§18.18 Sums

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

18.18.1		$$a_n=\frac{n!(2n+\alpha +\beta +1)\mathrm{\Gamma }\left(n+\alpha +\beta +1\right)}{2^{\alpha +\beta +1}\mathrm{\Gamma }\left(n+\alpha +1\right)\mathrm{\Gamma }\left(n+\beta +1\right)}{\int }_{-1}^{1}f(x)P_n^{(\alpha ,\beta )}\left(x\right){(1-x)}^{\alpha }{(1+x)}^{\beta }dx.$$
ⓘ Defines: $a_n$ (locally) Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, $P_n^{(\alpha ,\beta )}\left(x\right)$: Jacobi polynomial, $dx$: differential of $x$, $!$: factorial (as in $n!$), $\int$: integral, $n$: 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.E1 Encodings: TeX, pMML, png See also: Annotations for §18.18(i), §18.18(i), §18.18 and Ch.18

Then

18.18.2		$$f(z)=\sum _{n=0}^{\mathrm{\infty }}{a}_n{P}_n^{(\alpha ,\beta )}\left(z\right),$$
ⓘ Symbols: $P_n^{(\alpha ,\beta )}\left(x\right)$: 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 ${\int }_{-1}^1{(f(x))}^2{(1-x)}^{\alpha }{(1+x)}^{\beta }dx$ and ${\int }_{-1}^1{(f^{\prime }(x))}^2{(1-x)}^{\alpha +1}{(1+x)}^{\beta +1}dx$ converge. Then (18.18.2), with $z$ replaced by $x$, applies when ; moreover, the convergence is uniform on any compact interval within $(-1,1)$.

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

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

18.18.3		$$a_n=\left(n+\frac{1}{2}\right){\int }_{-1}^{1}f(x)P_n\left(x\right)dx.$$
ⓘ Symbols: $P_n\left(x\right)$: Legendre polynomial, $dx$: differential of $x$, $\int$: integral, $n$: nonnegative integer, $f(z)$: analytic function, $a_n$ and $x$: real variable Permalink: http://dlmf.nist.gov/18.18.E3 Encodings: TeX, pMML, png See also: Annotations for §18.18(i), §18.18(i), §18.18 and Ch.18

Assume $f(x)$ is real and continuous and $f^{\prime }(x)$ is piecewise continuous on $(0,\mathrm{\infty })$. Assume also ${\int }_0^{\mathrm{\infty }}{(f(x))}^2{\mathrm{e}}^{-x}{x}^{\alpha }dx$ converges. Then

18.18.4		$$f(x)=\sum _{n=0}^{\mathrm{\infty }}{b}_n{L}_n^{(\alpha )}\left(x\right),$$
,
ⓘ Symbols: $L_n^{(\alpha )}\left(x\right)$: 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		$$b_n=\frac{n!}{\mathrm{\Gamma }\left(n+\alpha +1\right)}{\int }_0^{\mathrm{\infty }}f(x)L_n^{(\alpha )}\left(x\right){\mathrm{e}}^{-x}{x}^{\alpha }dx.$$
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, $L_n^{(\alpha )}\left(x\right)$: Laguerre (or generalized Laguerre) polynomial, $dx$: differential of $x$, $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $\int$: integral, $n$: nonnegative integer, $f(z)$: analytic function and $x$: real variable Referenced by: §18.18(i) Permalink: http://dlmf.nist.gov/18.18.E5 Encodings: TeX, pMML, png See also: Annotations for §18.18(i), §18.18(i), §18.18 and Ch.18

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

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

18.18.6		$$f(x)=\sum _{n=0}^{\mathrm{\infty }}{d}_n{H}_n\left(x\right),$$
,
ⓘ Symbols: $H_n\left(x\right)$: 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		$$d_n=\frac{1}{\sqrt{\pi }{2}^{n}n!}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}f(x)H_n\left(x\right){\mathrm{e}}^{-x^2}dx.$$
ⓘ Defines: $d_n$ (locally) Symbols: $H_n\left(x\right)$: Hermite polynomial, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$, $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $\int$: integral, $n$: nonnegative integer, $f(z)$: analytic function and $x$: real variable Referenced by: §18.18(i) Permalink: http://dlmf.nist.gov/18.18.E7 Encodings: TeX, pMML, png See also: Annotations for §18.18(i), §18.18(i), §18.18 and Ch.18

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

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).

