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

§15.9 Relations to Other Functions

Permalink:: http://dlmf.nist.gov/15.9

§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(i) Orthogonal Polynomials

Notes:: See §§18.5(ii), 18.20(ii).
A&S Ref:: 15.4
Keywords:: classical orthogonal polynomials, hypergeometric function, orthogonal polynomials
Permalink:: http://dlmf.nist.gov/15.9.i

For the notation see §§18.3 and 18.19.

¶ Jacobi

Keywords:: Jacobi polynomials, hypergeometric function

15.9.1 $\mathop{P^{{(\alpha,\beta)}}_{{n}}\/}\nolimits\!\left(x\right)=\frac{\left(% \alpha+1\right)_{{n}}}{n!}\mathop{F\/}\nolimits\!\left({-n,n+\alpha+\beta+1% \atop\alpha+1};\frac{1-x}{2}\right).$

Symbols:: $\mathop{F\/}\nolimits\!\left(a,b;c;z\right)$ : hypergeometric function, $\mathop{P^{{(\alpha,\beta)}}_{{n}}\/}\nolimits\!\left(x\right)$ : Jacobi polynomial, $\left(a\right)_{{n}}$ : Pochhammer’s symbol, : : factorial, : real variable and : integer
Referenced by:: §15.12(iii)
Permalink:: http://dlmf.nist.gov/15.9.E1
Encodings:: TeX, pMML, png

¶ Gegenbauer (or Ultraspherical)

Keywords:: hypergeometric function, ultraspherical polynomials

15.9.2 $\mathop{C^{{(\lambda)}}_{{n}}\/}\nolimits\!\left(x\right)=\frac{\left(2\lambda% \right)_{{n}}}{n!}\mathop{F\/}\nolimits\!\left({-n,n+2\lambda\atop\lambda+% \frac{1}{2}};\frac{1-x}{2}\right).$

Symbols:: $\mathop{F\/}\nolimits\!\left(a,b;c;z\right)$ : hypergeometric function, $\left(a\right)_{{n}}$ : Pochhammer’s symbol, : : factorial, $\mathop{C^{{(\lambda)}}_{{n}}\/}\nolimits\!\left(x\right)$ : ultraspherical (or Gegenbauer) polynomial, : real variable and : integer
Permalink:: http://dlmf.nist.gov/15.9.E2
Encodings:: TeX, pMML, png

15.9.3 $\mathop{C^{{(\lambda)}}_{{n}}\/}\nolimits\!\left(x\right)=(2x)^{n}\frac{\left(% \lambda\right)_{{n}}}{n!}\mathop{F\/}\nolimits\!\left({-\frac{1}{2}n,\frac{1}{% 2}(1-n)\atop 1-\lambda-n};\frac{1}{x^{2}}\right).$

Symbols:: $\mathop{F\/}\nolimits\!\left(a,b;c;z\right)$ : hypergeometric function, $\left(a\right)_{{n}}$ : Pochhammer’s symbol, : : factorial, $\mathop{C^{{(\lambda)}}_{{n}}\/}\nolimits\!\left(x\right)$ : ultraspherical (or Gegenbauer) polynomial, : real variable and : integer
Permalink:: http://dlmf.nist.gov/15.9.E3
Encodings:: TeX, pMML, png

15.9.4 $\mathop{C^{{(\lambda)}}_{{n}}\/}\nolimits\!\left(\mathop{\cos\/}\nolimits% \theta\right)=e^{{ni\theta}}\frac{\left(\lambda\right)_{{n}}}{n!}\mathop{F\/}% \nolimits\!\left({-n,\lambda\atop 1-\lambda-n};e^{{-2i\theta}}\right).$

Symbols:: $\mathop{F\/}\nolimits\!\left(a,b;c;z\right)$ : hypergeometric function, $\left(a\right)_{{n}}$ : Pochhammer’s symbol, $\mathop{\cos\/}\nolimits z$ : cosine function, : base of exponential function, : : factorial, $\mathop{C^{{(\lambda)}}_{{n}}\/}\nolimits\!\left(x\right)$ : ultraspherical (or Gegenbauer) polynomial and : integer
Permalink:: http://dlmf.nist.gov/15.9.E4
Encodings:: TeX, pMML, png

