§15.9 Relations to Other Functions
Contents
- §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
¶ Jacobi
![\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).](https://webarchive.library.unt.edu/web/20130309161112im_/http://dlmf.nist.gov/15/9/E1.png)
¶ Gegenbauer (or Ultraspherical)
![\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).](https://webarchive.library.unt.edu/web/20130309161112im_/http://dlmf.nist.gov/15/9/E2.png)
![\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).](https://webarchive.library.unt.edu/web/20130309161112im_/http://dlmf.nist.gov/15/9/E3.png)
![\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).](https://webarchive.library.unt.edu/web/20130309161112im_/http://dlmf.nist.gov/15/9/E4.png)
¶ Chebyshev
![\mathop{T_{{n}}\/}\nolimits\!\left(x\right)=\mathop{F\/}\nolimits\!\left({-n,n\atop\frac{1}{2}};\frac{1-x}{2}\right).](https://webarchive.library.unt.edu/web/20130309161112im_/http://dlmf.nist.gov/15/9/E5.png)
![\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).](https://webarchive.library.unt.edu/web/20130309161112im_/http://dlmf.nist.gov/15/9/E6.png)
¶ Legendre
![\mathop{P_{{n}}\/}\nolimits\!\left(x\right)=\mathop{F\/}\nolimits\!\left({-n,n+1\atop 1};\frac{1-x}{2}\right).](https://webarchive.library.unt.edu/web/20130309161112im_/http://dlmf.nist.gov/15/9/E7.png)
¶ Krawtchouk
![\mathop{K_{{n}}\/}\nolimits\!\left(x;p,N\right)=\mathop{F\/}\nolimits\!\left({-n,-x\atop-N};\frac{1}{p}\right),](https://webarchive.library.unt.edu/web/20130309161112im_/http://dlmf.nist.gov/15/9/E8.png)
![n=0,1,2,\dots,N](https://webarchive.library.unt.edu/web/20130309161112im_/http://dlmf.nist.gov/15/9/m57.png)
compare also §15.2(ii).
¶ Meixner
![\mathop{M_{{n}}\/}\nolimits\!\left(x;\beta,c\right)=\mathop{F\/}\nolimits\!\left({-n,-x\atop\beta};1-\frac{1}{c}\right).](https://webarchive.library.unt.edu/web/20130309161112im_/http://dlmf.nist.gov/15/9/E9.png)
¶ Meixner–Pollaczek
![\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).](https://webarchive.library.unt.edu/web/20130309161112im_/http://dlmf.nist.gov/15/9/E10.png)
§15.9(ii) Jacobi Function
This is a generalization of Jacobi polynomials (§18.3) and has the representation
![\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).](https://webarchive.library.unt.edu/web/20130309161112im_/http://dlmf.nist.gov/15/9/E11.png)
The Jacobi transform is defined as
![\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,](https://webarchive.library.unt.edu/web/20130309161112im_/http://dlmf.nist.gov/15/9/E12.png)
with inverse
![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,](https://webarchive.library.unt.edu/web/20130309161112im_/http://dlmf.nist.gov/15/9/E13.png)
where the contour of integration is located to the right of the poles of the gamma functions in the integrand, and
![\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).](https://webarchive.library.unt.edu/web/20130309161112im_/http://dlmf.nist.gov/15/9/E14.png)
For this result, together with restrictions on the functions and
, see Koornwinder (1984a).
§15.9(iii) Gegenbauer Function
This is a generalization of Gegenbauer (or ultraspherical) polynomials (§18.3). It is defined by:
![\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).](https://webarchive.library.unt.edu/web/20130309161112im_/http://dlmf.nist.gov/15/9/E15.png)
§15.9(iv) Associated Legendre Functions; Ferrers Functions
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.