18.18.8		$$\begin{array}{l}{C}_n^{(\lambda )}\left(\cos {\theta }_1\cos {\theta }_2+\sin {\theta }_1\sin {\theta }_2\cos \varphi \right)\\ =\sum _{\mathrm{\ell }=0}^n\begin{array}{l}{2}^{2\mathrm{\ell }}(n-\mathrm{\ell })!\frac{2\lambda +2\mathrm{\ell }-1}{2\lambda -1}\frac{{({\left(\lambda \right)}_{\mathrm{\ell }})}^2}{{\left(2\lambda \right)}_{n+\mathrm{\ell }}}\\ \times {(\sin {\theta }_1)}^{\mathrm{\ell }}{C}_{n-\mathrm{\ell }}^{(\lambda +\mathrm{\ell })}\left(\cos {\theta }_1\right){(\sin {\theta }_2)}^{\mathrm{\ell }}\\ \times C_{n-\mathrm{\ell }}^{(\lambda +\mathrm{\ell })}\left(\cos {\theta }_2\right)C_{\mathrm{\ell }}^{(\lambda -\frac{1}{2})}\left(\cos \varphi \right),\end{array}\end{array}$$
$\lambda >0$, $\lambda \ne \frac{1}{2}$.
ⓘ Symbols: ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\cos z$: cosine function, $!$: factorial (as in $n!$), $\sin z$: sine function, $C_n^{(\lambda )}\left(x\right)$: ultraspherical (or Gegenbauer) polynomial, $\mathrm{\ell }$: nonnegative integer and $n$: nonnegative integer Referenced by: §18.17(ii), §18.18(ii), §18.18(ii) Permalink: http://dlmf.nist.gov/18.18.E8 Encodings: TeX, pMML, png See also: Annotations for §18.18(ii), §18.18(ii), §18.18 and Ch.18

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

18.18.9		$$\begin{array}{l}{P}_n\left(\cos {\theta }_1\cos {\theta }_2+\sin {\theta }_1\sin {\theta }_2\cos \varphi \right)\\ =\begin{array}{l}{P}_n\left(\cos {\theta }_1\right)P_n\left(\cos {\theta }_2\right)\\ +2\sum _{\mathrm{\ell }=1}^n\begin{array}{l}\frac{(n-\mathrm{\ell })!(n+\mathrm{\ell })!}{2^{2\mathrm{\ell }}{(n!)}^2}{(\sin {\theta }_1)}^{\mathrm{\ell }}{P}_{n-\mathrm{\ell }}^{(\mathrm{\ell },\mathrm{\ell })}\left(\cos {\theta }_1\right)\\ \times {(\sin {\theta }_2)}^{\mathrm{\ell }}{P}_{n-\mathrm{\ell }}^{(\mathrm{\ell },\mathrm{\ell })}\left(\cos {\theta }_2\right)\cos \left(\mathrm{\ell }\varphi \right).\end{array}\end{array}\end{array}$$
ⓘ Symbols: $P_n^{(\alpha ,\beta )}\left(x\right)$: Jacobi polynomial, $P_n\left(x\right)$: Legendre polynomial, $\cos z$: cosine function, $!$: factorial (as in $n!$), $\sin z$: sine function, $\mathrm{\ell }$: nonnegative integer and $n$: nonnegative integer Referenced by: §14.30(iii), §18.17(ii), §18.18(ii), §18.18(ii), §18.18(ii) Permalink: http://dlmf.nist.gov/18.18.E9 Encodings: TeX, pMML, png See also: Annotations for §18.18(ii), §18.18(ii), §18.18 and Ch.18

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).

