# §15.9(i) Orthogonal Polynomials

For the notation see §§18.3 and 18.19.

# ¶ Jacobi

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

# ¶ Gegenbauer (or Ultraspherical)

 15.9.2 $\displaystyle\mathop{C^{(\lambda)}_{n}\/}\nolimits\!\left(x\right)$ $\displaystyle=\frac{\left(2\lambda\right)_{n}}{n!}\mathop{F\/}\nolimits\!\left% ({-n,n+2\lambda\atop\lambda+\frac{1}{2}};\frac{1-x}{2}\right).$ 15.9.3 $\displaystyle\mathop{C^{(\lambda)}_{n}\/}\nolimits\!\left(x\right)$ $\displaystyle=(2x)^{n}\frac{\left(\lambda\right)_{n}}{n!}\mathop{F\/}\nolimits% \!\left({-\frac{1}{2}n,\frac{1}{2}(1-n)\atop 1-\lambda-n};\frac{1}{x^{2}}% \right).$
 15.9.4 $\mathop{C^{(\lambda)}_{n}\/}\nolimits\!\left(\mathop{\cos\/}\nolimits\theta% \right)=e^{ni\theta}\frac{\left(\lambda\right)_{n}}{n!}\mathop{F\/}\nolimits\!% \left({-n,\lambda\atop 1-\lambda-n};e^{-2i\theta}\right).$

# ¶ Chebyshev

 15.9.5 $\displaystyle\mathop{T_{n}\/}\nolimits\!\left(x\right)$ $\displaystyle=\mathop{F\/}\nolimits\!\left({-n,n\atop\frac{1}{2}};\frac{1-x}{2% }\right).$ 15.9.6 $\displaystyle\mathop{U_{n}\/}\nolimits\!\left(x\right)$ $\displaystyle=(n+1)\mathop{F\/}\nolimits\!\left({-n,n+2\atop\frac{3}{2}};\frac% {1-x}{2}\right).$

# ¶ Legendre

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

# ¶ Krawtchouk

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

compare also §15.2(ii).

# ¶ Meixner

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

# ¶ Meixner–Pollaczek

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

# §15.9(ii) Jacobi Function

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

with inverse

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

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

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

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

# §15.9(iii) Gegenbauer Function

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

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

# §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.

 15.9.16 $\displaystyle\mathop{\mathbf{F}\/}\nolimits\!\left({a,b\atop 2b};z\right)$ $\displaystyle=\frac{\sqrt{\pi}}{\mathop{\Gamma\/}\nolimits\!\left(b\right)}z^{% -b+(\ifrac{1}{2})}(1-z)^{(b-a-(\ifrac{1}{2}))/2}\*\mathop{P^{-b+(\ifrac{1}{2})% }_{a-b-(\ifrac{1}{2})}\/}\nolimits\!\left(\frac{2-z}{2\sqrt{1-z}}\right),$ $b\neq 0,-1,-2,\dots$, $|\mathop{\mathrm{ph}\/}\nolimits(1-z)|<\pi$ and $|1-z|<1$. 15.9.17 $\displaystyle\mathop{\mathbf{F}\/}\nolimits\!\left({a,a+\tfrac{1}{2}\atop c};z\right)$ $\displaystyle=2^{c-1}z^{\ifrac{(1-c)}{2}}(1-z)^{-a+(\ifrac{(c-1)}{2})}\*% \mathop{P^{1-c}_{2a-c}\/}\nolimits\!\left(\frac{1}{\sqrt{1-z}}\right),$ $|\mathop{\mathrm{ph}\/}\nolimits z|<\pi$ and $|\mathop{\mathrm{ph}\/}\nolimits(1-z)|<\pi$. 15.9.18 $\displaystyle\mathop{\mathbf{F}\/}\nolimits\!\left({a,b\atop a+b+\tfrac{1}{2}}% ;z\right)$ $\displaystyle=2^{a+b-(\ifrac{1}{2})}(-z)^{(-a-b+(\ifrac{1}{2}))/2}\*\mathop{P^% {-a-b+(\ifrac{1}{2})}_{a-b-(\ifrac{1}{2})}\/}\nolimits\!\left(\sqrt{1-z}\right),$ $\left|\mathop{\mathrm{ph}\/}\nolimits\!\left(-z\right)\right|<\pi$. 15.9.19 $\displaystyle\mathop{\mathbf{F}\/}\nolimits\!\left({a,b\atop a-b+1};z\right)$ $\displaystyle=z^{\ifrac{(b-a)}{2}}(1-z)^{-b}\*\mathop{P^{b-a}_{-b}\/}\nolimits% \!\left(\frac{1+z}{1-z}\right),$ $|\mathop{\mathrm{ph}\/}\nolimits z|<\pi$ and $|\mathop{\mathrm{ph}\/}\nolimits(1-z)|<\pi$. 15.9.20 $\displaystyle\mathop{\mathbf{F}\/}\nolimits\!\left({a,b\atop\tfrac{1}{2}(a+b+1% )};z\right)$ $\displaystyle=\left(-z(1-z)\right)^{\ifrac{(1-a-b)}{4}}\*\mathop{P^{\ifrac{(1-% a-b)}{2}}_{\ifrac{(a-b-1)}{2}}\/}\nolimits\!\left(1-2z\right),$ $\left|\mathop{\mathrm{ph}\/}\nolimits\!\left(-z\right)\right|<\pi$. 15.9.21 $\displaystyle\mathop{\mathbf{F}\/}\nolimits\!\left({a,1-a\atop c};z\right)$ $\displaystyle=\left(\frac{-z}{1-z}\right)^{\ifrac{(1-c)}{2}}\*\mathop{P^{1-c}_% {-a}\/}\nolimits\!\left(1-2z\right),$ $\left|\mathop{\mathrm{ph}\/}\nolimits\!\left(-z\right)\right|<\pi$.
 15.9.22 $\mathop{\mathbf{F}\/}\nolimits\!\left({a,b\atop\tfrac{1}{2}};z\right)=\frac{2^% {a+b-(\ifrac{3}{2})}}{\pi}\mathop{\Gamma\/}\nolimits\!\left(a+\tfrac{1}{2}% \right)\mathop{\Gamma\/}\nolimits\!\left(b+\tfrac{1}{2}\right)\*(z-1)^{(-a-b+(% \ifrac{1}{2}))/2}\*\left(e^{\pm\pi i(a+b-(\ifrac{1}{2}))}\mathop{P^{-a-b+(% \ifrac{1}{2})}_{a-b-(\ifrac{1}{2})}\/}\nolimits\!\left(-\sqrt{z}\right)+% \mathop{P^{-a-b+(\ifrac{1}{2})}_{a-b-(\ifrac{1}{2})}\/}\nolimits\!\left(\sqrt{% z}\right)\right),$ $a,b\neq-\frac{1}{2},-\frac{3}{2},-\frac{5}{2},\ldots$, $0<|\mathop{\mathrm{ph}\/}\nolimits z|<\pi$,

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

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

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