§18.18 Sums

§18.18(i) Series Expansions of Arbitrary Functions

Jacobi

Let $f(z)$ be analytic within an ellipse $E$ with foci $z=\pm 1$, and

 18.18.1 $a_{n}=\frac{n!(2n+\alpha+\beta+1)\Gamma\left(n+\alpha+\beta+1\right)}{2^{% \alpha+\beta+1}\Gamma\left(n+\alpha+1\right)\Gamma\left(n+\beta+1\right)}\*% \int_{-1}^{1}f(x)P^{(\alpha,\beta)}_{n}\left(x\right)(1-x)^{\alpha}(1+x)^{% \beta}\mathrm{d}x.$

Then

 18.18.2 $f(z)=\sum_{n=0}^{\infty}a_{n}P^{(\alpha,\beta)}_{n}\left(z\right),$ ⓘ Symbols: $P^{(\NVar{\alpha},\NVar{\beta})}_{\NVar{n}}\left(\NVar{x}\right)$: Jacobi polynomial, $z$: complex variable, $n$: nonnegative integer, $f(z)$: analytic function and $a_{n}$ Referenced by: §18.18(i), §18.18(i), §18.18(i) Permalink: http://dlmf.nist.gov/18.18.E2 Encodings: TeX, pMML, png See also: Annotations for §18.18(i), §18.18(i), §18.18 and Ch.18

when $z$ lies in the interior of $E$. Moreover, the series (18.18.2) converges uniformly on any compact domain within $E$.

Alternatively, assume $f(x)$ is real and continuous and $f^{\prime}(x)$ is piecewise continuous on $(-1,1)$. Assume also the integrals $\int_{-1}^{1}(f(x))^{2}(1-x)^{\alpha}(1+x)^{\beta}\mathrm{d}x$ and $\int_{-1}^{1}(f^{\prime}(x))^{2}(1-x)^{\alpha+1}(1+x)^{\beta+1}\mathrm{d}x$ converge. Then (18.18.2), with $z$ replaced by $x$, applies when $-1; moreover, the convergence is uniform on any compact interval within $(-1,1)$.

Chebyshev

See §3.11(ii), or set $\alpha=\beta=\pm\tfrac{1}{2}$ in the above results for Jacobi and refer to (18.7.3)–(18.7.6).

Legendre

This is the case $\alpha=\beta=0$ of Jacobi. Equation (18.18.1) becomes

 18.18.3 $a_{n}=\left(n+\tfrac{1}{2}\right)\int_{-1}^{1}f(x)P_{n}\left(x\right)\mathrm{d% }x.$

Laguerre

Assume $f(x)$ is real and continuous and $f^{\prime}(x)$ is piecewise continuous on $(0,\infty)$. Assume also $\int_{0}^{\infty}(f(x))^{2}e^{-x}x^{\alpha}\mathrm{d}x$ converges. Then

 18.18.4 $f(x)=\sum_{n=0}^{\infty}b_{n}L^{(\alpha)}_{n}\left(x\right),$ $0, ⓘ Symbols: $L^{(\NVar{\alpha})}_{\NVar{n}}\left(\NVar{x}\right)$: Laguerre (or generalized Laguerre) polynomial, $n$: nonnegative integer, $f(z)$: analytic function and $x$: real variable Referenced by: §18.18(i) Permalink: http://dlmf.nist.gov/18.18.E4 Encodings: TeX, pMML, png See also: Annotations for §18.18(i), §18.18(i), §18.18 and Ch.18

where

 18.18.5 $b_{n}=\frac{n!}{\Gamma\left(n+\alpha+1\right)}\int_{0}^{\infty}f(x)L^{(\alpha)% }_{n}\left(x\right)e^{-x}x^{\alpha}\mathrm{d}x.$

The convergence of the series (18.18.4) is uniform on any compact interval in $(0,\infty)$.