18.18.10		$$L_n^{({\alpha }_1+\mathrm{\cdots }+{\alpha }_r+r-1)}\left(x_1+\mathrm{\cdots }+x_r\right)=\sum _{m_1+\mathrm{\cdots }+m_r=n}{L}_{m_1}^{({\alpha }_1)}\left(x_1\right)\mathrm{\cdots }{L}_{m_r}^{({\alpha }_r)}\left(x_r\right).$$
ⓘ Symbols: $L_n^{(\alpha )}\left(x\right)$: 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

18.18.11		$$\frac{{(a_1^2+\mathrm{\cdots }+a_r^2)}^{\frac{1}{2}n}}{n!}{H}_n\left(\frac{a_1{x}_1+\mathrm{\cdots }+a_r{x}_r}{{(a_1^2+\mathrm{\cdots }+a_r^2)}^{\frac{1}{2}}}\right)=\sum _{m_1+\mathrm{\cdots }+m_r=n}\frac{a_1^{m_1}\mathrm{\cdots }{a}_r^{m_r}}{m_1!\mathrm{\cdots }{m}_r!}{H}_{m_1}\left(x_1\right)\mathrm{\cdots }{H}_{m_r}\left(x_r\right).$$
ⓘ Symbols: $H_n\left(x\right)$: Hermite polynomial, $!$: factorial (as in $n!$), $m$: nonnegative integer, $n$: nonnegative integer, $a_n$ and $x$: real variable Referenced by: §18.18(ii) Permalink: http://dlmf.nist.gov/18.18.E11 Encodings: TeX, pMML, png See also: Annotations for §18.18(ii), §18.18(ii), §18.18 and Ch.18

18.18.12		$$\frac{L_n^{(\alpha )}\left(\lambda x\right)}{L_n^{(\alpha )}\left(0\right)}=\sum _{\mathrm{\ell }=0}^n\left(\genfrac{}{}{0.0pt}{}{n}{\mathrm{\ell }}\right){\lambda }^{\mathrm{\ell }}{(1-\lambda )}^{n-\mathrm{\ell }}\frac{L_{\mathrm{\ell }}^{(\alpha )}\left(x\right)}{L_{\mathrm{\ell }}^{(\alpha )}\left(0\right)}.$$
ⓘ Symbols: $L_n^{(\alpha )}\left(x\right)$: Laguerre (or generalized Laguerre) polynomial, $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $\mathrm{\ell }$: nonnegative integer, $n$: nonnegative integer and $x$: real variable Referenced by: (18.17.34_5), §18.18(iii), §18.18(iv) Permalink: http://dlmf.nist.gov/18.18.E12 Encodings: TeX, pMML, png See also: Annotations for §18.18(iii), §18.18(iii), §18.18 and Ch.18

18.18.13		$$H_n\left(\lambda x\right)={\lambda }^n\sum _{\mathrm{\ell }=0}^{\lfloor n/2\rfloor }\frac{{\left(-n\right)}_{2\mathrm{\ell }}}{\mathrm{\ell }!}{(1-{\lambda }^{-2})}^{\mathrm{\ell }}{H}_{n-2\mathrm{\ell }}\left(x\right).$$
ⓘ Symbols: $H_n\left(x\right)$: Hermite polynomial, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $!$: factorial (as in $n!$), $\lfloor x\rfloor$: floor of $x$, $\mathrm{\ell }$: nonnegative integer, $n$: nonnegative integer and $x$: real variable Referenced by: §18.18(iii) Permalink: http://dlmf.nist.gov/18.18.E13 Encodings: TeX, pMML, png See also: Annotations for §18.18(iii), §18.18(iii), §18.18 and Ch.18