![\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),](https://webarchive.library.unt.edu/web/20130309161112im_/http://dlmf.nist.gov/15/9/E16.png)
![b\neq 0,-1,-2,\dots](https://webarchive.library.unt.edu/web/20130309161112im_/http://dlmf.nist.gov/15/9/m51.png)
![|\mathop{\mathrm{ph}\/}\nolimits(1-z)|<\pi](https://webarchive.library.unt.edu/web/20130309161112im_/http://dlmf.nist.gov/15/9/m63.png)
![|1-z|<1](https://webarchive.library.unt.edu/web/20130309161112im_/http://dlmf.nist.gov/15/9/m61.png)
![\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),](https://webarchive.library.unt.edu/web/20130309161112im_/http://dlmf.nist.gov/15/9/E17.png)
![|\mathop{\mathrm{ph}\/}\nolimits z|<\pi](https://webarchive.library.unt.edu/web/20130309161112im_/http://dlmf.nist.gov/15/9/m62.png)
![|\mathop{\mathrm{ph}\/}\nolimits(1-z)|<\pi](https://webarchive.library.unt.edu/web/20130309161112im_/http://dlmf.nist.gov/15/9/m63.png)
![\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),](https://webarchive.library.unt.edu/web/20130309161112im_/http://dlmf.nist.gov/15/9/E18.png)
![\left|\mathop{\mathrm{ph}\/}\nolimits\!\left(-z\right)\right|<\pi](https://webarchive.library.unt.edu/web/20130309161112im_/http://dlmf.nist.gov/15/9/m25.png)
![\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),](https://webarchive.library.unt.edu/web/20130309161112im_/http://dlmf.nist.gov/15/9/E19.png)
![|\mathop{\mathrm{ph}\/}\nolimits z|<\pi](https://webarchive.library.unt.edu/web/20130309161112im_/http://dlmf.nist.gov/15/9/m62.png)
![|\mathop{\mathrm{ph}\/}\nolimits(1-z)|<\pi](https://webarchive.library.unt.edu/web/20130309161112im_/http://dlmf.nist.gov/15/9/m63.png)
![\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),](https://webarchive.library.unt.edu/web/20130309161112im_/http://dlmf.nist.gov/15/9/E20.png)
![\left|\mathop{\mathrm{ph}\/}\nolimits\!\left(-z\right)\right|<\pi](https://webarchive.library.unt.edu/web/20130309161112im_/http://dlmf.nist.gov/15/9/m25.png)
![\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),](https://webarchive.library.unt.edu/web/20130309161112im_/http://dlmf.nist.gov/15/9/E21.png)
![\left|\mathop{\mathrm{ph}\/}\nolimits\!\left(-z\right)\right|<\pi](https://webarchive.library.unt.edu/web/20130309161112im_/http://dlmf.nist.gov/15/9/m25.png)
![\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),](https://webarchive.library.unt.edu/web/20130309161112im_/http://dlmf.nist.gov/15/9/E22.png)
![a,b\neq-\frac{1}{2},-\frac{3}{2},-\frac{5}{2},\ldots](https://webarchive.library.unt.edu/web/20130309161112im_/http://dlmf.nist.gov/15/9/m47.png)
![0<|\mathop{\mathrm{ph}\/}\nolimits z|<\pi](https://webarchive.library.unt.edu/web/20130309161112im_/http://dlmf.nist.gov/15/9/m21.png)
where the sign in the exponential is according as
.
![\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),](https://webarchive.library.unt.edu/web/20130309161112im_/http://dlmf.nist.gov/15/9/E23.png)
![a,b\neq\frac{1}{2},-\frac{1}{2},-\frac{3}{2},\dots](https://webarchive.library.unt.edu/web/20130309161112im_/http://dlmf.nist.gov/15/9/m48.png)
![0<|\mathop{\mathrm{ph}\/}\nolimits z|<\pi](https://webarchive.library.unt.edu/web/20130309161112im_/http://dlmf.nist.gov/15/9/m21.png)
where the sign in the exponential is according as
.