Hermite

Assume $f(x)$ is real and continuous and $f^{\prime}(x)$ is piecewise continuous on $(-\infty,\infty)$. Assume also $\int_{-\infty}^{\infty}(f(x))^{2}e^{-x^{2}}\mathrm{d}x$ converges. Then

 18.18.6 $f(x)=\sum_{n=0}^{\infty}d_{n}H_{n}\left(x\right),$ $-\infty, ⓘ Symbols: $H_{\NVar{n}}\left(\NVar{x}\right)$: Hermite polynomial, $n$: nonnegative integer, $f(z)$: analytic function, $d_{n}$ and $x$: real variable Referenced by: §18.18(i) Permalink: http://dlmf.nist.gov/18.18.E6 Encodings: TeX, pMML, png See also: Annotations for §18.18(i), §18.18(i), §18.18 and Ch.18

where

 18.18.7 $d_{n}=\frac{1}{\sqrt{\pi}2^{n}n!}\int_{-\infty}^{\infty}f(x)H_{n}\left(x\right% )e^{-x^{2}}\mathrm{d}x.$

The convergence of the series (18.18.6) is uniform on any compact interval in $(-\infty,\infty)$.

Ultraspherical

 18.18.8 $C^{(\lambda)}_{n}\left(\cos\theta_{1}\cos\theta_{2}+\sin\theta_{1}\sin\theta_{% 2}\cos\phi\right)=\sum_{\ell=0}^{n}2^{2\ell}(n-\ell)!\frac{2\lambda+2\ell-1}{2% \lambda-1}\frac{({\left(\lambda\right)_{\ell}})^{2}}{{\left(2\lambda\right)_{n% +\ell}}}(\sin\theta_{1})^{\ell}C^{(\lambda+\ell)}_{n-\ell}\left(\cos\theta_{1}% \right)(\sin\theta_{2})^{\ell}C^{(\lambda+\ell)}_{n-\ell}\left(\cos\theta_{2}% \right)C^{(\lambda-\frac{1}{2})}_{\ell}\left(\cos\phi\right),$ $\lambda>0$, $\lambda\neq\frac{1}{2}$.

For the case $\lambda=\frac{1}{2}$ use (18.18.9); compare (18.7.9).

Legendre

 18.18.9 $P_{n}\left(\cos\theta_{1}\cos\theta_{2}+\sin\theta_{1}\sin\theta_{2}\cos\phi% \right)={P_{n}\left(\cos\theta_{1}\right)P_{n}\left(\cos\theta_{2}\right)+2% \sum_{\ell=1}^{n}\frac{(n-\ell)!\;(n+\ell)!}{2^{2\ell}(n!)^{2}}(\sin\theta_{1}% )^{\ell}P^{(\ell,\ell)}_{n-\ell}\left(\cos\theta_{1}\right)(\sin\theta_{2})^{% \ell}P^{(\ell,\ell)}_{n-\ell}\left(\cos\theta_{2}\right)\cos\left(\ell\phi% \right)}.$

For (18.18.8), (18.18.9), and the corresponding formula for Jacobi polynomials see Koornwinder (1975b). See also (14.30.9).

Laguerre

 18.18.10 $L^{(\alpha_{1}+\dots+\alpha_{r}+r-1)}_{n}\left(x_{1}+\dots+x_{r}\right)=\sum_{% m_{1}+\dots+m_{r}=n}L^{(\alpha_{1})}_{m_{1}}\left(x_{1}\right)\cdots L^{(% \alpha_{r})}_{m_{r}}\left(x_{r}\right).$ ⓘ Symbols: $L^{(\NVar{\alpha})}_{\NVar{n}}\left(\NVar{x}\right)$: Laguerre (or generalized Laguerre) polynomial, $m$: nonnegative integer, $n$: nonnegative integer and $x$: real variable Referenced by: §18.18(ii), §18.18(viii) Permalink: http://dlmf.nist.gov/18.18.E10 Encodings: TeX, pMML, png See also: Annotations for §18.18(ii), §18.18(ii), §18.18 and Ch.18