18.18.14		$P_n^{(\gamma ,\beta )}\left(x\right)$	$={\displaystyle \frac{{\left(\beta +1\right)}_n}{{\left(\alpha +\beta +2\right)}_n}}{\displaystyle \sum _{\mathrm{\ell }=0}^n}{\displaystyle \frac{\alpha +\beta +2\mathrm{\ell }+1}{\alpha +\beta +1}}{\displaystyle \frac{{\left(\alpha +\beta +1\right)}_{\mathrm{\ell }}{\left(n+\beta +\gamma +1\right)}_{\mathrm{\ell }}}{{\left(\beta +1\right)}_{\mathrm{\ell }}{\left(n+\alpha +\beta +2\right)}_{\mathrm{\ell }}}}{\displaystyle \frac{{\left(\gamma -\alpha \right)}_{n-\mathrm{\ell }}}{(n-\mathrm{\ell })!}}{P}_{\mathrm{\ell }}^{(\alpha ,\beta )}\left(x\right),$
ⓘ Symbols: $P_n^{(\alpha ,\beta )}\left(x\right)$: Jacobi polynomial, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $!$: factorial (as in $n!$), $\mathrm{\ell }$: nonnegative integer, $n$: nonnegative integer and $x$: real variable Referenced by: §18.17(vi), §18.18(iv) Permalink: http://dlmf.nist.gov/18.18.E14 Encodings: TeX, pMML, png See also: Annotations for §18.18(iv), §18.18(iv), §18.18 and Ch.18
18.18.15		${\left({\displaystyle \frac{1+x}{2}}\right)}^n$	$={\displaystyle \frac{{\left(\beta +1\right)}_n}{{\left(\alpha +\beta +2\right)}_n}}{\displaystyle \sum _{\mathrm{\ell }=0}^n}{\displaystyle \frac{\alpha +\beta +2\mathrm{\ell }+1}{\alpha +\beta +1}}{\displaystyle \frac{{\left(\alpha +\beta +1\right)}_{\mathrm{\ell }}{\left(n-\mathrm{\ell }+1\right)}_{\mathrm{\ell }}}{{\left(\beta +1\right)}_{\mathrm{\ell }}{\left(n+\alpha +\beta +2\right)}_{\mathrm{\ell }}}}{P}_{\mathrm{\ell }}^{(\alpha ,\beta )}\left(x\right),$
ⓘ Symbols: $P_n^{(\alpha ,\beta )}\left(x\right)$: Jacobi polynomial, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\mathrm{\ell }$: nonnegative integer, $n$: nonnegative integer and $x$: real variable Referenced by: §18.18(iv) Permalink: http://dlmf.nist.gov/18.18.E15 Encodings: TeX, pMML, png See also: Annotations for §18.18(iv), §18.18(iv), §18.18 and Ch.18

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 $P_n^{(\gamma ,\delta )}\left(x\right)$ in terms of $P_n^{(\alpha ,\beta )}\left(x\right)$.

18.18.16		$C_n^{(\mu )}\left(x\right)$	$={\displaystyle \sum _{\mathrm{\ell }=0}^{\lfloor n/2\rfloor }}{\displaystyle \frac{\lambda +n-2\mathrm{\ell }}{\lambda }}{\displaystyle \frac{{\left(\mu \right)}_{n-\mathrm{\ell }}}{{\left(\lambda +1\right)}_{n-\mathrm{\ell }}}}{\displaystyle \frac{{\left(\mu -\lambda \right)}_{\mathrm{\ell }}}{\mathrm{\ell }!}}{C}_{n-2\mathrm{\ell }}^{(\lambda )}\left(x\right),$
ⓘ Symbols: ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $!$: factorial (as in $n!$), $\lfloor x\rfloor$: floor of $x$, $C_n^{(\lambda )}\left(x\right)$: ultraspherical (or Gegenbauer) polynomial, $\mathrm{\ell }$: nonnegative integer, $n$: nonnegative integer and $x$: real variable Referenced by: §18.18(iv), §18.18(iv) Permalink: http://dlmf.nist.gov/18.18.E16 Encodings: TeX, pMML, png See also: Annotations for §18.18(iv), §18.18(iv), §18.18 and Ch.18
18.18.17		${(2x)}^n$	$=n!{\displaystyle \sum _{\mathrm{\ell }=0}^{\lfloor n/2\rfloor }}{\displaystyle \frac{\lambda +n-2\mathrm{\ell }}{\lambda }}{\displaystyle \frac{1}{{\left(\lambda +1\right)}_{n-\mathrm{\ell }}\mathrm{\ell }!}}{C}_{n-2\mathrm{\ell }}^{(\lambda )}\left(x\right).$
ⓘ Symbols: ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $!$: factorial (as in $n!$), $\lfloor x\rfloor$: floor of $x$, $C_n^{(\lambda )}\left(x\right)$: ultraspherical (or Gegenbauer) polynomial, $\mathrm{\ell }$: nonnegative integer, $n$: nonnegative integer and $x$: real variable Referenced by: §18.18(iv) Permalink: http://dlmf.nist.gov/18.18.E17 Encodings: TeX, pMML, png See also: Annotations for §18.18(iv), §18.18(iv), §18.18 and Ch.18