¶ Chebyshev

Keywords:: Chebyshev polynomials, hypergeometric function

15.9.5 $\mathop{T_{{n}}\/}\nolimits\!\left(x\right)=\mathop{F\/}\nolimits\!\left({-n,n% \atop\frac{1}{2}};\frac{1-x}{2}\right).$

Symbols:: $\mathop{T_{{n}}\/}\nolimits\!\left(x\right)$ : Chebyshev polynomial of the first kind, $\mathop{F\/}\nolimits\!\left(a,b;c;z\right)$ : hypergeometric function, : real variable and : integer
Permalink:: http://dlmf.nist.gov/15.9.E5
Encodings:: TeX, pMML, png

15.9.6 $\mathop{U_{{n}}\/}\nolimits\!\left(x\right)=(n+1)\mathop{F\/}\nolimits\!\left(% {-n,n+2\atop\frac{3}{2}};\frac{1-x}{2}\right).$

Symbols:: $\mathop{U_{{n}}\/}\nolimits\!\left(x\right)$ : Chebyshev polynomial of the second kind, $\mathop{F\/}\nolimits\!\left(a,b;c;z\right)$ : hypergeometric function, : real variable and : integer
Permalink:: http://dlmf.nist.gov/15.9.E6
Encodings:: TeX, pMML, png

¶ Legendre

Keywords:: Legendre polynomials, hypergeometric function

15.9.7 $\mathop{P_{{n}}\/}\nolimits\!\left(x\right)=\mathop{F\/}\nolimits\!\left({-n,n% +1\atop 1};\frac{1-x}{2}\right).$

Symbols:: $\mathop{F\/}\nolimits\!\left(a,b;c;z\right)$ : hypergeometric function, $\mathop{P_{{n}}\/}\nolimits\!\left(x\right)$ : Legendre polynomial, : real variable and : integer
Permalink:: http://dlmf.nist.gov/15.9.E7
Encodings:: TeX, pMML, png

¶ Krawtchouk

Keywords:: Hahn class orthogonal polynomials, Krawtchouk polynomials

15.9.8 $\mathop{K_{{n}}\/}\nolimits\!\left(x;p,N\right)=\mathop{F\/}\nolimits\!\left({% -n,-x\atop-N};\frac{1}{p}\right),$ $n=0,1,2,\dots,N$ ;

Symbols:: $\mathop{F\/}\nolimits\!\left(a,b;c;z\right)$ : hypergeometric function, $\mathop{K_{{n}}\/}\nolimits\!\left(x;p,N\right)$ : Krawtchouk polynomial, : real variable and : integer
Permalink:: http://dlmf.nist.gov/15.9.E8
Encodings:: TeX, pMML, png

compare also §15.2(ii).

¶ Meixner

Keywords:: Meixner polynomials

15.9.9 $\mathop{M_{{n}}\/}\nolimits\!\left(x;\beta,c\right)=\mathop{F\/}\nolimits\!% \left({-n,-x\atop\beta};1-\frac{1}{c}\right).$

Symbols:: $\mathop{F\/}\nolimits\!\left(a,b;c;z\right)$ : hypergeometric function, $\mathop{M_{{n}}\/}\nolimits\!\left(x;\beta,c\right)$ : Meixner polynomial, : real variable, : integer and : real or complex parameter
Permalink:: http://dlmf.nist.gov/15.9.E9
Encodings:: TeX, pMML, png

¶ Meixner–Pollaczek

Keywords:: Meixner–Pollaczek polynomials, classical orthogonal polynomials, hypergeometric function, orthogonal polynomials

15.9.10 $\mathop{P^{{(\lambda)}}_{{n}}\/}\nolimits\!\left(x;\phi\right)=\frac{\left(2% \lambda\right)_{{n}}}{n!}e^{{ni\phi}}\mathop{F\/}\nolimits\!\left({-n,\lambda+% ix\atop 2\lambda};1-e^{{-2i\phi}}\right).$

