§18.12 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 Szegö (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 Szegö (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 Szegö (1975, (5.1.16)). For (18.12.15) see Andrews et al. (1999, (6.1.7)).
Keywords:: classical orthogonal polynomials
Referenced by:: §18.10(iii)
Permalink:: http://dlmf.nist.gov/18.12

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

¶ Jacobi

Keywords:: Jacobi polynomials

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:: $\mathop{P^{{(\alpha,\beta)}}_{{n}}\/}\nolimits\!\left(x\right)$ : 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

18.12.2 $\left(\tfrac{1}{2}(1-x)z\right)^{{-\frac{1}{2}\alpha}}\mathop{J_{{\alpha}}\/}\nolimits\!\left(\sqrt{2(1-x)z}\right)\*\left(\tfrac{1}{2}(1+x)z\right)^{{-\frac{1}{2}\beta}}\mathop{I_{{\beta}}\/}\nolimits\!\left(\sqrt{2(1+x)z}\right)=\sum _{{n=0}}^{\infty}\frac{\mathop{P^{{(\alpha,\beta)}}_{{n}}\/}\nolimits\!\left(x\right)}{\mathop{\Gamma\/}\nolimits\!\left(n+\alpha+1\right)\mathop{\Gamma\/}\nolimits\!\left(n+\beta+1\right)}z^{n}.$

Symbols:: $\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the first kind, $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{P^{{(\alpha,\beta)}}_{{n}}\/}\nolimits\!\left(x\right)$ : Jacobi polynomial, $\mathop{I_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified Bessel function, $z$ : complex variable, $n$ : nonnegative integer and $x$ : real variable
Referenced by:: §18.12
Permalink:: http://dlmf.nist.gov/18.12.E2
Encodings:: TeX, pMML, png

18.12.3 $(1+z)^{{-\alpha-\beta-1}}\*\mathop{{{}_{{2}}F_{{1}}}\/}\nolimits\!\left({\tfrac{1}{2}(\alpha+\beta+1),\tfrac{1}{2}(\alpha+\beta+2)\atop\beta+1};\frac{2(x+1)z}{(1+z)^{2}}\right)=\sum _{{n=0}}^{\infty}\frac{\left(\alpha+\beta+1\right)_{{n}}}{\left(\beta+1\right)_{{n}}}\mathop{P^{{(\alpha,\beta)}}_{{n}}\/}\nolimits\!\left(x\right)z^{n},$ $|z|<1$ ,

Symbols:: $\mathop{{{}_{{p}}F_{{q}}}\/}\nolimits\!\left({a_{1},\dots,a_{p}\atop b_{1},\dots,b_{q}};z\right)$ : generalized hypergeometric function, $\mathop{P^{{(\alpha,\beta)}}_{{n}}\/}\nolimits\!\left(x\right)$ : Jacobi polynomial, $\left(a\right)_{{n}}$ : Pochhammer’s symbol, $z$ : complex variable, $n$ : nonnegative integer and $x$ : real variable
Referenced by:: §18.12
Permalink:: http://dlmf.nist.gov/18.12.E3
Encodings:: TeX, pMML, png

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

Keywords:: ultraspherical polynomials

18.12.4 $(1-2xz+z^{2})^{{-\lambda}}=\sum _{{n=0}}^{\infty}\mathop{C^{{(\lambda)}}_{{n}}\/}\nolimits\!\left(x\right)z^{n}=\sum _{{n=0}}^{\infty}\frac{\left(2\lambda\right)_{{n}}}{\left(\lambda+\tfrac{1}{2}\right)_{{n}}}\mathop{P^{{(\lambda-\frac{1}{2},\lambda-\frac{1}{2})}}_{{n}}\/}\nolimits\!\left(x\right)z^{n},$ $|z|<1$ .

Symbols:: $\mathop{P^{{(\alpha,\beta)}}_{{n}}\/}\nolimits\!\left(x\right)$ : Jacobi polynomial, $\left(a\right)_{{n}}$ : Pochhammer’s symbol, $\mathop{C^{{(\lambda)}}_{{n}}\/}\nolimits\!\left(x\right)$ : 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

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:: $\mathop{C^{{(\lambda)}}_{{n}}\/}\nolimits\!\left(x\right)$ : 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

18.12.6 $\mathop{\Gamma\/}\nolimits\!\left(\lambda+\tfrac{1}{2}\right)e^{{z\mathop{\cos\/}\nolimits\theta}}(\tfrac{1}{2}z\mathop{\sin\/}\nolimits\theta)^{{\frac{1}{2}-\lambda}}\mathop{J_{{\lambda-\frac{1}{2}}}\/}\nolimits\!\left(z\mathop{\sin\/}\nolimits\theta\right)=\sum _{{n=0}}^{\infty}\frac{\mathop{C^{{(\lambda)}}_{{n}}\/}\nolimits\!\left(\mathop{\cos\/}\nolimits\theta\right)}{\left(2\lambda\right)_{{n}}}z^{n},$ $0\leq\theta\leq\pi$ .

