# §18.12 Generating Functions

With the notation of §§10.2(ii), 10.25(ii), and 15.2,

## Jacobi

 18.12.1 $\frac{2^{\alpha+\beta}}{R(1+R-z)^{\alpha}(1+R+z)^{\beta}}=\sum_{n=0}^{\infty}% \mathop{P^{(\alpha,\beta)}_{n}\/}\nolimits\!\left(x\right)z^{n},$ $R=\sqrt{1-2xz+z^{2}}$, $|z|<1$. Symbols: $\mathop{P^{(\NVar{\alpha},\NVar{\beta})}_{\NVar{n}}\/}\nolimits\!\left(\NVar{x% }\right)$: Jacobi polynomial, $z$: complex variable, $n$: nonnegative integer and $x$: real variable A&S Ref: 22.9.1 Referenced by: §18.12 Permalink: http://dlmf.nist.gov/18.12.E1 Encodings: TeX, pMML, png See also: Annotations for 18.12
 18.12.2 $\left(\tfrac{1}{2}(1-x)z\right)^{-\frac{1}{2}\alpha}\mathop{J_{\alpha}\/}% \nolimits\!\left(\sqrt{2(1-x)z}\right)\*\left(\tfrac{1}{2}(1+x)z\right)^{-% \frac{1}{2}\beta}\mathop{I_{\beta}\/}\nolimits\!\left(\sqrt{2(1+x)z}\right)=% \sum_{n=0}^{\infty}\frac{\mathop{P^{(\alpha,\beta)}_{n}\/}\nolimits\!\left(x% \right)}{\mathop{\Gamma\/}\nolimits\!\left(n+\alpha+1\right)\mathop{\Gamma\/}% \nolimits\!\left(n+\beta+1\right)}z^{n}.$
 18.12.3 $(1+z)^{-\alpha-\beta-1}\*\mathop{{{}_{2}F_{1}}\/}\nolimits\!\left({\tfrac{1}{2% }(\alpha+\beta+1),\tfrac{1}{2}(\alpha+\beta+2)\atop\beta+1};\frac{2(x+1)z}{(1+% z)^{2}}\right)=\sum_{n=0}^{\infty}\frac{{\left(\alpha+\beta+1\right)_{n}}}{{% \left(\beta+1\right)_{n}}}\mathop{P^{(\alpha,\beta)}_{n}\/}\nolimits\!\left(x% \right)z^{n},$ $|z|<1$,

and a similar formula by symmetry; compare the second row in Table 18.6.1. For the hypergeometric function $\mathop{{{}_{2}F_{1}}\/}\nolimits$ see §§15.1, 15.2(i).

## Ultraspherical

 18.12.4 $(1-2xz+z^{2})^{-\lambda}=\sum_{n=0}^{\infty}\mathop{C^{(\lambda)}_{n}\/}% \nolimits\!\left(x\right)z^{n}=\sum_{n=0}^{\infty}\frac{{\left(2\lambda\right)% _{n}}}{{\left(\lambda+\tfrac{1}{2}\right)_{n}}}\mathop{P^{(\lambda-\frac{1}{2}% ,\lambda-\frac{1}{2})}_{n}\/}\nolimits\!\left(x\right)z^{n},$ $|z|<1$.
 18.12.5 $\frac{1-xz}{(1-2xz+z^{2})^{\lambda+1}}=\sum_{n=0}^{\infty}\frac{n+2\lambda}{2% \lambda}\mathop{C^{(\lambda)}_{n}\/}\nolimits\!\left(x\right)z^{n},$ $|z|<1$. Symbols: $\mathop{C^{(\NVar{\lambda})}_{\NVar{n}}\/}\nolimits\!\left(\NVar{x}\right)$: ultraspherical (or Gegenbauer) polynomial, $z$: complex variable, $n$: nonnegative integer and $x$: real variable Referenced by: §18.12, §18.18(viii) Permalink: http://dlmf.nist.gov/18.12.E5 Encodings: TeX, pMML, png See also: Annotations for 18.12
 18.12.6 $\mathop{\Gamma\/}\nolimits\!\left(\lambda+\tfrac{1}{2}\right)e^{z\mathop{\cos% \/}\nolimits\theta}(\tfrac{1}{2}z\mathop{\sin\/}\nolimits\theta)^{\frac{1}{2}-% \lambda}\mathop{J_{\lambda-\frac{1}{2}}\/}\nolimits\!\left(z\mathop{\sin\/}% \nolimits\theta\right)=\sum_{n=0}^{\infty}\frac{\mathop{C^{(\lambda)}_{n}\/}% \nolimits\!\left(\mathop{\cos\/}\nolimits\theta\right)}{{\left(2\lambda\right)% _{n}}}z^{n},$ $0\leq\theta\leq\pi$.

