15 Hypergeometric Function Properties 15.8 Transformations of Variable 15.10 Hypergeometric Differential Equation

§15.9 Relations to Other Functions

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Permalink:: http://dlmf.nist.gov/15.9
Addition (effective with 1.1.2):: Subsection 15.9(v) on hypergeometric representations of complete elliptic integrals has been added.
See also:: Annotations for Ch.15

§15.9(i) Orthogonal Polynomials
§15.9(ii) Jacobi Function
§15.9(iii) Gegenbauer Function
§15.9(iv) Associated Legendre Functions; Ferrers Functions
§15.9(v) Complete Elliptic Integrals

§15.9(i) Orthogonal Polynomials

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Keywords:: classical orthogonal polynomials, hypergeometric function, orthogonal polynomials, relations to hypergeometric function, relations to other functions
A&S Ref:: 15.4
Notes:: See §§18.5(ii), 18.20(ii).
Permalink:: http://dlmf.nist.gov/15.9.i
See also:: Annotations for §15.9 and Ch.15

For the notation see §§18.3 and 18.19.

Jacobi

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Keywords:: Jacobi polynomials, hypergeometric function, relations to other functions
See also:: Annotations for §15.9(i), §15.9 and Ch.15

15.9.1		$P^{(\alpha,\beta)}_{n}\left(x\right)=\frac{{\left(\alpha+1\right)_{n}}}{n!}F% \left({-n,n+\alpha+\beta+1\atop\alpha+1};\frac{1-x}{2}\right).$
ⓘ 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, $P^{(\NVar{\alpha},\NVar{\beta})}_{\NVar{n}}\left(\NVar{x}\right)$ : Jacobi polynomial, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $!$ : factorial (as in $n!$ ), $x$ : real variable and $n$ : integer Referenced by: §15.12(iii) Permalink: http://dlmf.nist.gov/15.9.E1 Encodings: TeX, pMML, png See also: Annotations for §15.9(i), §15.9(i), §15.9 and Ch.15

Gegenbauer (or Ultraspherical)

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Keywords:: hypergeometric function, relations to other functions, ultraspherical polynomials
See also:: Annotations for §15.9(i), §15.9 and Ch.15

15.9.2	$\displaystyle C^{(\lambda)}_{n}\left(x\right)$	$\displaystyle=\frac{{\left(2\lambda\right)_{n}}}{n!}F\left({-n,n+2\lambda\atop% \lambda+\frac{1}{2}};\frac{1-x}{2}\right).$
ⓘ 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, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $!$ : factorial (as in $n!$ ), $C^{(\NVar{\lambda})}_{\NVar{n}}\left(\NVar{x}\right)$ : ultraspherical (or Gegenbauer) polynomial, $x$ : real variable and $n$ : integer Permalink: http://dlmf.nist.gov/15.9.E2 Encodings: TeX, pMML, png See also: Annotations for §15.9(i), §15.9(i), §15.9 and Ch.15
15.9.3	$\displaystyle C^{(\lambda)}_{n}\left(x\right)$	$\displaystyle=(2x)^{n}\frac{{\left(\lambda\right)_{n}}}{n!}F\left({-\frac{1}{2% }n,\frac{1}{2}(1-n)\atop 1-\lambda-n};\frac{1}{x^{2}}\right).$
ⓘ 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, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $!$ : factorial (as in $n!$ ), $C^{(\NVar{\lambda})}_{\NVar{n}}\left(\NVar{x}\right)$ : ultraspherical (or Gegenbauer) polynomial, $x$ : real variable and $n$ : integer Permalink: http://dlmf.nist.gov/15.9.E3 Encodings: TeX, pMML, png See also: Annotations for §15.9(i), §15.9(i), §15.9 and Ch.15

