# §18.12 Generating Functions

The $z$-radii of convergence will depend on $x$, and in first instance we will assume $x\in[-1,1]$ for Jacobi, ultraspherical, Chebyshev and Legendre, $x\in[0,\infty)$ for Laguerre, and $x\in\mathbb{R}$ for Hermite. With the notation of §§10.2(ii), 10.25(ii), 15.2, and 16.2,

## Jacobi

 18.12.1 $\frac{2^{\alpha+\beta}}{R(1+R-z)^{\alpha}(1+R+z)^{\beta}}=\sum_{n=0}^{\infty}P% ^{(\alpha,\beta)}_{n}\left(x\right)z^{n},$ $R=\sqrt{1-2xz+z^{2}}$, $|z|<1$, ⓘ Symbols: $P^{(\NVar{\alpha},\NVar{\beta})}_{\NVar{n}}\left(\NVar{x}\right)$: Jacobi polynomial, $z$: complex variable, $n$: nonnegative integer and $x$: real variable A&S Ref: 22.9.1 Referenced by: §18.12, §18.12 Permalink: http://dlmf.nist.gov/18.12.E1 Encodings: TeX, pMML, png See also: Annotations for §18.12, §18.12 and Ch.18
 18.12.2 ${{}_{0}{\mathbf{F}}_{1}}\left({-\atop\alpha+1};\frac{(x-1)z}{2}\right)\*{{}_{0% }{\mathbf{F}}_{1}}\left({-\atop\beta+1};\frac{(x+1)z}{2}\right)=\left(\tfrac{1% }{2}(1-x)z\right)^{-\frac{1}{2}\alpha}J_{\alpha}\left(\sqrt{2(1-x)z}\right)\*% \left(\tfrac{1}{2}(1+x)z\right)^{-\frac{1}{2}\beta}I_{\beta}\left(\sqrt{2(1+x)% z}\right)=\sum_{n=0}^{\infty}\frac{P^{(\alpha,\beta)}_{n}\left(x\right)}{% \Gamma\left(n+\alpha+1\right)\Gamma\left(n+\beta+1\right)}z^{n},$
 18.12.2_5 ${{}_{2}F_{1}}\left({\gamma,\alpha+\beta+1-\gamma\atop\alpha+1};\frac{1-R-z}{2}% \right)\*{{}_{2}F_{1}}\left({\gamma,\alpha+\beta+1-\gamma\atop\beta+1};\frac{1% -R+z}{2}\right)=\sum_{n=0}^{\infty}\frac{{\left(\gamma\right)_{n}}{\left(% \alpha+\beta+1-\gamma\right)_{n}}}{{\left(\alpha+1\right)_{n}}{\left(\beta+1% \right)_{n}}}P^{(\alpha,\beta)}_{n}\left(x\right)z^{n},$ $R=\sqrt{1-2xz+z^{2}}$, $|z|<1$, ⓘ Symbols: ${{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$: $=F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ notation for 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), $z$: complex variable, $n$: nonnegative integer and $x$: real variable Proved: Rainville (1960, §141, (2)); Brafman’s generating function(proved) Source: Koekoek et al. (2010, (9.8.15)) Referenced by: §18.12, §18.12, Erratum (V1.2.0) §18.12 Permalink: http://dlmf.nist.gov/18.12.E2_5 Encodings: TeX, pMML, png Addition (effective with 1.2.0): This equation was added. See also: Annotations for §18.12, §18.12 and Ch.18

with $\gamma$ arbitrary. Note that (18.12.2_5) yields (18.12.1) by putting $\gamma=0$ and (18.12.2) by replacing $z$ by $-\gamma^{-2}z$ and next letting $\gamma\to\infty$.

