18 Orthogonal Polynomials Classical Orthogonal Polynomials 18.17 Integrals 18.19 Hahn Class: Definitions

§18.18 Sums

Keywords:: Jacobi polynomials, classical orthogonal polynomials
Referenced by:: §14.18(iv)
Permalink:: http://dlmf.nist.gov/18.18

§18.18(i) Series Expansions of Arbitrary Functions
§18.18(ii) Addition Theorems
§18.18(iii) Multiplication Theorems
§18.18(iv) Connection Formulas
§18.18(v) Linearization Formulas
§18.18(vi) Bateman-Type Sums
§18.18(vii) Poisson Kernels
§18.18(viii) Other Sums
§18.18(ix) Compendia

§18.18(i) Series Expansions of Arbitrary Functions

Notes:: See Lebedev (1965, pp. 68–71 and 88–89), and Nikiforov and Uvarov (1988, pp. 21 and 59). For (18.18.2) see Szegö (1975, Theorem 9.1.2 and the Remarks on p. 248).
Referenced by:: §14.18(i), §18.40
Permalink:: http://dlmf.nist.gov/18.18.i

¶ Jacobi

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

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

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{P^{{(\alpha,\beta)}}_{{n}}\/}\nolimits\!\left(x\right)$ : Jacobi polynomial, : differential of , : : factorial, $\int$ : integral, : nonnegative integer, : analytic function, $a_{n}$ and : real variable
Referenced by:: ¶ ‣ §18.18(i)
Permalink:: http://dlmf.nist.gov/18.18.E1
Encodings:: TeX, pMML, png

Then

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

Symbols:: $\mathop{P^{{(\alpha,\beta)}}_{{n}}\/}\nolimits\!\left(x\right)$ : Jacobi polynomial, : complex variable, : nonnegative integer, : 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

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

Alternatively, assume is real and continuous and $f^{{\prime}}(x)$ is piecewise continuous on . 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 replaced by , applies when ; moreover, the convergence is uniform on any compact interval within .

¶ Chebyshev

Keywords:: Chebyshev polynomials

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

Keywords:: Legendre polynomials

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

18.18.3 $a_{n}=\left(n+\tfrac{1}{2}\right)\int_{{-1}}^{1}f(x)\mathop{P_{{n}}\/}% \nolimits\!\left(x\right)dx.$

Symbols:: $\mathop{P^{{\mu}}_{{\nu}}\/}\nolimits\!\left(z\right)$ : associated Legendre function of the first kind, : differential of , $\int$ : integral, : nonnegative integer, : analytic function, $a_{n}$ and : real variable
Permalink:: http://dlmf.nist.gov/18.18.E3
Encodings:: TeX, pMML, png

¶ Laguerre

Keywords:: Laguerre polynomials

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

18.18.4 $f(x)=\sum_{{n=0}}^{\infty}b_{n}\mathop{L^{{(\alpha)}}_{{n}}\/}\nolimits\!\left% (x\right),$ $0<x<\infty$ ,

Symbols:: $\mathop{L^{{(\alpha)}}_{{n}}\/}\nolimits\!\left(x\right)$ : Laguerre (or generalized Laguerre) polynomial, : nonnegative integer, : analytic function and : real variable
Referenced by:: ¶ ‣ §18.18(i)
Permalink:: http://dlmf.nist.gov/18.18.E4
Encodings:: TeX, pMML, png

where

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

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, : differential of , : base of exponential function, : : factorial, $\mathop{L^{{(\alpha)}}_{{n}}\/}\nolimits\!\left(x\right)$ : Laguerre (or generalized Laguerre) polynomial, $\int$ : integral, : nonnegative integer, : analytic function and : real variable
Permalink:: http://dlmf.nist.gov/18.18.E5
Encodings:: TeX, pMML, png

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

¶ Hermite

Keywords:: Hermite polynomials

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

18.18.6 $f(x)=\sum_{{n=0}}^{\infty}d_{n}\mathop{H_{{n}}\/}\nolimits\!\left(x\right),$ $-\infty<x<\infty$ ,

Symbols:: $\mathop{H_{{n}}\/}\nolimits\!\left(x\right)$ : Hermite polynomial, : nonnegative integer, : analytic function, $d_{n}$ and : real variable
Referenced by:: ¶ ‣ §18.18(i)
Permalink:: http://dlmf.nist.gov/18.18.E6
Encodings:: TeX, pMML, png

where

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

Symbols:: $\mathop{H_{{n}}\/}\nolimits\!\left(x\right)$ : Hermite polynomial, : differential of , : base of exponential function, : : factorial, $\int$ : integral, : nonnegative integer, : analytic function, $d_{n}$ and : real variable
Permalink:: http://dlmf.nist.gov/18.18.E7
Encodings:: TeX, pMML, png

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

§18.18(ii) Addition Theorems

Notes:: For (18.18.8) see Carlson (1971). (18.18.9) is the case $\alpha=0$ of (18.18.8). (18.18.10) follows from (18.12.13). (18.18.11) follows from (18.12.15).
Keywords:: classical orthogonal polynomials
Permalink:: http://dlmf.nist.gov/18.18.ii