15.9.4		$C^{(\lambda)}_{n}\left(\cos\theta\right)={\mathrm{e}}^{n\mathrm{i}\theta}\frac% {{\left(\lambda\right)_{n}}}{n!}F\left({-n,\lambda\atop 1-\lambda-n};{\mathrm{% e}}^{-2\mathrm{i}\theta}\right).$
ⓘ 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, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : 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 and $n$ : integer Permalink: http://dlmf.nist.gov/15.9.E4 Encodings: TeX, pMML, png See also: Annotations for §15.9(i), §15.9(i), §15.9 and Ch.15

Chebyshev

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Keywords:: Chebyshev polynomials, hypergeometric function, relations to other functions
See also:: Annotations for §15.9(i), §15.9 and Ch.15

15.9.5	$\displaystyle T_{n}\left(x\right)$	$\displaystyle=F\left({-n,n\atop\frac{1}{2}};\frac{1-x}{2}\right).$
ⓘ Symbols: $T_{\NVar{n}}\left(\NVar{x}\right)$ : Chebyshev polynomial of the first kind, $F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ 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, $x$ : real variable and $n$ : integer Permalink: http://dlmf.nist.gov/15.9.E5 Encodings: TeX, pMML, png See also: Annotations for §15.9(i), §15.9(i), §15.9 and Ch.15
15.9.6	$\displaystyle U_{n}\left(x\right)$	$\displaystyle=(n+1)F\left({-n,n+2\atop\frac{3}{2}};\frac{1-x}{2}\right).$
ⓘ Symbols: $U_{\NVar{n}}\left(\NVar{x}\right)$ : Chebyshev polynomial of the second kind, $F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ 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, $x$ : real variable and $n$ : integer Permalink: http://dlmf.nist.gov/15.9.E6 Encodings: TeX, pMML, png See also: Annotations for §15.9(i), §15.9(i), §15.9 and Ch.15

Legendre

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Keywords:: Legendre polynomials, hypergeometric function, relations to other functions
See also:: Annotations for §15.9(i), §15.9 and Ch.15

15.9.7		$P_{n}\left(x\right)=F\left({-n,n+1\atop 1};\frac{1-x}{2}\right).$
ⓘ 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, $P_{\NVar{n}}\left(\NVar{x}\right)$ : Legendre polynomial, $x$ : real variable and $n$ : integer Permalink: http://dlmf.nist.gov/15.9.E7 Encodings: TeX, pMML, png See also: Annotations for §15.9(i), §15.9(i), §15.9 and Ch.15

Krawtchouk

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Keywords:: Hahn class orthogonal polynomials, Krawtchouk polynomials, relation to hypergeometric function, relations to hypergeometric function
See also:: Annotations for §15.9(i), §15.9 and Ch.15

15.9.8		$K_{n}\left(x;p,N\right)=F\left({-n,-x\atop-N};\frac{1}{p}\right),$
$n=0,1,2,\dots,N$ ;
ⓘ 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, $K_{\NVar{n}}\left(\NVar{x};\NVar{p},\NVar{N}\right)$ : Krawtchouk polynomial, $x$ : real variable and $n$ : integer Permalink: http://dlmf.nist.gov/15.9.E8 Encodings: TeX, pMML, png See also: Annotations for §15.9(i), §15.9(i), §15.9 and Ch.15

compare also §15.2(ii).

Meixner

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Keywords:: Meixner polynomials, relation to hypergeometric function
See also:: Annotations for §15.9(i), §15.9 and Ch.15

15.9.9		$M_{n}\left(x;\beta,c\right)=F\left({-n,-x\atop\beta};1-\frac{1}{c}\right).$
ⓘ 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, $M_{\NVar{n}}\left(\NVar{x};\NVar{\beta},\NVar{c}\right)$ : Meixner polynomial, $x$ : real variable, $n$ : integer and $c$ : real or complex parameter Permalink: http://dlmf.nist.gov/15.9.E9 Encodings: TeX, pMML, png See also: Annotations for §15.9(i), §15.9(i), §15.9 and Ch.15