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

18.18.18		$L_n^{(\beta )}\left(x\right)$	$={\displaystyle \sum _{\mathrm{\ell }=0}^n}{\displaystyle \frac{{\left(\beta -\alpha \right)}_{n-\mathrm{\ell }}}{(n-\mathrm{\ell })!}}{L}_{\mathrm{\ell }}^{(\alpha )}\left(x\right),$
ⓘ Symbols: $L_n^{(\alpha )}\left(x\right)$: Laguerre (or generalized Laguerre) polynomial, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $!$: factorial (as in $n!$), $\mathrm{\ell }$: nonnegative integer, $n$: nonnegative integer and $x$: real variable Referenced by: §18.18(iv) Permalink: http://dlmf.nist.gov/18.18.E18 Encodings: TeX, pMML, png See also: Annotations for §18.18(iv), §18.18(iv), §18.18 and Ch.18
18.18.19		$x^n$	$={\left(\alpha +1\right)}_n{\displaystyle \sum _{\mathrm{\ell }=0}^n}{\displaystyle \frac{{\left(-n\right)}_{\mathrm{\ell }}}{{\left(\alpha +1\right)}_{\mathrm{\ell }}}}{L}_{\mathrm{\ell }}^{(\alpha )}\left(x\right).$
ⓘ Symbols: $L_n^{(\alpha )}\left(x\right)$: Laguerre (or generalized Laguerre) polynomial, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\mathrm{\ell }$: nonnegative integer, $n$: nonnegative integer and $x$: real variable Referenced by: §18.18(iv) Permalink: http://dlmf.nist.gov/18.18.E19 Encodings: TeX, pMML, png See also: Annotations for §18.18(iv), §18.18(iv), §18.18 and Ch.18

18.18.20		$${(2x)}^n=\sum _{\mathrm{\ell }=0}^{\lfloor n/2\rfloor }\frac{{\left(-n\right)}_{2\mathrm{\ell }}}{\mathrm{\ell }!}{H}_{n-2\mathrm{\ell }}\left(x\right).$$
ⓘ Symbols: $H_n\left(x\right)$: Hermite polynomial, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $!$: factorial (as in $n!$), $\lfloor x\rfloor$: floor of $x$, $\mathrm{\ell }$: nonnegative integer, $n$: nonnegative integer and $x$: real variable Referenced by: §18.18(iv) Permalink: http://dlmf.nist.gov/18.18.E20 Encodings: TeX, pMML, png See also: Annotations for §18.18(iv), §18.18(iv), §18.18 and Ch.18

18.18.21		$$T_m\left(x\right)T_n\left(x\right)=\frac{1}{2}(T_{m+n}\left(x\right)+T_{m-n}\left(x\right)).$$
ⓘ Symbols: $T_n\left(x\right)$: Chebyshev polynomial of the first kind, $m$: nonnegative integer, $n$: nonnegative integer and $x$: real variable Permalink: http://dlmf.nist.gov/18.18.E21 Encodings: TeX, pMML, png See also: Annotations for §18.18(v), §18.18(v), §18.18 and Ch.18