Symbols:: $\mathop{F\/}\nolimits\!\left(a,b;c;z\right)$ : hypergeometric function, $\mathop{P^{{(\lambda)}}_{{n}}\/}\nolimits\!\left(x;\phi\right)$ : Meixner–Pollaczek polynomial, $\left(a\right)_{{n}}$ : Pochhammer’s symbol, : base of exponential function, : : factorial, : real variable and : integer
Permalink:: http://dlmf.nist.gov/15.9.E10
Encodings:: TeX, pMML, png

§15.9(ii) Jacobi Function

Keywords:: Jacobi function, Jacobi transform, hypergeometric function
Referenced by:: §14.3(iv), §15.14, §15.17(iii)
Permalink:: http://dlmf.nist.gov/15.9.ii

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

15.9.11 $\mathop{\phi^{{(\alpha,\beta)}}_{{\lambda}}\/}\nolimits\!\left(t\right)=% \mathop{F\/}\nolimits\!\left({\tfrac{1}{2}(\alpha+\beta+1-i\lambda),\tfrac{1}{% 2}(\alpha+\beta+1+i\lambda)\atop\alpha+1};-{\mathop{\sinh\/}\nolimits^{{2}}}t% \right).$

Defines:: $\mathop{\phi^{{(\alpha,\beta)}}_{{\lambda}}\/}\nolimits\!\left(t\right)$ : Jacobi function
Symbols:: $\mathop{F\/}\nolimits\!\left(a,b;c;z\right)$ : hypergeometric function and $\mathop{\sinh\/}\nolimits z$ : hyperbolic sine function
Permalink:: http://dlmf.nist.gov/15.9.E11
Encodings:: TeX, pMML, png

The Jacobi transform is defined as

15.9.12 $\widetilde{f}(\lambda)=\int_{0}^{\infty}f(t)\mathop{\phi^{{(\alpha,\beta)}}_{{% \lambda}}\/}\nolimits\!\left(t\right)(2\mathop{\sinh\/}\nolimits t)^{{2\alpha+% 1}}(2\mathop{\cosh\/}\nolimits t)^{{2\beta+1}}dt,$

Symbols:: $\mathop{\phi^{{(\alpha,\beta)}}_{{\lambda}}\/}\nolimits\!\left(t\right)$ : Jacobi function, : differential of , $\mathop{\cosh\/}\nolimits z$ : hyperbolic cosine function, $\mathop{\sinh\/}\nolimits z$ : hyperbolic sine function, $\int$ : integral and : function
Permalink:: http://dlmf.nist.gov/15.9.E12
Encodings:: TeX, pMML, png

with inverse

15.9.13 $f(t)=\frac{1}{2\pi i}\int_{{-i\infty}}^{{i\infty}}\widetilde{f}(i\lambda)% \mathop{\Phi^{{(\alpha,\beta)}}_{{i\lambda}}\/}\nolimits\!\left(t\right)\frac{% \mathop{\Gamma\/}\nolimits\!\left(\tfrac{1}{2}(\alpha+\beta+1+\lambda)\right)% \mathop{\Gamma\/}\nolimits\!\left(\tfrac{1}{2}(\alpha-\beta+1+\lambda)\right)}% {\mathop{\Gamma\/}\nolimits\!\left(\alpha+1\right)\mathop{\Gamma\/}\nolimits\!% \left(\lambda\right)2^{{\alpha+\beta+1-\lambda}}}d\lambda,$

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, : differential of , $\int$ : integral and : function
Permalink:: http://dlmf.nist.gov/15.9.E13
Encodings:: TeX, pMML, png

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

15.9.14 $\mathop{\Phi^{{(\alpha,\beta)}}_{{\lambda}}\/}\nolimits\!\left(t\right)=(2% \mathop{\cosh\/}\nolimits t)^{{i\lambda-\alpha-\beta-1}}\mathop{F\/}\nolimits% \!\left({\tfrac{1}{2}(\alpha+\beta+1-i\lambda),\tfrac{1}{2}(\alpha-\beta+1-i% \lambda)\atop 1-i\lambda};{\mathop{\mathrm{sech}\/}\nolimits^{{2}}}t\right).$