 18.12.3 $(1+z)^{-\alpha-\beta-1}\*{{}_{2}F_{1}}\left({\tfrac{1}{2}(\alpha+\beta+1),% \tfrac{1}{2}(\alpha+\beta+2)\atop\beta+1};\frac{2(x+1)z}{(1+z)^{2}}\right)=% \sum_{n=0}^{\infty}\frac{{\left(\alpha+\beta+1\right)_{n}}}{{\left(\beta+1% \right)_{n}}}P^{(\alpha,\beta)}_{n}\left(x\right)z^{n},$ $|z|<1$,
 18.12.3_5 $\frac{1+z}{(1-2xz+z^{2})^{\beta+\frac{3}{2}}}=\sum_{n=0}^{\infty}\frac{{\left(% 2\beta+2\right)_{n}}}{{\left(\beta+1\right)_{n}}}P^{(\beta+1,\beta)}_{n}\left(% x\right)z^{n},$ $|z|<1$, ⓘ Symbols: $P^{(\NVar{\alpha},\NVar{\beta})}_{\NVar{n}}\left(\NVar{x}\right)$: Jacobi polynomial, ${\left(\NVar{a}\right)_{\NVar{n}}}$: Pochhammer’s symbol (or shifted factorial), $z$: complex variable, $n$: nonnegative integer and $x$: real variable Proof sketch: Substitute $\alpha=\beta+1$ in (18.12.3) and use (15.4.6). Referenced by: §18.12, §18.12, Erratum (V1.2.0) §18.12 Permalink: http://dlmf.nist.gov/18.12.E3_5 Encodings: TeX, pMML, png Addition (effective with 1.2.0): This equation was added. See also: Annotations for §18.12, §18.12 and Ch.18

and similar formulas as (18.12.3) and (18.12.3_5) by symmetry; compare the second row in Table 18.6.1. See Ismail (2009, (4.3.2)) for another variant of (18.12.3).

## Ultraspherical

 18.12.4 $(1-2xz+z^{2})^{-\lambda}=\sum_{n=0}^{\infty}C^{(\lambda)}_{n}\left(x\right)z^{% n}=\sum_{n=0}^{\infty}\frac{{\left(2\lambda\right)_{n}}}{{\left(\lambda+\tfrac% {1}{2}\right)_{n}}}P^{(\lambda-\frac{1}{2},\lambda-\frac{1}{2})}_{n}\left(x% \right)z^{n},$ $|z|<1$.
 18.12.5 $\frac{1-xz}{(1-2xz+z^{2})^{\lambda+1}}=\sum_{n=0}^{\infty}\frac{n+2\lambda}{2% \lambda}C^{(\lambda)}_{n}\left(x\right)z^{n},$ $|z|<1$. ⓘ Symbols: $C^{(\NVar{\lambda})}_{\NVar{n}}\left(\NVar{x}\right)$: 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 and Ch.18
 18.12.6 $\Gamma\left(\lambda+\tfrac{1}{2}\right){\mathrm{e}}^{z\cos\theta}(\tfrac{1}{2}% z\sin\theta)^{\frac{1}{2}-\lambda}J_{\lambda-\frac{1}{2}}\left(z\sin\theta% \right)=\sum_{n=0}^{\infty}\frac{C^{(\lambda)}_{n}\left(\cos\theta\right)}{{% \left(2\lambda\right)_{n}}}z^{n},$ $0\leq\theta\leq\pi$.

## Chebyshev

 18.12.7 $\displaystyle\frac{1-z^{2}}{1-2xz+z^{2}}$ $\displaystyle=1+2\sum_{n=1}^{\infty}T_{n}\left(x\right)z^{n},$ $|z|<1$. ⓘ Symbols: $T_{\NVar{n}}\left(\NVar{x}\right)$: 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 and Ch.18 18.12.8 $\displaystyle\frac{1-xz}{1-2xz+z^{2}}$ $\displaystyle=\sum_{n=0}^{\infty}T_{n}\left(x\right)z^{n},$ $|z|<1$. ⓘ Symbols: $T_{\NVar{n}}\left(\NVar{x}\right)$: 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 and Ch.18
 18.12.9 $-\ln\left(1-2xz+z^{2}\right)=2\sum_{n=1}^{\infty}\frac{T_{n}\left(x\right)}{n}% z^{n},$ $|z|<1$. ⓘ Symbols: $T_{\NVar{n}}\left(\NVar{x}\right)$: Chebyshev polynomial of the first kind, $\ln\NVar{z}$: principal branch of logarithm function, $z$: complex variable, $n$: nonnegative integer and $x$: real variable A&S Ref: 22.9.8 Referenced by: §18.12 Permalink: http://dlmf.nist.gov/18.12.E9 Encodings: TeX, pMML, png See also: Annotations for §18.12, §18.12 and Ch.18
 18.12.10 $\frac{1}{1-2xz+z^{2}}=\sum_{n=0}^{\infty}U_{n}\left(x\right)z^{n},$ $|z|<1$. ⓘ Symbols: $U_{\NVar{n}}\left(\NVar{x}\right)$: 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, §18.12 and Ch.18