Hermite

 18.18.11 $\frac{(a_{1}^{2}+\dots+a_{r}^{2})^{\frac{1}{2}n}}{n!}H_{n}\left(\frac{a_{1}x_{% 1}+\cdots+a_{r}x_{r}}{(a_{1}^{2}+\cdots+a_{r}^{2})^{\frac{1}{2}}}\right)=\sum_% {m_{1}+\cdots+m_{r}=n}\frac{a_{1}^{m_{1}}\cdots a_{r}^{m_{r}}}{m_{1}!\cdots m_% {r}!}H_{m_{1}}\left(x_{1}\right)\cdots H_{m_{r}}\left(x_{r}\right).$

§18.18(iii) Multiplication Theorems

Laguerre

 18.18.12 $\frac{L^{(\alpha)}_{n}\left(\lambda x\right)}{L^{(\alpha)}_{n}\left(0\right)}=% \sum_{\ell=0}^{n}\genfrac{(}{)}{0.0pt}{}{n}{\ell}\lambda^{\ell}(1-\lambda)^{n-% \ell}\frac{L^{(\alpha)}_{\ell}\left(x\right)}{L^{(\alpha)}_{\ell}\left(0\right% )}.$

Hermite

 18.18.13 $H_{n}\left(\lambda x\right)=\lambda^{n}\sum_{\ell=0}^{\left\lfloor n/2\right% \rfloor}\frac{{\left(-n\right)_{2\ell}}}{\ell!}(1-\lambda^{-2})^{\ell}H_{n-2% \ell}\left(x\right).$

§18.18(iv) Connection Formulas

Jacobi

 18.18.14 $\displaystyle P^{(\gamma,\beta)}_{n}\left(x\right)$ $\displaystyle=\dfrac{{\left(\beta+1\right)_{n}}}{{\left(\alpha+\beta+2\right)_% {n}}}\sum_{\ell=0}^{n}\dfrac{\alpha+\beta+2\ell+1}{\alpha+\beta+1}\dfrac{{% \left(\alpha+\beta+1\right)_{\ell}}{\left(n+\beta+\gamma+1\right)_{\ell}}}{{% \left(\beta+1\right)_{\ell}}{\left(n+\alpha+\beta+2\right)_{\ell}}}\dfrac{{% \left(\gamma-\alpha\right)_{n-\ell}}}{(n-\ell)!}P^{(\alpha,\beta)}_{\ell}\left% (x\right),$ 18.18.15 $\displaystyle\left(\frac{1+x}{2}\right)^{n}$ $\displaystyle=\frac{{\left(\beta+1\right)_{n}}}{{\left(\alpha+\beta+2\right)_{% n}}}\sum_{\ell=0}^{n}\frac{\alpha+\beta+2\ell+1}{\alpha+\beta+1}\frac{{\left(% \alpha+\beta+1\right)_{\ell}}{\left(n-\ell+1\right)_{\ell}}}{{\left(\beta+1% \right)_{\ell}}{\left(n+\alpha+\beta+2\right)_{\ell}}}P^{(\alpha,\beta)}_{\ell% }\left(x\right),$

and a similar pair of equations by symmetry; compare the second row in Table 18.6.1.

Ultraspherical

 18.18.16 $\displaystyle C^{(\mu)}_{n}\left(x\right)$ $\displaystyle=\sum_{\ell=0}^{\left\lfloor n/2\right\rfloor}\frac{\lambda+n-2% \ell}{\lambda}\frac{{\left(\mu\right)_{n-\ell}}}{{\left(\lambda+1\right)_{n-% \ell}}}\frac{{\left(\mu-\lambda\right)_{\ell}}}{\ell!}C^{(\lambda)}_{n-2\ell}% \left(x\right),$ 18.18.17 $\displaystyle(2x)^{n}$ $\displaystyle=n!\sum_{\ell=0}^{\left\lfloor n/2\right\rfloor}\frac{\lambda+n-2% \ell}{\lambda}\frac{1}{{\left(\lambda+1\right)_{n-\ell}}\,\ell!}C^{(\lambda)}_% {n-2\ell}\left(x\right).$