Symbols:: $\mathop{F\/}\nolimits\!\left(a,b;c;z\right)$ : hypergeometric function, $\mathop{\cosh\/}\nolimits z$ : hyperbolic cosine function and $\mathop{\mathrm{sech}\/}\nolimits z$ : hyperbolic secant function
Permalink:: http://dlmf.nist.gov/15.9.E14
Encodings:: TeX, pMML, png

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

§15.9(iii) Gegenbauer Function

Notes:: See Erdélyi et al. (1953a, §§3.15.1 and 3.15.2).
Keywords:: Gegenbauer function, hypergeometric function
Referenced by:: §14.3(iv)
Permalink:: http://dlmf.nist.gov/15.9.iii

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

15.9.15 $\mathop{C^{{(\lambda)}}_{{\alpha}}\/}\nolimits\!\left(z\right)=\frac{\mathop{% \Gamma\/}\nolimits\!\left(\alpha+2\lambda\right)}{\mathop{\Gamma\/}\nolimits\!% \left(2\lambda\right)\mathop{\Gamma\/}\nolimits\!\left(\alpha+1\right)}\mathop% {F\/}\nolimits\!\left({-\alpha,\alpha+2\lambda\atop\lambda+\tfrac{1}{2}};\frac% {1-z}{2}\right).$

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{F\/}\nolimits\!\left(a,b;c;z\right)$ : hypergeometric function, $\mathop{C^{{(\lambda)}}_{{n}}\/}\nolimits\!\left(x\right)$ : ultraspherical (or Gegenbauer) polynomial and : complex variable
Permalink:: http://dlmf.nist.gov/15.9.E15
Encodings:: TeX, pMML, png

§15.9(iv) Associated Legendre Functions; Ferrers Functions

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).
A&S Ref:: 15.4
Keywords:: Ferrers functions, associated Legendre functions, hypergeometric function
Permalink:: http://dlmf.nist.gov/15.9.iv

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 $\mathop{\mathbf{F}\/}\nolimits\!\left({a,b\atop 2b};z\right)=\frac{\sqrt{\pi}}% {\mathop{\Gamma\/}\nolimits\!\left(b\right)}z^{{-b+(\ifrac{1}{2})}}(1-z)^{{(b-% a-(\ifrac{1}{2}))/2}}\*\mathop{P^{{-b+(\ifrac{1}{2})}}_{{a-b-(\ifrac{1}{2})}}% \/}\nolimits\!\left(\frac{2-z}{2\sqrt{1-z}}\right),$ $b\neq 0,-1,-2,\dots$ , $|\mathop{\mathrm{ph}\/}\nolimits(1-z)|<\pi$ and

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{P^{{\mu}}_{{\nu}}\/}\nolimits\!\left(z\right)$ : associated Legendre function of the first kind, $\mathop{\mathrm{ph}\/}\nolimits$ : phase, $\mathop{\mathbf{F}\/}\nolimits\!\left(a,b;c;z\right)$ : Olver’s hypergeometric function, : complex variable, : real or complex parameter and : real or complex parameter
Permalink:: http://dlmf.nist.gov/15.9.E16
Encodings:: TeX, pMML, png

15.9.17 $\mathop{\mathbf{F}\/}\nolimits\!\left({a,a+\tfrac{1}{2}\atop c};z\right)=2^{{c% -1}}z^{{\ifrac{(1-c)}{2}}}(1-z)^{{-a+(\ifrac{(c-1)}{2})}}\*\mathop{P^{{1-c}}_{% {2a-c}}\/}\nolimits\!\left(\frac{1}{\sqrt{1-z}}\right),$ $|\mathop{\mathrm{ph}\/}\nolimits z|<\pi$ and $|\mathop{\mathrm{ph}\/}\nolimits(1-z)|<\pi$ .

