# §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), §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 $\int_{0}^{\infty}(f(x))^{2}{\mathrm{e}}^{-x}x^{\alpha}\,\mathrm{d}x$ 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), §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 $b_{n}=\frac{n!}{\Gamma\left(n+\alpha+1\right)}\int_{0}^{\infty}f(x)L^{(\alpha)% }_{n}\left(x\right){\mathrm{e}}^{-x}x^{\alpha}\,\mathrm{d}x.$

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

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

### Expansion of $L^{2}$ functions

In all three cases of Jacobi, Laguerre and Hermite, if $f(x)$ is $L^{2}$ on the corresponding interval with respect to the corresponding weight function and if $a_{n},b_{n},d_{n}$ are given by (18.18.1), (18.18.5), (18.18.7), respectively, then the respective series expansions (18.18.2), (18.18.4), (18.18.6) are valid with the sums converging in $L^{2}$ sense. See Szegő (1975, Theorems 3.1.5 and 5.7.1). See also (18.2.24), (18.2.25).

## §18.18(ii) Addition Theorems

### 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 integral representations for products implied by (18.18.8) and (18.18.9) see (18.17.5) and (18.17.6), respectively. For (18.18.8) see also (14.30.9). For formulas for Jacobi and Laguerre polynomials analogous to (18.18.8) and (18.18.9), see (Koornwinder, 1975b, 1977).

### 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 and Inversion 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. See Andrews et al. (1999, Lemma 7.1.1) for the more general expansion of $P^{(\gamma,\delta)}_{n}\left(x\right)$ in terms of $P^{(\alpha,\beta)}_{n}\left(x\right)$.

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

See (18.5.11) for the limit case $\lambda\to 0$ of (18.18.16).

### 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 $\lambda>0$. See (18.17.45) and (18.17.49) for integrated forms of (18.18.22) and (18.18.23), respectively. See Rahman (1981) for the linearization formula for Jacobi polynomials and Zeng (1992) for the linearization coefficients for Laguerre polynomials.

## §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

See (18.2.41) for the Poisson kernel in case of general OP’s.

### 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). Formula (18.18.27) is known as the Hille–Hardy formula.

### Hermite

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

Formula (18.18.28) is known as the Mehler formula. See Ismail (2009, Theorem 4.7.2) for a formula called Kibble–Slepian formula, which generalizes (18.18.28).

These Poisson kernels are positive, provided that $x,y$ are real, $0\leq z<1$, and in the case of (18.18.27) $x,y\geq 0$. For the Poisson kernel of Jacobi polynomials (the Bailey formula) see Bailey (1938).

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

See also (18.38.3) for a finite sum of Jacobi polynomials.

## §18.18(ix) Compendia

For further sums see Hansen (1975, pp. 292-330), Gradshteyn and Ryzhik (2015, §§8.92–8.98), and Prudnikov et al. (1986b, pp. 637-644 and 700-718).