## Chebyshev

 18.12.7 $\displaystyle\frac{1-z^{2}}{1-2xz+z^{2}}$ $\displaystyle=1+2\sum_{n=1}^{\infty}\mathop{T_{n}\/}\nolimits\!\left(x\right)z% ^{n},$ $|z|<1$. Symbols: $\mathop{T_{\NVar{n}}\/}\nolimits\!\left(\NVar{x}\right)$: Chebyshev polynomial of the first kind, $z$: complex variable, $n$: nonnegative integer and $x$: real variable Referenced by: §18.12 Permalink: http://dlmf.nist.gov/18.12.E7 Encodings: TeX, pMML, png See also: Annotations for 18.12 18.12.8 $\displaystyle\frac{1-xz}{1-2xz+z^{2}}$ $\displaystyle=\sum_{n=0}^{\infty}\mathop{T_{n}\/}\nolimits\!\left(x\right)z^{n},$ $|z|<1$. Symbols: $\mathop{T_{\NVar{n}}\/}\nolimits\!\left(\NVar{x}\right)$: Chebyshev polynomial of the first kind, $z$: complex variable, $n$: nonnegative integer and $x$: real variable A&S Ref: 22.9.6 Referenced by: §18.12 Permalink: http://dlmf.nist.gov/18.12.E8 Encodings: TeX, pMML, png See also: Annotations for 18.12
 18.12.9 $-\mathop{\ln\/}\nolimits\!\left(1-2xz+z^{2}\right)=2\sum_{n=1}^{\infty}\frac{% \mathop{T_{n}\/}\nolimits\!\left(x\right)}{n}z^{n},$ $|z|<1$.
 18.12.10 $\frac{1}{1-2xz+z^{2}}=\sum_{n=0}^{\infty}\mathop{U_{n}\/}\nolimits\!\left(x% \right)z^{n},$ $|z|<1$. Symbols: $\mathop{U_{\NVar{n}}\/}\nolimits\!\left(\NVar{x}\right)$: Chebyshev polynomial of the second kind, $z$: complex variable, $n$: nonnegative integer and $x$: real variable A&S Ref: 22.9.10 Referenced by: §18.12 Permalink: http://dlmf.nist.gov/18.12.E10 Encodings: TeX, pMML, png See also: Annotations for 18.12

## Legendre

 18.12.11 $\frac{1}{\sqrt{1-2xz+z^{2}}}=\sum_{n=0}^{\infty}\mathop{P_{n}\/}\nolimits\!% \left(x\right)z^{n},$ $|z|<1$. Symbols: $\mathop{P_{\NVar{n}}\/}\nolimits\!\left(\NVar{x}\right)$: Legendre polynomial, $z$: complex variable, $n$: nonnegative integer and $x$: real variable A&S Ref: 22.9.12 Referenced by: §18.12 Permalink: http://dlmf.nist.gov/18.12.E11 Encodings: TeX, pMML, png See also: Annotations for 18.12
 18.12.12 $e^{xz}\mathop{J_{0}\/}\nolimits\!\left(z\sqrt{1-x^{2}}\right)=\sum_{n=0}^{% \infty}\frac{\mathop{P_{n}\/}\nolimits\!\left(x\right)}{n!}z^{n}.$

## Laguerre

 18.12.13 $(1-z)^{-\alpha-1}\mathop{\exp\/}\nolimits\!\left(\frac{xz}{z-1}\right)=\sum_{n% =0}^{\infty}\mathop{L^{(\alpha)}_{n}\/}\nolimits\!\left(x\right)z^{n},$ $|z|<1$.
 18.12.14 $\mathop{\Gamma\/}\nolimits\!\left(\alpha+1\right)(xz)^{-\frac{1}{2}\alpha}e^{z% }\mathop{J_{\alpha}\/}\nolimits\!\left(2\sqrt{xz}\right)=\sum_{n=0}^{\infty}% \frac{\mathop{L^{(\alpha)}_{n}\/}\nolimits\!\left(x\right)}{{\left(\alpha+1% \right)_{n}}}z^{n}.$

## Hermite

 18.12.15 $e^{2xz-z^{2}}=\sum_{n=0}^{\infty}\frac{\mathop{H_{n}\/}\nolimits\!\left(x% \right)}{n!}z^{n},$
 18.12.16 $e^{xz-\frac{1}{2}z^{2}}=\sum_{n=0}^{\infty}\frac{\mathop{\mathit{He}_{n}\/}% \nolimits\!\left(x\right)}{n!}z^{n}.$