8(iv)
Permalink:: http://dlmf.nist.gov/15.8.E13
Encodings:: pMML, png, TeX
See also:: Annotations for §15.8(iii), §15.8(iii), §15.8 and Ch.15

►

15.8.14

F\left({a,b\atop 2b};z\right)=\left(1-z\right)^{-\ifrac{a}{2}}F\left({\tfrac{1% }{2}a,b-\tfrac{1}{2}a\atop b+\tfrac{1}{2}};\frac{z^{2}}{4z-4}\right),

|\operatorname{ph}\left(1-z\right)|<\pi

ⓘ

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, $\pi$ : the ratio of the circumference of a circle to its diameter, $\operatorname{ph}$ : phase, $z$ : complex variable, $a$ : real or complex parameter and $b$ : real or complex parameter
Permalink:: http://dlmf.nist.gov/15.8.E14
Encodings:: pMML, png, TeX
See also:: Annotations for §15.8(iii), §15.8(iii), §15.8 and Ch.15

… ►

15.8.15

F\left({a,b\atop a-b+1};z\right)=(1+z)^{-a}F\left({\frac{1}{2}a,\frac{1}{2}a+% \frac{1}{2}\atop a-b+1};\frac{4z}{(1+z)^{2}}\right),

|z|<1

ⓘ

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 and $b$ : real or complex parameter
Referenced by:: §15.4(i), §15.8(iv)
Permalink:: http://dlmf.nist.gov/15.8.E15
Encodings:: pMML, png, TeX
See also:: Annotations for §15.8(iii), §15.8(iii), §15.8 and Ch.15

►

15.8.16

F\left({a,b\atop a-b+1};z\right)=(1-z)^{-a}F\left({\frac{1}{2}a,\frac{1}{2}a-b% +\frac{1}{2}\atop a-b+1};\frac{-4z}{(1-z)^{2}}\right),

|z|<1

ⓘ

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 and $b$ : real or complex parameter
Permalink:: http://dlmf.nist.gov/15.8.E16
Encodings:: pMML, png, TeX
See also:: Annotations for §15.8(iii), §15.8(iii), §15.8 and Ch.15

…

Gauss%E2%80%93Jacobi%20formula

1—10 of 435 matching pages

1: 22.16 Related Functions

§22.16(i) Jacobi’s Amplitude ( $\operatorname{am}$ ) Function

§22.16(ii) Jacobi’s Epsilon Function

Quasi-Addition and Quasi-Periodic Formulas

§22.16(iii) Jacobi’s Zeta Function

Properties

2: 18.3 Definitions

§18.3 Definitions

Jacobi on Other Intervals

3: 15.5 Derivatives and Contiguous Functions

§15.5(i) Differentiation Formulas

4: 16.12 Products

5: 15.4 Special Cases

6: 15.2 Definitions and Analytical Properties

§15.2(i) Gauss Series

7: 18.5 Explicit Representations

§18.5(ii) Rodrigues Formulas

Jacobi

8: 20.11 Generalizations and Analogs

§20.11(i) Gauss Sum

9: 16.3 Derivatives and Contiguous Functions

§16.3(i) Differentiation Formulas

10: 15.8 Transformations of Variable

Gauss%E2%80%93Jacobi%20formula

1—10 of 435 matching pages

1: 22.16 Related Functions

§22.16(i) Jacobi’s Amplitude ( am ) Function

§22.16(ii) Jacobi’s Epsilon Function

Quasi-Addition and Quasi-Periodic Formulas

§22.16(iii) Jacobi’s Zeta Function

Properties

2: 18.3 Definitions

§18.3 Definitions

Jacobi on Other Intervals

3: 15.5 Derivatives and Contiguous Functions

§15.5(i) Differentiation Formulas

4: 16.12 Products

5: 15.4 Special Cases

6: 15.2 Definitions and Analytical Properties

§15.2(i) Gauss Series

7: 18.5 Explicit Representations

§18.5(ii) Rodrigues Formulas

Jacobi

8: 20.11 Generalizations and Analogs

§20.11(i) Gauss Sum

9: 16.3 Derivatives and Contiguous Functions

§16.3(i) Differentiation Formulas

10: 15.8 Transformations of Variable

§22.16(i) Jacobi’s Amplitude ( $\operatorname{am}$ ) Function