Symbols:: $\mathop{P^{{\mu}}_{{\nu}}\/}\nolimits\!\left(z\right)$ : associated Legendre function of the first kind, $\mathop{\mathrm{ph}\/}\nolimits$ : phase, $\mathop{\mathbf{F}\/}\nolimits\!\left(a,b;c;z\right)$ : Olver’s hypergeometric function, : complex variable, : real or complex parameter and : real or complex parameter
Permalink:: http://dlmf.nist.gov/15.9.E17
Encodings:: TeX, pMML, png

15.9.18 $\mathop{\mathbf{F}\/}\nolimits\!\left({a,b\atop a+b+\tfrac{1}{2}};z\right)=2^{% {a+b-(\ifrac{1}{2})}}(-z)^{{(-a-b+(\ifrac{1}{2}))/2}}\*\mathop{P^{{-a-b+(% \ifrac{1}{2})}}_{{a-b-(\ifrac{1}{2})}}\/}\nolimits\!\left(\sqrt{1-z}\right),$ $\left|\mathop{\mathrm{ph}\/}\nolimits\!\left(-z\right)\right|<\pi$ .

Symbols:: $\mathop{P^{{\mu}}_{{\nu}}\/}\nolimits\!\left(z\right)$ : associated Legendre function of the first kind, $\mathop{\mathrm{ph}\/}\nolimits$ : phase, $\mathop{\mathbf{F}\/}\nolimits\!\left(a,b;c;z\right)$ : Olver’s hypergeometric function, : complex variable, : real or complex parameter and : real or complex parameter
Permalink:: http://dlmf.nist.gov/15.9.E18
Encodings:: TeX, pMML, png

15.9.19 $\mathop{\mathbf{F}\/}\nolimits\!\left({a,b\atop a-b+1};z\right)=z^{{\ifrac{(b-% a)}{2}}}(1-z)^{{-b}}\*\mathop{P^{{b-a}}_{{-b}}\/}\nolimits\!\left(\frac{1+z}{1% -z}\right),$ $|\mathop{\mathrm{ph}\/}\nolimits z|<\pi$ and $|\mathop{\mathrm{ph}\/}\nolimits(1-z)|<\pi$ .

Symbols:: $\mathop{P^{{\mu}}_{{\nu}}\/}\nolimits\!\left(z\right)$ : associated Legendre function of the first kind, $\mathop{\mathrm{ph}\/}\nolimits$ : phase, $\mathop{\mathbf{F}\/}\nolimits\!\left(a,b;c;z\right)$ : Olver’s hypergeometric function, : complex variable, : real or complex parameter and : real or complex parameter
Permalink:: http://dlmf.nist.gov/15.9.E19
Encodings:: TeX, pMML, png

15.9.20 $\mathop{\mathbf{F}\/}\nolimits\!\left({a,b\atop\tfrac{1}{2}(a+b+1)};z\right)=% \left(-z(1-z)\right)^{{\ifrac{(1-a-b)}{4}}}\*\mathop{P^{{\ifrac{(1-a-b)}{2}}}_% {{\ifrac{(a-b-1)}{2}}}\/}\nolimits\!\left(1-2z\right),$ $\left|\mathop{\mathrm{ph}\/}\nolimits\!\left(-z\right)\right|<\pi$ .

Symbols:: $\mathop{P^{{\mu}}_{{\nu}}\/}\nolimits\!\left(z\right)$ : associated Legendre function of the first kind, $\mathop{\mathrm{ph}\/}\nolimits$ : phase, $\mathop{\mathbf{F}\/}\nolimits\!\left(a,b;c;z\right)$ : Olver’s hypergeometric function, : complex variable, : real or complex parameter and : real or complex parameter
Permalink:: http://dlmf.nist.gov/15.9.E20
Encodings:: TeX, pMML, png

15.9.21 $\mathop{\mathbf{F}\/}\nolimits\!\left({a,1-a\atop c};z\right)=\left(\frac{-z}{% 1-z}\right)^{{\ifrac{(1-c)}{2}}}\*\mathop{P^{{1-c}}_{{-a}}\/}\nolimits\!\left(% 1-2z\right),$ $\left|\mathop{\mathrm{ph}\/}\nolimits\!\left(-z\right)\right|<\pi$ .