18.18.22		$$C_m^{(\lambda )}\left(x\right)C_n^{(\lambda )}\left(x\right)=\sum _{\mathrm{\ell }=0}^{\min (m,n)}\begin{array}{l}\frac{(m+n+\lambda -2\mathrm{\ell })(m+n-2\mathrm{\ell })!}{(m+n+\lambda -\mathrm{\ell })\mathrm{\ell }!(m-\mathrm{\ell })!(n-\mathrm{\ell })!}\\ \times \frac{{\left(\lambda \right)}_{\mathrm{\ell }}{\left(\lambda \right)}_{m-\mathrm{\ell }}{\left(\lambda \right)}_{n-\mathrm{\ell }}{\left(2\lambda \right)}_{m+n-\mathrm{\ell }}}{{\left(\lambda \right)}_{m+n-\mathrm{\ell }}{\left(2\lambda \right)}_{m+n-2\mathrm{\ell }}}{C}_{m+n-2\mathrm{\ell }}^{(\lambda )}\left(x\right).\end{array}$$
ⓘ Symbols: ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $!$: factorial (as in $n!$), $C_n^{(\lambda )}\left(x\right)$: ultraspherical (or Gegenbauer) polynomial, $\mathrm{\ell }$: nonnegative integer, $m$: nonnegative integer, $n$: nonnegative integer and $x$: real variable Referenced by: (18.17.41_5), §18.17(viii), §18.18(v) Permalink: http://dlmf.nist.gov/18.18.E22 Encodings: TeX, pMML, png See also: Annotations for §18.18(v), §18.18(v), §18.18 and Ch.18

18.18.23		$$H_m\left(x\right)H_n\left(x\right)=\sum _{\mathrm{\ell }=0}^{\min (m,n)}\left(\genfrac{}{}{0.0pt}{}{m}{\mathrm{\ell }}\right)\left(\genfrac{}{}{0.0pt}{}{n}{\mathrm{\ell }}\right)2^{\mathrm{\ell }}\mathrm{\ell }!H_{m+n-2\mathrm{\ell }}\left(x\right).$$
ⓘ Symbols: $H_n\left(x\right)$: Hermite polynomial, $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $!$: factorial (as in $n!$), $\mathrm{\ell }$: nonnegative integer, $m$: nonnegative integer, $n$: nonnegative integer and $x$: real variable Referenced by: §18.17(viii), §18.17(v), §18.18(v) Permalink: http://dlmf.nist.gov/18.18.E23 Encodings: TeX, pMML, png See also: Annotations for §18.18(v), §18.18(v), §18.18 and Ch.18

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.

With

18.18.24		$$b_{n,\mathrm{\ell }}=\left(\genfrac{}{}{0.0pt}{}{n}{\mathrm{\ell }}\right)\frac{{\left(n+\alpha +\beta +1\right)}_{\mathrm{\ell }}{\left(-\beta -n\right)}_{n-\mathrm{\ell }}}{2^{\mathrm{\ell }}{\left(\alpha +1\right)}_n},$$
ⓘ Symbols: ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $\mathrm{\ell }$: nonnegative integer and $n$: nonnegative integer Permalink: http://dlmf.nist.gov/18.18.E24 Encodings: TeX, pMML, png See also: Annotations for §18.18(vi), §18.18(vi), §18.18 and Ch.18

18.18.25		$$\frac{P_n^{(\alpha ,\beta )}\left(x\right)}{P_n^{(\alpha ,\beta )}\left(1\right)}\frac{P_n^{(\alpha ,\beta )}\left(y\right)}{P_n^{(\alpha ,\beta )}\left(1\right)}=\sum _{\mathrm{\ell }=0}^n{b}_{n,\mathrm{\ell }}{(x+y)}^{\mathrm{\ell }}\frac{P_{\mathrm{\ell }}^{(\alpha ,\beta )}\left((1+xy)/(x+y)\right)}{P_{\mathrm{\ell }}^{(\alpha ,\beta )}\left(1\right)},$$
ⓘ Symbols: $P_n^{(\alpha ,\beta )}\left(x\right)$: Jacobi polynomial, $y$: real variable, $\mathrm{\ell }$: nonnegative integer, $n$: nonnegative integer and $x$: real variable Permalink: http://dlmf.nist.gov/18.18.E25 Encodings: TeX, pMML, png See also: Annotations for §18.18(vi), §18.18(vi), §18.18 and Ch.18

