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

Then

 18.18.2 $f(z)=\sum_{n=0}^{\infty}a_{n}\mathop{P^{(\alpha,\beta)}_{n}\/}\nolimits\!\left% (z\right),$

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

### 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 $f(x)=\sum_{n=0}^{\infty}b_{n}\mathop{L^{(\alpha)}_{n}\/}\nolimits\!\left(x% \right),$ $0,

where

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

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 $f(x)=\sum_{n=0}^{\infty}d_{n}\mathop{H_{n}\/}\nolimits\!\left(x\right),$ $-\infty,

where

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

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

### Ultraspherical

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

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

### Legendre

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

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 $\mathop{L^{(\alpha_{1}+\dots+\alpha_{r}+r-1)}_{n}\/}\nolimits\!\left(x_{1}+% \dots+x_{r}\right)=\sum_{m_{1}+\dots+m_{r}=n}\mathop{L^{(\alpha_{1})}_{m_{1}}% \/}\nolimits\!\left(x_{1}\right)\cdots\mathop{L^{(\alpha_{r})}_{m_{r}}\/}% \nolimits\!\left(x_{r}\right).$

### Hermite

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

## §18.18(iii) Multiplication Theorems

### Laguerre

 18.18.12 $\frac{\mathop{L^{(\alpha)}_{n}\/}\nolimits\!\left(\lambda x\right)}{\mathop{L^% {(\alpha)}_{n}\/}\nolimits\!\left(0\right)}=\sum_{\ell=0}^{n}\binom{n}{\ell}% \lambda^{\ell}(1-\lambda)^{n-\ell}\frac{\mathop{L^{(\alpha)}_{\ell}\/}% \nolimits\!\left(x\right)}{\mathop{L^{(\alpha)}_{\ell}\/}\nolimits\!\left(0% \right)}.$

### Hermite

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

## §18.18(iv) Connection Formulas

### Jacobi

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

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

### Ultraspherical

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

### Laguerre

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

### Hermite

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

## §18.18(v) Linearization Formulas

### Chebyshev

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

### Ultraspherical

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

### Hermite

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

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 $b_{n,\ell}=\binom{n}{\ell}\frac{{\left(n+\alpha+\beta+1\right)_{\ell}}{\left(-% \beta-n\right)_{n-\ell}}}{2^{\ell}{\left(\alpha+1\right)_{n}}},$
 18.18.25 $\frac{\mathop{P^{(\alpha,\beta)}_{n}\/}\nolimits\!\left(x\right)}{\mathop{P^{(% \alpha,\beta)}_{n}\/}\nolimits\!\left(1\right)}\frac{\mathop{P^{(\alpha,\beta)% }_{n}\/}\nolimits\!\left(y\right)}{\mathop{P^{(\alpha,\beta)}_{n}\/}\nolimits% \!\left(1\right)}=\sum_{\ell=0}^{n}b_{n,\ell}(x+y)^{\ell}\*\frac{\mathop{P^{(% \alpha,\beta)}_{\ell}\/}\nolimits\!\left(\ifrac{(1+xy)}{(x+y)}\right)}{\mathop% {P^{(\alpha,\beta)}_{\ell}\/}\nolimits\!\left(1\right)},$
 18.18.26 $\frac{\mathop{P^{(\alpha,\beta)}_{n}\/}\nolimits\!\left(x\right)}{\mathop{P^{(% \alpha,\beta)}_{n}\/}\nolimits\!\left(1\right)}=\sum_{\ell=0}^{n}b_{n,\ell}(x+% 1)^{\ell}.$

## §18.18(vii) Poisson Kernels

### Laguerre

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

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

### Hermite

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

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 $\sum_{\ell=0}^{n}\mathop{C^{(\lambda)}_{\ell}\/}\nolimits\!\left(x\right)% \mathop{C^{(\mu)}_{n-\ell}\/}\nolimits\!\left(x\right)=\mathop{C^{(\lambda+\mu% )}_{n}\/}\nolimits\!\left(x\right).$
 18.18.30 $\sum_{\ell=0}^{n}\frac{\ell+2\lambda}{2\lambda}\mathop{C^{(\lambda)}_{\ell}\/}% \nolimits\!\left(x\right)x^{n-\ell}=\mathop{C^{(\lambda+1)}_{n}\/}\nolimits\!% \left(x\right).$

### Chebyshev

 18.18.31 $\displaystyle\sum_{\ell=0}^{n}\mathop{T_{\ell}\/}\nolimits\!\left(x\right)x^{n% -\ell}$ $\displaystyle=\mathop{U_{n}\/}\nolimits\!\left(x\right).$ 18.18.32 $\displaystyle 2\sum_{\ell=0}^{n}\mathop{T_{2\ell}\/}\nolimits\!\left(x\right)$ $\displaystyle=1+\mathop{U_{2n}\/}\nolimits\!\left(x\right),$ 18.18.33 $\displaystyle 2\sum_{\ell=0}^{n}\mathop{T_{2\ell+1}\/}\nolimits\!\left(x\right)$ $\displaystyle=\mathop{U_{2n+1}\/}\nolimits\!\left(x\right).$ 18.18.34 $\displaystyle 2(1-x^{2})\sum_{\ell=0}^{n}\mathop{U_{2\ell}\/}\nolimits\!\left(% x\right)$ $\displaystyle=1-\mathop{T_{2n+2}\/}\nolimits\!\left(x\right),$ 18.18.35 $\displaystyle 2(1-x^{2})\sum_{\ell=0}^{n}\mathop{U_{2\ell+1}\/}\nolimits\!% \left(x\right)$ $\displaystyle=x-\mathop{T_{2n+3}\/}\nolimits\!\left(x\right).$

### Legendre and Chebyshev

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

### Laguerre

 18.18.37 $\sum_{\ell=0}^{n}\mathop{L^{(\alpha)}_{\ell}\/}\nolimits\!\left(x\right)=% \mathop{L^{(\alpha+1)}_{n}\/}\nolimits\!\left(x\right),$
 18.18.38 $\sum_{\ell=0}^{n}\mathop{L^{(\alpha)}_{\ell}\/}\nolimits\!\left(x\right)% \mathop{L^{(\beta)}_{n-\ell}\/}\nolimits\!\left(y\right)=\mathop{L^{(\alpha+% \beta+1)}_{n}\/}\nolimits\!\left(x+y\right).$ Symbols: $\mathop{L^{(\NVar{\alpha})}_{\NVar{n}}\/}\nolimits\!\left(\NVar{x}\right)$: 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)

### Hermite and Laguerre

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

## §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).