## Legendre

 18.12.11 $\frac{1}{\sqrt{1-2xz+z^{2}}}=\sum_{n=0}^{\infty}P_{n}\left(x\right)z^{n},$ $|z|<1$. ⓘ Symbols: $P_{\NVar{n}}\left(\NVar{x}\right)$: 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 and Ch.18
 18.12.12 ${\mathrm{e}}^{xz}J_{0}\left(z\sqrt{1-x^{2}}\right)=\sum_{n=0}^{\infty}\frac{P_% {n}\left(x\right)}{n!}z^{n}.$

## Laguerre

 18.12.13 $(1-z)^{-\alpha-1}\exp\left(\frac{xz}{z-1}\right)=\sum_{n=0}^{\infty}L^{(\alpha% )}_{n}\left(x\right)z^{n},$ $|z|<1$. ⓘ Symbols: $L^{(\NVar{\alpha})}_{\NVar{n}}\left(\NVar{x}\right)$: Laguerre (or generalized Laguerre) polynomial, $\exp\NVar{z}$: exponential function, $z$: complex variable, $n$: nonnegative integer and $x$: real variable A&S Ref: 22.9.15 Referenced by: §18.12, §18.18(ii), §18.18(iv), §18.18(viii), §18.2(xii), (8.7.6) Permalink: http://dlmf.nist.gov/18.12.E13 Encodings: TeX, pMML, png See also: Annotations for §18.12, §18.12 and Ch.18
 18.12.14 $\Gamma\left(\alpha+1\right)(xz)^{-\frac{1}{2}\alpha}{\mathrm{e}}^{z}J_{\alpha}% \left(2\sqrt{xz}\right)=\sum_{n=0}^{\infty}\frac{L^{(\alpha)}_{n}\left(x\right% )}{{\left(\alpha+1\right)_{n}}}z^{n}.$

## Hermite

 18.12.15 ${\mathrm{e}}^{2xz-z^{2}}=\sum_{n=0}^{\infty}\frac{H_{n}\left(x\right)}{n!}z^{n},$ ⓘ Symbols: $H_{\NVar{n}}\left(\NVar{x}\right)$: Hermite polynomial, $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $z$: complex variable, $n$: nonnegative integer and $x$: real variable A&S Ref: 22.9.17 Referenced by: §18.12, §18.17(v), §18.18(ii), §18.2(xii) Permalink: http://dlmf.nist.gov/18.12.E15 Encodings: TeX, pMML, png See also: Annotations for §18.12, §18.12 and Ch.18
 18.12.16 ${\mathrm{e}}^{xz-\frac{1}{2}z^{2}}=\sum_{n=0}^{\infty}\frac{\mathit{He}_{n}% \left(x\right)}{n!}z^{n},$
 18.12.17 $\frac{1+2xz+4z^{2}}{\left(1+4z^{2}\right)^{\frac{3}{2}}}\exp\left(\frac{4x^{2}% z^{2}}{1+4z^{2}}\right)=\sum_{n=0}^{\infty}\frac{H_{n}\left(x\right)}{\left% \lfloor n/2\right\rfloor!}z^{n},$ $|z|<1$. ⓘ Symbols: $H_{\NVar{n}}\left(\NVar{x}\right)$: Hermite polynomial, $\exp\NVar{z}$: exponential function, $!$: factorial (as in $n!$), $\left\lfloor\NVar{x}\right\rfloor$: floor of $x$, $z$: complex variable, $n$: nonnegative integer and $x$: real variable Source: Koekoek et al. (2010, (9.15.14)) Referenced by: §18.12, Erratum (V1.2.0) §18.12 Permalink: http://dlmf.nist.gov/18.12.E17 Encodings: TeX, pMML, png Addition (effective with 1.2.0): This equation was added. See also: Annotations for §18.12, §18.12 and Ch.18

See §18.18(vii) for Poisson kernels; these are special cases of bilateral generating functions.