18.18.26		$$\frac{P_n^{(\alpha ,\beta )}\left(x\right)}{P_n^{(\alpha ,\beta )}\left(1\right)}=\sum _{\mathrm{\ell }=0}^n{b}_{n,\mathrm{\ell }}{(x+1)}^{\mathrm{\ell }}.$$
ⓘ Symbols: $P_n^{(\alpha ,\beta )}\left(x\right)$: Jacobi polynomial, $\mathrm{\ell }$: nonnegative integer, $n$: nonnegative integer and $x$: real variable Permalink: http://dlmf.nist.gov/18.18.E26 Encodings: TeX, pMML, png See also: Annotations for §18.18(vi), §18.18(vi), §18.18 and Ch.18

18.18.27		$$\sum _{n=0}^{\mathrm{\infty }}\frac{n!L_n^{(\alpha )}\left(x\right)L_n^{(\alpha )}\left(y\right)}{{\left(\alpha +1\right)}_n}{z}^n=\frac{\mathrm{\Gamma }\left(\alpha +1\right){(xyz)}^{-\frac{1}{2}\alpha }}{1-z}\exp \left(\frac{-(x+y)z}{1-z}\right)I_{\alpha }\left(\frac{2{(xyz)}^{\frac{1}{2}}}{1-z}\right),$$
.
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, $L_n^{(\alpha )}\left(x\right)$: Laguerre (or generalized Laguerre) polynomial, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\exp z$: exponential function, $!$: factorial (as in $n!$), $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $y$: real variable, $z$: complex variable, $n$: nonnegative integer and $x$: real variable Referenced by: §18.18(vii), §18.18(vii) Permalink: http://dlmf.nist.gov/18.18.E27 Encodings: TeX, pMML, png See also: Annotations for §18.18(vii), §18.18(vii), §18.18 and Ch.18

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.

18.18.28		$$\sum _{n=0}^{\mathrm{\infty }}\frac{H_n\left(x\right)H_n\left(y\right)}{2^{n}n!}{z}^n={(1-z^2)}^{-\frac{1}{2}}\exp \left(\frac{2xyz-(x^2+y^2)z^2}{1-z^2}\right),$$
.
ⓘ Symbols: $H_n\left(x\right)$: Hermite polynomial, $\exp z$: exponential function, $!$: factorial (as in $n!$), $y$: real variable, $z$: complex variable, $n$: nonnegative integer and $x$: real variable Referenced by: §18.18(vii) Permalink: http://dlmf.nist.gov/18.18.E28 Encodings: TeX, pMML, png See also: Annotations for §18.18(vii), §18.18(vii), §18.18 and Ch.18

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, , and in the case of (18.18.27) $x,y\ge 0$. For the Poisson kernel of Jacobi polynomials (the Bailey formula) see Bailey (1938).

18.18.29		$$\sum _{\mathrm{\ell }=0}^n{C}_{\mathrm{\ell }}^{(\lambda )}\left(x\right)C_{n-\mathrm{\ell }}^{(\mu )}\left(x\right)=C_n^{(\lambda +\mu )}\left(x\right),$$
ⓘ Symbols: $C_n^{(\lambda )}\left(x\right)$: ultraspherical (or Gegenbauer) polynomial, $\mathrm{\ell }$: 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		$$\sum _{\mathrm{\ell }=0}^n\frac{\mathrm{\ell }+2\lambda }{2\lambda }{C}_{\mathrm{\ell }}^{(\lambda )}\left(x\right)x^{n-\mathrm{\ell }}=C_n^{(\lambda +1)}\left(x\right).$$
ⓘ Symbols: $C_n^{(\lambda )}\left(x\right)$: ultraspherical (or Gegenbauer) polynomial, $\mathrm{\ell }$: 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

