18 Orthogonal Polynomials Classical Orthogonal Polynomials 18.11 Relations to Other Functions 18.13 Continued Fractions

§18.12 Generating Functions

ⓘ

Keywords:: classical orthogonal polynomials, generating functions
Notes:: For (18.12.1) see Andrews et al. (1999, (6.4.3)). For (18.12.2) see Bateman (1905, pp. 113–114) and Koornwinder (1974, p. 128). For (18.12.3) see Andrews et al. (1999, (6.4.7)). For (18.12.4) see Szegő (1975, (4.7.23)). (18.12.5) follows by combining (18.12.4) and its $z$ -differentiated form. For (18.12.6) see Rainville (1960, §144, (7)). For (18.12.7) see Andrews et al. (1999, (5.1.16)). (18.12.8) is an immediate consequence of (18.12.7). For (18.12.9) see Szegő (1975, (4.7.25)). (18.12.10) is the special case $\lambda=1$ of (18.12.4) in view of (18.7.4). (18.12.11) is the special case $\lambda=\frac{1}{2}$ of (18.12.4). (18.12.12) is the special case $\lambda=\frac{1}{2}$ of (18.12.6). For (18.12.13) see Andrews et al. (1999, (6.2.4)). For (18.12.14) see Szegő (1975, (5.1.16)). For (18.12.15) see Andrews et al. (1999, (6.1.7)).
Referenced by:: §18.10(iii), Erratum (V1.2.0) §18.12
Permalink:: http://dlmf.nist.gov/18.12
Addition (effective with 1.2.0):: Equations (18.12.2_5), (18.12.3_5), (18.12.17) were added.
See also:: Annotations for Ch.18

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

ⓘ

Keywords:: Jacobi polynomials, generating functions
See also:: Annotations for §18.12 and Ch.18

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		${{}_{0}{\mathbf{F}}_{1}}\left({-\atop\alpha+1};\frac{(x-1)z}{2}\right)\{{}_{0% }{\mathbf{F}}_{1}}\left({-\atop\beta+1};\frac{(x+1)z}{2}\right)=\left(\tfrac{1% }{2}(1-x)z\right)^{-\frac{1}{2}\alpha}J_{\alpha}\left(\sqrt{2(1-x)z}\right)\% \left(\tfrac{1}{2}(1+x)z\right)^{-\frac{1}{2}\beta}I_{\beta}\left(\sqrt{2(1+x)% z}\right)=\sum_{n=0}^{\infty}\frac{P^{(\alpha,\beta)}_{n}\left(x\right)}{% \Gamma\left(n+\alpha+1\right)\Gamma\left(n+\beta+1\right)}z^{n},$
ⓘ Symbols: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $\Gamma\left(\NVar{z}\right)$ : gamma function, $P^{(\NVar{\alpha},\NVar{\beta})}_{\NVar{n}}\left(\NVar{x}\right)$ : Jacobi polynomial, ${{}_{\NVar{p}}{\mathbf{F}}_{\NVar{q}}}\left(\NVar{\mathbf{a}};\NVar{\mathbf{b}% };\NVar{z}\right)$ or ${{}_{\NVar{p}}{\mathbf{F}}_{\NVar{q}}}\left({\NVar{\mathbf{a}}\atop\NVar{% \mathbf{b}}};\NVar{z}\right)$ : scaled (or Olver’s) generalized hypergeometric function, $I_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Bessel function of the first kind, $z$ : complex variable, $n$ : nonnegative integer and $x$ : real variable Source: Koekoek et al. (2010, (9.8.12)) Referenced by: §18.12, §18.12, Erratum (V1.2.0) for Equation (18.12.2) Permalink: http://dlmf.nist.gov/18.12.E2 Encodings: TeX, pMML, png See also: Annotations for §18.12, §18.12 and Ch.18

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		$(1+z)^{-\alpha-\beta-1}\*{{}_{2}F_{1}}\left({\tfrac{1}{2}(\alpha+\beta+1),% \tfrac{1}{2}(\alpha+\beta+2)\atop\beta+1};\frac{2(x+1)z}{(1+z)^{2}}\right)=% \sum_{n=0}^{\infty}\frac{{\left(\alpha+\beta+1\right)_{n}}}{{\left(\beta+1% \right)_{n}}}P^{(\alpha,\beta)}_{n}\left(x\right)z^{n},$
$\|z\|<1$ ,
ⓘ 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 Referenced by: (18.12.3_5), §18.12, §18.12 Permalink: http://dlmf.nist.gov/18.12.E3 Encodings: TeX, pMML, png See also: Annotations for §18.12, §18.12 and Ch.18

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

ⓘ

Keywords:: generating functions, ultraspherical polynomials
See also:: Annotations for §18.12 and Ch.18