Meixner–Pollaczek

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Keywords:: Meixner–Pollaczek polynomials, classical orthogonal polynomials, hypergeometric function, orthogonal polynomials, relation to hypergeometric function, relations to hypergeometric function, relations to other functions
See also:: Annotations for §15.9(i), §15.9 and Ch.15

15.9.10		$P^{(\lambda)}_{n}\left(x;\phi\right)=\frac{{\left(2\lambda\right)_{n}}}{n!}{% \mathrm{e}}^{n\mathrm{i}\phi}F\left({-n,\lambda+\mathrm{i}x\atop 2\lambda};1-{% \mathrm{e}}^{-2\mathrm{i}\phi}\right).$
ⓘ 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, $P^{(\NVar{\lambda})}_{\NVar{n}}\left(\NVar{x};\NVar{\phi}\right)$ : Meixner–Pollaczek polynomial, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $\mathrm{e}$ : base of natural logarithm, $!$ : factorial (as in $n!$ ), $\mathrm{i}$ : imaginary unit, $x$ : real variable and $n$ : integer Permalink: http://dlmf.nist.gov/15.9.E10 Encodings: TeX, pMML, png See also: Annotations for §15.9(i), §15.9(i), §15.9 and Ch.15

§15.9(ii) Jacobi Function

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Keywords:: Jacobi function, Jacobi transform, definition, hypergeometric function, inversion, relations to other functions
Referenced by:: §14.3(iv), §15.14, §15.17(iii)
Permalink:: http://dlmf.nist.gov/15.9.ii
See also:: Annotations for §15.9 and Ch.15

This is a generalization of Jacobi polynomials (§18.3) and has the representation

15.9.11		$\phi^{(\alpha,\beta)}_{\lambda}\left(t\right)=F\left({\tfrac{1}{2}(\alpha+% \beta+1-\mathrm{i}\lambda),\tfrac{1}{2}(\alpha+\beta+1+\mathrm{i}\lambda)\atop% \alpha+1};-{\sinh}^{2}t\right).$
ⓘ Defines: $\phi^{(\NVar{\alpha},\NVar{\beta})}_{\NVar{\lambda}}\left(\NVar{t}\right)$ : Jacobi function 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, $\sinh\NVar{z}$ : hyperbolic sine function and $\mathrm{i}$ : imaginary unit Permalink: http://dlmf.nist.gov/15.9.E11 Encodings: TeX, pMML, png See also: Annotations for §15.9(ii), §15.9 and Ch.15

The Jacobi transform is defined as

15.9.12		$\widetilde{f}(\lambda)=\int_{0}^{\infty}f(t)\phi^{(\alpha,\beta)}_{\lambda}% \left(t\right)(2\sinh t)^{2\alpha+1}(2\cosh t)^{2\beta+1}\,\mathrm{d}t,$
ⓘ Symbols: $\phi^{(\NVar{\alpha},\NVar{\beta})}_{\NVar{\lambda}}\left(\NVar{t}\right)$ : Jacobi function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\cosh\NVar{z}$ : hyperbolic cosine function, $\sinh\NVar{z}$ : hyperbolic sine function, $\int$ : integral and $f(t)$ : function Permalink: http://dlmf.nist.gov/15.9.E12 Encodings: TeX, pMML, png See also: Annotations for §15.9(ii), §15.9 and Ch.15

with inverse

15.9.13		$f(t)=\frac{1}{2\pi\mathrm{i}}\int_{-\mathrm{i}\infty}^{\mathrm{i}\infty}% \widetilde{f}(\mathrm{i}\lambda)\Phi^{(\alpha,\beta)}_{\mathrm{i}\lambda}(t)% \frac{\Gamma\left(\tfrac{1}{2}(\alpha+\beta+1+\lambda)\right)\Gamma\left(% \tfrac{1}{2}(\alpha-\beta+1+\lambda)\right)}{\Gamma\left(\alpha+1\right)\Gamma% \left(\lambda\right)2^{\alpha+\beta+1-\lambda}}\,\mathrm{d}\lambda,$
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{i}$ : imaginary unit, $\int$ : integral, $(\NVar{a},\NVar{b})$ : open interval, $f(t)$ : function and $\Phi^{(\alpha,\beta)}_{\lambda}(t)$ : function Permalink: http://dlmf.nist.gov/15.9.E13 Encodings: TeX, pMML, png See also: Annotations for §15.9(ii), §15.9 and Ch.15

where the contour of integration is located to the right of the poles of the gamma functions in the integrand, and