Laguerre

 18.18.18 $\displaystyle L^{(\beta)}_{n}\left(x\right)$ $\displaystyle=\sum_{\ell=0}^{n}\frac{{\left(\beta-\alpha\right)_{n-\ell}}}{(n-% \ell)!}L^{(\alpha)}_{\ell}\left(x\right),$ 18.18.19 $\displaystyle x^{n}$ $\displaystyle={\left(\alpha+1\right)_{n}}\sum_{\ell=0}^{n}\frac{{\left(-n% \right)_{\ell}}}{{\left(\alpha+1\right)_{\ell}}}L^{(\alpha)}_{\ell}\left(x% \right).$

Hermite

 18.18.20 $(2x)^{n}=\sum_{\ell=0}^{\left\lfloor n/2\right\rfloor}\frac{{\left(-n\right)_{% 2\ell}}}{\ell!}H_{n-2\ell}\left(x\right).$

§18.18(v) Linearization Formulas

Chebyshev

 18.18.21 $T_{m}\left(x\right)T_{n}\left(x\right)=\tfrac{1}{2}(T_{m+n}\left(x\right)+T_{m% -n}\left(x\right)).$

Ultraspherical

 18.18.22 $C^{(\lambda)}_{m}\left(x\right)C^{(\lambda)}_{n}\left(x\right)=\sum_{\ell=0}^{% \min(m,n)}\frac{(m+n+\lambda-2\ell)(m+n-2\ell)!}{(m+n+\lambda-\ell)\ell!\,(m-% \ell)!\,(n-\ell)!}\*\frac{{\left(\lambda\right)_{\ell}}{\left(\lambda\right)_{% m-\ell}}{\left(\lambda\right)_{n-\ell}}{\left(2\lambda\right)_{m+n-\ell}}}{{% \left(\lambda\right)_{m+n-\ell}}{\left(2\lambda\right)_{m+n-2\ell}}}C^{(% \lambda)}_{m+n-2\ell}\left(x\right).$

Hermite

 18.18.23 $H_{m}\left(x\right)H_{n}\left(x\right)=\sum_{\ell=0}^{\min(m,n)}\genfrac{(}{)}% {0.0pt}{}{m}{\ell}\genfrac{(}{)}{0.0pt}{}{n}{\ell}2^{\ell}\ell!H_{m+n-2\ell}% \left(x\right).$

The coefficients in the expansions (18.18.22) and (18.18.23) are positive, provided that in the former case $\lambda>0$.

§18.18(vi) Bateman-Type Sums

Jacobi

With

 18.18.24 $b_{n,\ell}=\genfrac{(}{)}{0.0pt}{}{n}{\ell}\frac{{\left(n+\alpha+\beta+1\right% )_{\ell}}{\left(-\beta-n\right)_{n-\ell}}}{2^{\ell}{\left(\alpha+1\right)_{n}}},$
 18.18.25 $\frac{P^{(\alpha,\beta)}_{n}\left(x\right)}{P^{(\alpha,\beta)}_{n}\left(1% \right)}\frac{P^{(\alpha,\beta)}_{n}\left(y\right)}{P^{(\alpha,\beta)}_{n}% \left(1\right)}=\sum_{\ell=0}^{n}b_{n,\ell}(x+y)^{\ell}\*\frac{P^{(\alpha,% \beta)}_{\ell}\left(\ifrac{(1+xy)}{(x+y)}\right)}{P^{(\alpha,\beta)}_{\ell}% \left(1\right)},$
 18.18.26 $\frac{P^{(\alpha,\beta)}_{n}\left(x\right)}{P^{(\alpha,\beta)}_{n}\left(1% \right)}=\sum_{\ell=0}^{n}b_{n,\ell}(x+1)^{\ell}.$