18.12.4		$(1-2xz+z^{2})^{-\lambda}=\sum_{n=0}^{\infty}C^{(\lambda)}_{n}\left(x\right)z^{% n}=\sum_{n=0}^{\infty}\frac{{\left(2\lambda\right)_{n}}}{{\left(\lambda+\tfrac% {1}{2}\right)_{n}}}P^{(\lambda-\frac{1}{2},\lambda-\frac{1}{2})}_{n}\left(x% \right)z^{n},$
$\|z\|<1$ .
ⓘ 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), $C^{(\NVar{\lambda})}_{\NVar{n}}\left(\NVar{x}\right)$ : ultraspherical (or Gegenbauer) polynomial, $z$ : complex variable, $n$ : nonnegative integer and $x$ : real variable A&S Ref: 22.9.3 Referenced by: §18.12, §18.18(viii) Permalink: http://dlmf.nist.gov/18.12.E4 Encodings: TeX, pMML, png See also: Annotations for §18.12, §18.12 and Ch.18

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		$\Gamma\left(\lambda+\tfrac{1}{2}\right){\mathrm{e}}^{z\cos\theta}(\tfrac{1}{2}% z\sin\theta)^{\frac{1}{2}-\lambda}J_{\lambda-\frac{1}{2}}\left(z\sin\theta% \right)=\sum_{n=0}^{\infty}\frac{C^{(\lambda)}_{n}\left(\cos\theta\right)}{{% \left(2\lambda\right)_{n}}}z^{n},$
$0\leq\theta\leq\pi$ .
ⓘ Symbols: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $\Gamma\left(\NVar{z}\right)$ : gamma function, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\mathrm{e}$ : base of natural logarithm, $\sin\NVar{z}$ : sine function, $C^{(\NVar{\lambda})}_{\NVar{n}}\left(\NVar{x}\right)$ : ultraspherical (or Gegenbauer) polynomial, $z$ : complex variable and $n$ : nonnegative integer Referenced by: §18.12 Permalink: http://dlmf.nist.gov/18.12.E6 Encodings: TeX, pMML, png See also: Annotations for §18.12, §18.12 and Ch.18

Chebyshev

ⓘ

Keywords:: Chebyshev polynomials, generating functions
See also:: Annotations for §18.12 and Ch.18

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

ⓘ

Keywords:: Legendre polynomials, generating functions
See also:: Annotations for §18.12 and Ch.18

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		${\mathrm{e}}^{xz}J_{0}\left(z\sqrt{1-x^{2}}\right)=\sum_{n=0}^{\infty}\frac{P_% {n}\left(x\right)}{n!}z^{n}.$
ⓘ Symbols: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $P_{\NVar{n}}\left(\NVar{x}\right)$ : Legendre polynomial, $\mathrm{e}$ : base of natural logarithm, $!$ : factorial (as in $n!$ ), $z$ : complex variable, $n$ : nonnegative integer and $x$ : real variable Referenced by: §18.12 Permalink: http://dlmf.nist.gov/18.12.E12 Encodings: TeX, pMML, png See also: Annotations for §18.12, §18.12 and Ch.18

Laguerre

ⓘ

Keywords:: Laguerre polynomials, generating functions
See also:: Annotations for §18.12 and Ch.18

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		$\Gamma\left(\alpha+1\right)(xz)^{-\frac{1}{2}\alpha}{\mathrm{e}}^{z}J_{\alpha}% \left(2\sqrt{xz}\right)=\sum_{n=0}^{\infty}\frac{L^{(\alpha)}_{n}\left(x\right% )}{{\left(\alpha+1\right)_{n}}}z^{n}.$
ⓘ Symbols: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $\Gamma\left(\NVar{z}\right)$ : gamma function, $L^{(\NVar{\alpha})}_{\NVar{n}}\left(\NVar{x}\right)$ : Laguerre (or generalized Laguerre) polynomial, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $\mathrm{e}$ : base of natural logarithm, $z$ : complex variable, $n$ : nonnegative integer and $x$ : real variable A&S Ref: 22.9.16 Referenced by: §18.12 Permalink: http://dlmf.nist.gov/18.12.E14 Encodings: TeX, pMML, png See also: Annotations for §18.12, §18.12 and Ch.18

Hermite

ⓘ

Keywords:: Hermite polynomials, generating functions
See also:: Annotations for §18.12 and Ch.18

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		${\mathrm{e}}^{xz-\frac{1}{2}z^{2}}=\sum_{n=0}^{\infty}\frac{\mathit{He}_{n}% \left(x\right)}{n!}z^{n},$
ⓘ Symbols: $\mathit{He}_{\NVar{n}}\left(\NVar{x}\right)$ : Hermite polynomial, $\mathrm{e}$ : base of natural logarithm, $!$ : factorial (as in $n!$ ), $z$ : complex variable, $n$ : nonnegative integer and $x$ : real variable Referenced by: §18.17(v) Permalink: http://dlmf.nist.gov/18.12.E16 Encodings: TeX, pMML, png See also: Annotations for §18.12, §18.12 and Ch.18

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.

18.11 Relations to Other Functions 18.13 Continued Fractions

Site Privacy Accessibility Privacy Program Copyrights Vulnerability Disclosure No Fear Act Policy FOIA Environmental Policy Scientific Integrity Information Quality Standards Commerce.gov Science.gov USA.gov