18.18.31		${\displaystyle \sum _{\mathrm{\ell }=0}^n}{T}_{\mathrm{\ell }}\left(x\right)x^{n-\mathrm{\ell }}$	$=U_n\left(x\right),$
ⓘ Symbols: $T_n\left(x\right)$: Chebyshev polynomial of the first kind, $U_n\left(x\right)$: Chebyshev polynomial of the second kind, $\mathrm{\ell }$: nonnegative integer, $n$: nonnegative integer and $x$: real variable Referenced by: §18.18(viii) Permalink: http://dlmf.nist.gov/18.18.E31 Encodings: TeX, pMML, png See also: Annotations for §18.18(viii), §18.18(viii), §18.18 and Ch.18
18.18.32		$2{\displaystyle \sum _{\mathrm{\ell }=0}^n}{T}_{2\mathrm{\ell }}\left(x\right)$	$=1+U_{2n}\left(x\right),$
ⓘ Symbols: $T_n\left(x\right)$: Chebyshev polynomial of the first kind, $U_n\left(x\right)$: Chebyshev polynomial of the second kind, $\mathrm{\ell }$: 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		$2{\displaystyle \sum _{\mathrm{\ell }=0}^n}{T}_{2\mathrm{\ell }+1}\left(x\right)$	$=U_{2n+1}\left(x\right),$
ⓘ Symbols: $T_n\left(x\right)$: Chebyshev polynomial of the first kind, $U_n\left(x\right)$: Chebyshev polynomial of the second kind, $\mathrm{\ell }$: nonnegative integer, $n$: nonnegative integer and $x$: real variable A&S Ref: 22.12.3 Permalink: http://dlmf.nist.gov/18.18.E33 Encodings: TeX, pMML, png See also: Annotations for §18.18(viii), §18.18(viii), §18.18 and Ch.18
18.18.34		$2(1-x^2){\displaystyle \sum _{\mathrm{\ell }=0}^n}{U}_{2\mathrm{\ell }}\left(x\right)$	$=1-T_{2n+2}\left(x\right),$
ⓘ Symbols: $T_n\left(x\right)$: Chebyshev polynomial of the first kind, $U_n\left(x\right)$: Chebyshev polynomial of the second kind, $\mathrm{\ell }$: nonnegative integer, $n$: nonnegative integer and $x$: real variable A&S Ref: 22.12.4 Permalink: http://dlmf.nist.gov/18.18.E34 Encodings: TeX, pMML, png See also: Annotations for §18.18(viii), §18.18(viii), §18.18 and Ch.18
18.18.35		$2(1-x^2){\displaystyle \sum _{\mathrm{\ell }=0}^n}{U}_{2\mathrm{\ell }+1}\left(x\right)$	$=x-T_{2n+3}\left(x\right).$
ⓘ Symbols: $T_n\left(x\right)$: Chebyshev polynomial of the first kind, $U_n\left(x\right)$: Chebyshev polynomial of the second kind, $\mathrm{\ell }$: 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

18.18.36		$$\sum _{\mathrm{\ell }=0}^n{P}_{\mathrm{\ell }}\left(x\right)P_{n-\mathrm{\ell }}\left(x\right)=U_n\left(x\right).$$
ⓘ Symbols: $U_n\left(x\right)$: Chebyshev polynomial of the second kind, $P_n\left(x\right)$: Legendre polynomial, $\mathrm{\ell }$: nonnegative integer, $n$: nonnegative integer and $x$: real variable Referenced by: §18.18(viii) Permalink: http://dlmf.nist.gov/18.18.E36 Encodings: TeX, pMML, png See also: Annotations for §18.18(viii), §18.18(viii), §18.18 and Ch.18

18.18.37		$$\sum _{\mathrm{\ell }=0}^n{L}_{\mathrm{\ell }}^{(\alpha )}\left(x\right)=L_n^{(\alpha +1)}\left(x\right),$$
ⓘ Symbols: $L_n^{(\alpha )}\left(x\right)$: Laguerre (or generalized Laguerre) polynomial, $\mathrm{\ell }$: 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		$$\sum _{\mathrm{\ell }=0}^n{L}_{\mathrm{\ell }}^{(\alpha )}\left(x\right)L_{n-\mathrm{\ell }}^{(\beta )}\left(y\right)=L_n^{(\alpha +\beta +1)}\left(x+y\right).$$
ⓘ Symbols: $L_n^{(\alpha )}\left(x\right)$: Laguerre (or generalized Laguerre) polynomial, $y$: real variable, $\mathrm{\ell }$: 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