15.9.14		$\Phi^{(\alpha,\beta)}_{\lambda}(t)=(2\cosh t)^{\mathrm{i}\lambda-\alpha-\beta-% 1}F\left({\tfrac{1}{2}(\alpha+\beta+1-\mathrm{i}\lambda),\tfrac{1}{2}(\alpha-% \beta+1-\mathrm{i}\lambda)\atop 1-\mathrm{i}\lambda};{\operatorname{sech}}^{2}% t\right).$
ⓘ Defines: $\Phi^{(\alpha,\beta)}_{\lambda}(t)$ : function (locally) 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, $\cosh\NVar{z}$ : hyperbolic cosine function, $\operatorname{sech}\NVar{z}$ : hyperbolic secant function, $\mathrm{i}$ : imaginary unit and $(\NVar{a},\NVar{b})$ : open interval Permalink: http://dlmf.nist.gov/15.9.E14 Encodings: TeX, pMML, png See also: Annotations for §15.9(ii), §15.9 and Ch.15

For this result, together with restrictions on the functions $f(t)$ and $\widetilde{f}(\lambda)$ , see Koornwinder (1984a).

§15.9(iii) Gegenbauer Function

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Keywords:: Gegenbauer function, definition, hypergeometric function, relation to hypergeometric function, relations to other functions
Notes:: See Erdélyi et al. (1953a, §§3.15.1 and 3.15.2).
Referenced by:: §14.3(iv), Erratum (V1.0.28) for Chapters 14 Legendre and Related Functions, 15 Hypergeometric Function
Permalink:: http://dlmf.nist.gov/15.9.iii
See also:: Annotations for §15.9 and Ch.15

This is a generalization of Gegenbauer (or ultraspherical) polynomials (§18.3). It is defined by:

15.9.15		$C^{(\lambda)}_{\alpha}\left(z\right)=\frac{\Gamma\left(\alpha+2\lambda\right)}% {\Gamma\left(2\lambda\right)\Gamma\left(\alpha+1\right)}F\left({-\alpha,\alpha% +2\lambda\atop\lambda+\tfrac{1}{2}};\frac{1-z}{2}\right).$
ⓘ Defines: $C^{(\NVar{\lambda})}_{\NVar{\alpha}}\left(\NVar{z}\right)$ : Gegenbauer function Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ 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 and $z$ : complex variable Referenced by: §14.3(iv), §14.3(iv), Erratum (V1.1.1) for Subsection 14.3(iv) Permalink: http://dlmf.nist.gov/15.9.E15 Encodings: TeX, pMML, png See also: Annotations for §15.9(iii), §15.9 and Ch.15

§15.9(iv) Associated Legendre Functions; Ferrers Functions

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Keywords:: Ferrers functions, associated Legendre functions, hypergeometric function, relations to other functions
A&S Ref:: 15.4
Notes:: (15.9.21) is the definition of an associated Legendre function; compare (14.3.6) and §14.21(i). For the other results in this subsection apply the quadratic transformations in §15.8(iii).
Referenced by:: Erratum (V1.0.9) for Subsection 15.9(iv)
Permalink:: http://dlmf.nist.gov/15.9.iv
Addition (effective with 1.0.9):: The sentence below (15.9.21) with the cross-reference to (14.3.1) has been added.
See also:: Annotations for §15.9 and Ch.15

Any hypergeometric function for which a quadratic transformation exists can be expressed in terms of associated Legendre functions or Ferrers functions. For examples see §§14.3(i)–14.3(iii) and 14.21(iii).