Symbols:: $\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the first kind, $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\left(a\right)_{{n}}$ : Pochhammer’s symbol, $\mathop{\cos\/}\nolimits z$ : cosine function, $e$ : base of exponential function, $\mathop{\sin\/}\nolimits z$ : sine function, $\mathop{C^{{(\lambda)}}_{{n}}\/}\nolimits\!\left(x\right)$ : 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

¶ Chebyshev

Keywords:: Chebyshev polynomials

18.12.7 $\frac{1-z^{2}}{1-2xz+z^{2}}=1+2\sum _{{n=1}}^{\infty}\mathop{T_{{n}}\/}\nolimits\!\left(x\right)z^{n},$ $|z|<1$ .

Symbols:: $\mathop{T_{{n}}\/}\nolimits\!\left(x\right)$ : 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

18.12.8 $\frac{1-xz}{1-2xz+z^{2}}=\sum _{{n=0}}^{\infty}\mathop{T_{{n}}\/}\nolimits\!\left(x\right)z^{n},$ $|z|<1$ .

Symbols:: $\mathop{T_{{n}}\/}\nolimits\!\left(x\right)$ : 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

18.12.9 $-\mathop{\ln\/}\nolimits\!\left(1-2xz+z^{2}\right)=2\sum _{{n=1}}^{\infty}\frac{\mathop{T_{{n}}\/}\nolimits\!\left(x\right)}{n}z^{n},$ $|z|<1$ .

Symbols:: $\mathop{T_{{n}}\/}\nolimits\!\left(x\right)$ : Chebyshev polynomial of the first kind, $\mathop{\ln\/}\nolimits z$ : 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

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:: $\mathop{U_{{n}}\/}\nolimits\!\left(x\right)$ : 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

¶ Legendre

Keywords:: Legendre polynomials

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:: $\mathop{P_{{n}}\/}\nolimits\!\left(x\right)$ : 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

18.12.12 $e^{{xz}}\mathop{J_{{0}}\/}\nolimits\!\left(z\sqrt{1-x^{2}}\right)=\sum _{{n=0}}^{\infty}\frac{\mathop{P_{{n}}\/}\nolimits\!\left(x\right)}{n!}z^{n}.$

Symbols:: $\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the first kind, $\mathop{P_{{n}}\/}\nolimits\!\left(x\right)$ : Legendre polynomial, $e$ : base of exponential function, $!$ : $n!$ : factorial, $z$ : 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

¶ Laguerre

Keywords:: Laguerre polynomials

18.12.13 $(1-z)^{{-\alpha-1}}\mathop{\exp\/}\nolimits\!\left(\frac{xz}{z-1}\right)=\sum _{{n=0}}^{\infty}\mathop{L^{{(\alpha)}}_{{n}}\/}\nolimits\!\left(x\right)z^{n},$ $|z|<1$ .

Symbols:: $\mathop{\exp\/}\nolimits z$ : exponential function, $\mathop{L^{{(\alpha)}}_{{n}}\/}\nolimits\!\left(x\right)$ : Laguerre (or generalized Laguerre) polynomial, $z$ : 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)
Permalink:: http://dlmf.nist.gov/18.12.E13
Encodings:: TeX, pMML, png

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

Symbols:: $\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the first kind, $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\left(a\right)_{{n}}$ : Pochhammer’s symbol, $e$ : base of exponential function, $\mathop{L^{{(\alpha)}}_{{n}}\/}\nolimits\!\left(x\right)$ : Laguerre (or generalized Laguerre) polynomial, $z$ : 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

¶ Hermite

Keywords:: Hermite polynomials

18.12.15 $e^{{2xz-z^{2}}}=\sum _{{n=0}}^{\infty}\frac{\mathop{H_{{n}}\/}\nolimits\!\left(x\right)}{n!}z^{n},$

Symbols:: $\mathop{H_{{n}}\/}\nolimits\!\left(x\right)$ : Hermite polynomial, $e$ : base of exponential function, $!$ : $n!$ : factorial, $z$ : 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)
Permalink:: http://dlmf.nist.gov/18.12.E15
Encodings:: TeX, pMML, png

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

Symbols:: $\mathop{\mathit{He}_{{n}}\/}\nolimits\!\left(x\right)$ : Hermite polynomial, $e$ : base of exponential function, $!$ : $n!$ : factorial, $z$ : 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