§18.18(vii) Poisson Kernels

Laguerre

 18.18.27 $\sum_{n=0}^{\infty}\frac{n!\,L^{(\alpha)}_{n}\left(x\right)L^{(\alpha)}_{n}% \left(y\right)}{{\left(\alpha+1\right)_{n}}}z^{n}=\frac{\Gamma\left(\alpha+1% \right)(xyz)^{-\frac{1}{2}\alpha}}{1-z}\*\exp\left(\frac{-(x+y)z}{1-z}\right)I% _{\alpha}\left(\frac{2(xyz)^{\frac{1}{2}}}{1-z}\right),$ $|z|<1$.

For the modified Bessel function $I_{\nu}\left(z\right)$ see §10.25(ii).

Hermite

 18.18.28 $\sum_{n=0}^{\infty}\frac{H_{n}\left(x\right)H_{n}\left(y\right)}{2^{n}n!}z^{n}% =(1-z^{2})^{-\frac{1}{2}}\exp\left(\frac{2xyz-(x^{2}+y^{2})z^{2}}{1-z^{2}}% \right),$ $|z|<1$.

These Poisson kernels are positive, provided that $x,y$ are real, $0\leq z<1$, and in the case of (18.18.27) $x,y\geq 0$.

§18.18(viii) Other Sums

In this subsection the variables $x$ and $y$ are not confined to the closures of the intervals of orthogonality; compare §18.2(i).

Ultraspherical

 18.18.29 $\sum_{\ell=0}^{n}C^{(\lambda)}_{\ell}\left(x\right)C^{(\mu)}_{n-\ell}\left(x% \right)=C^{(\lambda+\mu)}_{n}\left(x\right).$ ⓘ Symbols: $C^{(\NVar{\lambda})}_{\NVar{n}}\left(\NVar{x}\right)$: ultraspherical (or Gegenbauer) polynomial, $\ell$: nonnegative integer, $n$: nonnegative integer and $x$: real variable Referenced by: §18.18(viii) Permalink: http://dlmf.nist.gov/18.18.E29 Encodings: TeX, pMML, png See also: Annotations for §18.18(viii), §18.18(viii), §18.18 and Ch.18
 18.18.30 $\sum_{\ell=0}^{n}\frac{\ell+2\lambda}{2\lambda}C^{(\lambda)}_{\ell}\left(x% \right)x^{n-\ell}=C^{(\lambda+1)}_{n}\left(x\right).$ ⓘ Symbols: $C^{(\NVar{\lambda})}_{\NVar{n}}\left(\NVar{x}\right)$: ultraspherical (or Gegenbauer) polynomial, $\ell$: nonnegative integer, $n$: nonnegative integer and $x$: real variable Referenced by: §18.18(viii) Permalink: http://dlmf.nist.gov/18.18.E30 Encodings: TeX, pMML, png See also: Annotations for §18.18(viii), §18.18(viii), §18.18 and Ch.18