The following formulas apply with principal branches of the hypergeometric functions, associated Legendre functions, and fractional powers.

15.9.16	$\displaystyle\mathbf{F}\left({a,b\atop 2b};z\right)$	$\displaystyle=\frac{\sqrt{\pi}}{\Gamma\left(b\right)}z^{-b+(\ifrac{1}{2})}(1-z% )^{(b-a-(\ifrac{1}{2}))/2}\*P^{-b+(\ifrac{1}{2})}_{a-b-(\ifrac{1}{2})}\left(% \frac{2-z}{2\sqrt{1-z}}\right),$
$b\neq 0,-1,-2,\dots$ , $\|\operatorname{ph}\left(1-z\right)\|<\pi$ and $\|1-z\|<1$ .
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $P^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{z}\right)$ : associated Legendre function of the first kind, $\pi$ : the ratio of the circumference of a circle to its diameter, $\operatorname{ph}$ : phase, $\mathbf{F}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ or $\mathbf{F}\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)$ : $={{}_{2}{\mathbf{F}}_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ Olver’s hypergeometric function, $z$ : complex variable, $a$ : real or complex parameter and $b$ : real or complex parameter Permalink: http://dlmf.nist.gov/15.9.E16 Encodings: TeX, pMML, png See also: Annotations for §15.9(iv), §15.9 and Ch.15
15.9.17	$\displaystyle\mathbf{F}\left({a,a+\tfrac{1}{2}\atop c};z\right)$	$\displaystyle=2^{c-1}z^{\ifrac{(1-c)}{2}}(1-z)^{-a+(\ifrac{(c-1)}{2})}\*P^{1-c% }_{2a-c}\left(\frac{1}{\sqrt{1-z}}\right),$
$\|\operatorname{ph}z\|<\pi$ and $\|\operatorname{ph}\left(1-z\right)\|<\pi$ .
ⓘ Symbols: $P^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{z}\right)$ : associated Legendre function of the first kind, $\pi$ : the ratio of the circumference of a circle to its diameter, $\operatorname{ph}$ : phase, $\mathbf{F}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ or $\mathbf{F}\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)$ : $={{}_{2}{\mathbf{F}}_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ Olver’s hypergeometric function, $z$ : complex variable, $a$ : real or complex parameter and $c$ : real or complex parameter Permalink: http://dlmf.nist.gov/15.9.E17 Encodings: TeX, pMML, png See also: Annotations for §15.9(iv), §15.9 and Ch.15
15.9.18	$\displaystyle\mathbf{F}\left({a,b\atop a+b+\tfrac{1}{2}};z\right)$	$\displaystyle=2^{a+b-(\ifrac{1}{2})}(-z)^{(-a-b+(\ifrac{1}{2}))/2}\*P^{-a-b+(% \ifrac{1}{2})}_{a-b-(\ifrac{1}{2})}\left(\sqrt{1-z}\right),$
$\left\|\operatorname{ph}\left(-z\right)\right\|<\pi$ .
ⓘ Symbols: $P^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{z}\right)$ : associated Legendre function of the first kind, $\pi$ : the ratio of the circumference of a circle to its diameter, $\operatorname{ph}$ : phase, $\mathbf{F}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ or $\mathbf{F}\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)$ : $={{}_{2}{\mathbf{F}}_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ Olver’s hypergeometric function, $z$ : complex variable, $a$ : real or complex parameter and $b$ : real or complex parameter Permalink: http://dlmf.nist.gov/15.9.E18 Encodings: TeX, pMML, png See also: Annotations for §15.9(iv), §15.9 and Ch.15
15.9.19	$\displaystyle\mathbf{F}\left({a,b\atop a-b+1};z\right)$	$\displaystyle=z^{\ifrac{(b-a)}{2}}(1-z)^{-b}\*P^{b-a}_{-b}\left(\frac{1+z}{1-z% }\right),$
$\|\operatorname{ph}z\|<\pi$ and $\|\operatorname{ph}\left(1-z\right)\|<\pi$ .
ⓘ Symbols: $P^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{z}\right)$ : associated Legendre function of the first kind, $\pi$ : the ratio of the circumference of a circle to its diameter, $\operatorname{ph}$ : phase, $\mathbf{F}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ or $\mathbf{F}\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)$ : $={{}_{2}{\mathbf{F}}_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ Olver’s hypergeometric function, $z$ : complex variable, $a$ : real or complex parameter and $b$ : real or complex parameter Permalink: http://dlmf.nist.gov/15.9.E19 Encodings: TeX, pMML, png See also: Annotations for §15.9(iv), §15.9 and Ch.15
15.9.20	$\displaystyle\mathbf{F}\left({a,b\atop\tfrac{1}{2}(a+b+1)};z\right)$	$\displaystyle=\left(-z(1-z)\right)^{\ifrac{(1-a-b)}{4}}\*P^{\ifrac{(1-a-b)}{2}% }_{\ifrac{(a-b-1)}{2}}\left(1-2z\right),$
$\left\|\operatorname{ph}\left(-z\right)\right\|<\pi$ .
ⓘ Symbols: $P^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{z}\right)$ : associated Legendre function of the first kind, $\pi$ : the ratio of the circumference of a circle to its diameter, $\operatorname{ph}$ : phase, $\mathbf{F}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ or $\mathbf{F}\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)$ : $={{}_{2}{\mathbf{F}}_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ Olver’s hypergeometric function, $z$ : complex variable, $a$ : real or complex parameter and $b$ : real or complex parameter Permalink: http://dlmf.nist.gov/15.9.E20 Encodings: TeX, pMML, png See also: Annotations for §15.9(iv), §15.9 and Ch.15
15.9.21	$\displaystyle\mathbf{F}\left({a,1-a\atop c};z\right)$	$\displaystyle=\left(\frac{-z}{1-z}\right)^{\ifrac{(1-c)}{2}}\*P^{1-c}_{-a}% \left(1-2z\right),$
$\left\|\operatorname{ph}\left(-z\right)\right\|<\pi$ .
ⓘ Symbols: $P^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{z}\right)$ : associated Legendre function of the first kind, $\pi$ : the ratio of the circumference of a circle to its diameter, $\operatorname{ph}$ : phase, $\mathbf{F}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ or $\mathbf{F}\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)$ : $={{}_{2}{\mathbf{F}}_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ Olver’s hypergeometric function, $z$ : complex variable, $a$ : real or complex parameter and $c$ : real or complex parameter Referenced by: §15.9(iv) Permalink: http://dlmf.nist.gov/15.9.E21 Encodings: TeX, pMML, png See also: Annotations for §15.9(iv), §15.9 and Ch.15
For the case $0<z<1$ see (14.3.1).