Symbols:: $\mathop{P^{{\mu}}_{{\nu}}\/}\nolimits\!\left(z\right)$ : associated Legendre function of the first kind, $\mathop{\mathrm{ph}\/}\nolimits$ : phase, $\mathop{\mathbf{F}\/}\nolimits\!\left(a,b;c;z\right)$ : Olver’s hypergeometric function, : complex variable, : real or complex parameter and : real or complex parameter
Referenced by:: §15.9(iv)
Permalink:: http://dlmf.nist.gov/15.9.E21
Encodings:: TeX, pMML, png

15.9.22 $\mathop{\mathbf{F}\/}\nolimits\!\left({a,b\atop\tfrac{1}{2}};z\right)=\frac{2^% {{a+b-(\ifrac{3}{2})}}}{\pi}\mathop{\Gamma\/}\nolimits\!\left(a+\tfrac{1}{2}% \right)\mathop{\Gamma\/}\nolimits\!\left(b+\tfrac{1}{2}\right)\*(z-1)^{{(-a-b+% (\ifrac{1}{2}))/2}}\*\left(e^{{\pm\pi i(a+b-(\ifrac{1}{2}))}}\mathop{P^{{-a-b+% (\ifrac{1}{2})}}_{{a-b-(\ifrac{1}{2})}}\/}\nolimits\!\left(-\sqrt{z}\right)+% \mathop{P^{{-a-b+(\ifrac{1}{2})}}_{{a-b-(\ifrac{1}{2})}}\/}\nolimits\!\left(% \sqrt{z}\right)\right),$ $a,b\neq-\frac{1}{2},-\frac{3}{2},-\frac{5}{2},\ldots$ , $0<|\mathop{\mathrm{ph}\/}\nolimits z|<\pi$ ,

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{P^{{\mu}}_{{\nu}}\/}\nolimits\!\left(z\right)$ : associated Legendre function of the first kind, : base of exponential function, $\mathop{\mathrm{ph}\/}\nolimits$ : phase, $\mathop{\mathbf{F}\/}\nolimits\!\left(a,b;c;z\right)$ : Olver’s hypergeometric function, : complex variable, : real or complex parameter and : real or complex parameter
Permalink:: http://dlmf.nist.gov/15.9.E22
Encodings:: TeX, pMML, png

where the sign in the exponential is $\pm$ according as $\imagpart{z}\gtrless 0$ .

15.9.23 $\mathop{\mathbf{F}\/}\nolimits\!\left({a,b\atop\tfrac{3}{2}};z\right)=\frac{2^% {{a+b-(\ifrac{5}{2})}}}{\pi\sqrt{z}}\mathop{\Gamma\/}\nolimits\!\left(a-\tfrac% {1}{2}\right)\mathop{\Gamma\/}\nolimits\!\left(b-\tfrac{1}{2}\right)\*(z-1)^{{% (-a-b+(\ifrac{3}{2}))/2}}\*\left(e^{{\pm\pi i(a+b-(\ifrac{3}{2}))}}\mathop{P^{% {-a-b+(\ifrac{3}{2})}}_{{a-b-(\ifrac{1}{2})}}\/}\nolimits\!\left(-\sqrt{z}% \right)-\mathop{P^{{-a-b+(\ifrac{3}{2})}}_{{a-b-(\ifrac{1}{2})}}\/}\nolimits\!% \left(\sqrt{z}\right)\right),$ $a,b\neq\frac{1}{2},-\frac{1}{2},-\frac{3}{2},\dots$ , $0<|\mathop{\mathrm{ph}\/}\nolimits z|<\pi$ ,

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{P^{{\mu}}_{{\nu}}\/}\nolimits\!\left(z\right)$ : associated Legendre function of the first kind, : base of exponential function, $\mathop{\mathrm{ph}\/}\nolimits$ : phase, $\mathop{\mathbf{F}\/}\nolimits\!\left(a,b;c;z\right)$ : Olver’s hypergeometric function, : complex variable, : real or complex parameter and : real or complex parameter
Permalink:: http://dlmf.nist.gov/15.9.E23
Encodings:: TeX, pMML, png