Chebyshev

 18.18.31 $\displaystyle\sum_{\ell=0}^{n}T_{\ell}\left(x\right)x^{n-\ell}$ $\displaystyle=U_{n}\left(x\right).$ 18.18.32 $\displaystyle 2\sum_{\ell=0}^{n}T_{2\ell}\left(x\right)$ $\displaystyle=1+U_{2n}\left(x\right),$ ⓘ Symbols: $T_{\NVar{n}}\left(\NVar{x}\right)$: Chebyshev polynomial of the first kind, $U_{\NVar{n}}\left(\NVar{x}\right)$: Chebyshev polynomial of the second kind, $\ell$: nonnegative integer, $n$: nonnegative integer and $x$: real variable A&S Ref: 22.12.2 Referenced by: §18.18(viii) Permalink: http://dlmf.nist.gov/18.18.E32 Encodings: TeX, pMML, png See also: Annotations for §18.18(viii), §18.18(viii), §18.18 and Ch.18 18.18.33 $\displaystyle 2\sum_{\ell=0}^{n}T_{2\ell+1}\left(x\right)$ $\displaystyle=U_{2n+1}\left(x\right).$ 18.18.34 $\displaystyle 2(1-x^{2})\sum_{\ell=0}^{n}U_{2\ell}\left(x\right)$ $\displaystyle=1-T_{2n+2}\left(x\right),$ 18.18.35 $\displaystyle 2(1-x^{2})\sum_{\ell=0}^{n}U_{2\ell+1}\left(x\right)$ $\displaystyle=x-T_{2n+3}\left(x\right).$ ⓘ Symbols: $T_{\NVar{n}}\left(\NVar{x}\right)$: Chebyshev polynomial of the first kind, $U_{\NVar{n}}\left(\NVar{x}\right)$: Chebyshev polynomial of the second kind, $\ell$: nonnegative integer, $n$: nonnegative integer and $x$: real variable A&S Ref: 22.12.5 Referenced by: §18.18(viii) Permalink: http://dlmf.nist.gov/18.18.E35 Encodings: TeX, pMML, png See also: Annotations for §18.18(viii), §18.18(viii), §18.18 and Ch.18

Legendre and Chebyshev

 18.18.36 $\sum_{\ell=0}^{n}P_{\ell}\left(x\right)P_{n-\ell}\left(x\right)=U_{n}\left(x% \right).$

Laguerre

 18.18.37 $\sum_{\ell=0}^{n}L^{(\alpha)}_{\ell}\left(x\right)=L^{(\alpha+1)}_{n}\left(x% \right),$ ⓘ Symbols: $L^{(\NVar{\alpha})}_{\NVar{n}}\left(\NVar{x}\right)$: Laguerre (or generalized Laguerre) polynomial, $\ell$: nonnegative integer, $n$: nonnegative integer and $x$: real variable Referenced by: §18.18(viii) Permalink: http://dlmf.nist.gov/18.18.E37 Encodings: TeX, pMML, png See also: Annotations for §18.18(viii), §18.18(viii), §18.18 and Ch.18
 18.18.38 $\sum_{\ell=0}^{n}L^{(\alpha)}_{\ell}\left(x\right)L^{(\beta)}_{n-\ell}\left(y% \right)=L^{(\alpha+\beta+1)}_{n}\left(x+y\right).$ ⓘ Symbols: $L^{(\NVar{\alpha})}_{\NVar{n}}\left(\NVar{x}\right)$: Laguerre (or generalized Laguerre) polynomial, $y$: real variable, $\ell$: nonnegative integer, $n$: nonnegative integer and $x$: real variable A&S Ref: 22.12.6 Referenced by: §18.18(viii) Permalink: http://dlmf.nist.gov/18.18.E38 Encodings: TeX, pMML, png See also: Annotations for §18.18(viii), §18.18(viii), §18.18 and Ch.18

Hermite and Laguerre

 18.18.39 $\sum_{\ell=0}^{n}\genfrac{(}{)}{0.0pt}{}{n}{\ell}H_{\ell}\left(2^{\frac{1}{2}}% x\right)H_{n-\ell}\left(2^{\frac{1}{2}}y\right)=2^{\frac{1}{2}n}H_{n}\left(x+y% \right),$
 18.18.40 $\sum_{\ell=0}^{n}\genfrac{(}{)}{0.0pt}{}{n}{\ell}H_{2\ell}\left(x\right)H_{2n-% 2\ell}\left(y\right)=(-1)^{n}2^{2n}n!L_{n}\left(x^{2}+y^{2}\right).$

§18.18(ix) Compendia

For further sums see Hansen (1975, pp. 292-330), Gradshteyn and Ryzhik (2000, pp. 978–993), and Prudnikov et al. (1986b, pp. 637-644 and 700-718).