15.9.22		$\mathbf{F}\left({a,b\atop\tfrac{1}{2}};z\right)=\frac{2^{a+b-(\ifrac{3}{2})}}{% \pi}\Gamma\left(a+\tfrac{1}{2}\right)\Gamma\left(b+\tfrac{1}{2}\right)\(z-1)^% {(-a-b+(\ifrac{1}{2}))/2}\\left({\mathrm{e}}^{\pm\pi\mathrm{i}(a+b-(\ifrac{1}% {2}))}P^{-a-b+(\ifrac{1}{2})}_{a-b-(\ifrac{1}{2})}\left(-\sqrt{z}\right)+P^{-a% -b+(\ifrac{1}{2})}_{a-b-(\ifrac{1}{2})}\left(\sqrt{z}\right)\right),$
$a,b\neq-\frac{1}{2},-\frac{3}{2},-\frac{5}{2},\dotsc$ , $0<\|\operatorname{ph}z\|<\pi$ ,
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $P^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{z}\right)$ : associated Legendre function of the first kind, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $\operatorname{ph}$ : phase, $\mathbf{F}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ or $\mathbf{F}\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)$ : $={{}_{2}{\mathbf{F}}_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ Olver’s hypergeometric function, $z$ : complex variable, $a$ : real or complex parameter and $b$ : real or complex parameter Permalink: http://dlmf.nist.gov/15.9.E22 Encodings: TeX, pMML, png See also: Annotations for §15.9(iv), §15.9 and Ch.15

where the sign in the exponential is $\pm$ according as $\Im z\gtrless 0$ .

15.9.23		$\mathbf{F}\left({a,b\atop\tfrac{3}{2}};z\right)=\frac{2^{a+b-(\ifrac{5}{2})}}{% \pi\sqrt{z}}\Gamma\left(a-\tfrac{1}{2}\right)\Gamma\left(b-\tfrac{1}{2}\right)% \(z-1)^{(-a-b+(\ifrac{3}{2}))/2}\\left({\mathrm{e}}^{\pm\pi\mathrm{i}(a+b-(% \ifrac{3}{2}))}P^{-a-b+(\ifrac{3}{2})}_{a-b-(\ifrac{1}{2})}\left(-\sqrt{z}% \right)-P^{-a-b+(\ifrac{3}{2})}_{a-b-(\ifrac{1}{2})}\left(\sqrt{z}\right)% \right),$
$a,b\neq\frac{1}{2},-\frac{1}{2},-\frac{3}{2},\dots$ , $0<\|\operatorname{ph}z\|<\pi$ ,
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $P^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{z}\right)$ : associated Legendre function of the first kind, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $\operatorname{ph}$ : phase, $\mathbf{F}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ or $\mathbf{F}\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)$ : $={{}_{2}{\mathbf{F}}_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ Olver’s hypergeometric function, $z$ : complex variable, $a$ : real or complex parameter and $b$ : real or complex parameter Permalink: http://dlmf.nist.gov/15.9.E23 Encodings: TeX, pMML, png See also: Annotations for §15.9(iv), §15.9 and Ch.15

where the sign in the exponential is $\pm$ according as $\Im z\gtrless 0$ .

§15.9(v) Complete Elliptic Integrals

ⓘ

Referenced by:: §15.9, Erratum (V1.1.2) for Additions
Permalink:: http://dlmf.nist.gov/15.9.v
Addition (effective with 1.1.2):: Equations (15.9.24), (15.9.25) and (15.9.26) were added.
See also:: Annotations for §15.9 and Ch.15

15.9.24		$K\left(k\right)=\frac{\pi}{2}F\left({\frac{1}{2},\frac{1}{2}\atop 1};k^{2}% \right),$
ⓘ 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, $\pi$ : 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 and $k$ : integer Referenced by: §15.9(v) Permalink: http://dlmf.nist.gov/15.9.E24 Encodings: TeX, pMML, png Addition (effective with 1.1.2): This equation was added. See also: Annotations for §15.9(v), §15.9 and Ch.15

15.9.25		$E\left(k\right)=\frac{\pi}{2}F\left({-\frac{1}{2},\frac{1}{2}\atop 1};k^{2}% \right),$
ⓘ 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, $\pi$ : 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 and $k$ : integer Referenced by: §15.9(v) Permalink: http://dlmf.nist.gov/15.9.E25 Encodings: TeX, pMML, png Addition (effective with 1.1.2): This equation was added. See also: Annotations for §15.9(v), §15.9 and Ch.15

15.9.26		$D\left(k\right)=\frac{\pi}{4}F\left({\frac{1}{2},\frac{3}{2}\atop 2};k^{2}% \right).$
ⓘ 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, $\pi$ : the ratio of the circumference of a circle to its diameter, $D\left(\NVar{k}\right)$ : complete elliptic integral of Legendre’s type and $k$ : integer Referenced by: §15.9(v) Permalink: http://dlmf.nist.gov/15.9.E26 Encodings: TeX, pMML, png Addition (effective with 1.1.2): This equation was added. See also: Annotations for §15.9(v), §15.9 and Ch.15