¶ Ultraspherical

Keywords:: ultraspherical polynomials

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

Symbols:: $\left(a\right)_{{n}}$ : Pochhammer’s symbol, $\mathop{\cos\/}\nolimits z$ : cosine function, : : factorial, $\mathop{\sin\/}\nolimits z$ : sine function, $\mathop{C^{{(\lambda)}}_{{n}}\/}\nolimits\!\left(x\right)$ : ultraspherical (or Gegenbauer) polynomial, $\ell$ : nonnegative integer and : nonnegative integer
Referenced by:: ¶ ‣ §18.18(ii), §18.18(ii)
Permalink:: http://dlmf.nist.gov/18.18.E8
Encodings:: TeX, pMML, png

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

¶ Legendre

Keywords:: Legendre polynomials

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

Symbols:: $\mathop{P^{{(\alpha,\beta)}}_{{n}}\/}\nolimits\!\left(x\right)$ : Jacobi polynomial, $\mathop{P_{{n}}\/}\nolimits\!\left(x\right)$ : Legendre polynomial, $\mathop{\cos\/}\nolimits z$ : cosine function, : : factorial, $\mathop{\sin\/}\nolimits z$ : sine function, $\ell$ : nonnegative integer and : nonnegative integer
Referenced by:: ¶ ‣ §14.30(iii), ¶ ‣ §18.18(ii), ¶ ‣ §18.18(ii), §18.18(ii)
Permalink:: http://dlmf.nist.gov/18.18.E9
Encodings:: TeX, pMML, png

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

¶ Laguerre

Keywords:: Laguerre polynomials

18.18.10 $\mathop{L^{{(\alpha_{1}+\dots+\alpha_{r}+r-1)}}_{{n}}\/}\nolimits\!\left(x_{1}% +\dots+x_{r}\right)=\sum_{{m_{1}+\dots+m_{r}=n}}\mathop{L^{{(\alpha_{1})}}_{{m% _{1}}}\/}\nolimits\!\left(x_{1}\right)\cdots\mathop{L^{{(\alpha_{r})}}_{{m_{r}% }}\/}\nolimits\!\left(x_{r}\right).$

Symbols:: $\mathop{L^{{(\alpha)}}_{{n}}\/}\nolimits\!\left(x\right)$ : Laguerre (or generalized Laguerre) polynomial, : nonnegative integer, : nonnegative integer and : real variable
Referenced by:: §18.18(ii), §18.18(viii)
Permalink:: http://dlmf.nist.gov/18.18.E10
Encodings:: TeX, pMML, png

¶ Hermite

Keywords:: Hermite polynomials

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

Symbols:: $\mathop{H_{{n}}\/}\nolimits\!\left(x\right)$ : Hermite polynomial, : : factorial, : nonnegative integer, : nonnegative integer, $a_{n}$ and : real variable
Referenced by:: §18.18(ii)
Permalink:: http://dlmf.nist.gov/18.18.E11
Encodings:: TeX, pMML, png

§18.18(iii) Multiplication Theorems

Notes:: (18.18.12) follows by computing $\int_{0}^{\infty}\mathop{L^{{(\alpha)}}_{{n}}\/}\nolimits\!\left(\lambda x% \right)\mathop{L^{{(\alpha)}}_{{\ell}}\/}\nolimits\!\left(x\right)e^{{-x}}x^{% \alpha}dx$ with use of the Rodrigues formula (Table 18.5.1), integration by parts, and (18.5.12). (18.18.13) follows from (18.18.12) for $\alpha=\pm\frac{1}{2}$ by (18.7.19), (18.7.20).
Keywords:: classical orthogonal polynomials
Permalink:: http://dlmf.nist.gov/18.18.iii

¶ Laguerre

Keywords:: Laguerre polynomials

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

Symbols:: $\binom{m}{n}$ : binomial coefficient, $\mathop{L^{{(\alpha)}}_{{n}}\/}\nolimits\!\left(x\right)$ : Laguerre (or generalized Laguerre) polynomial, $\ell$ : nonnegative integer, : nonnegative integer and : real variable
Referenced by:: §18.18(iii), §18.18(iv)
Permalink:: http://dlmf.nist.gov/18.18.E12
Encodings:: TeX, pMML, png

¶ Hermite

Keywords:: Hermite polynomials

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

Symbols:: $\mathop{H_{{n}}\/}\nolimits\!\left(x\right)$ : Hermite polynomial, $\left(a\right)_{{n}}$ : Pochhammer’s symbol, : : factorial, $\left\lfloor x\right\rfloor$ : floor of , $\ell$ : nonnegative integer, : nonnegative integer and : real variable
Referenced by:: §18.18(iii)
Permalink:: http://dlmf.nist.gov/18.18.E13
Encodings:: TeX, pMML, png

§18.18(iv) Connection Formulas

Notes:: For (18.18.14) see Andrews et al. (1999, Theorem 7.1.3). For (18.18.15) see Askey (1974). For (18.18.16) see Andrews et al. (1999, Theorem 7.1.4 ${}^{\prime}$ ). (18.18.17) follows from (18.18.16), (18.6.4), and the fourth row of Table 18.6.1. (18.18.18) follows from (18.12.13). (18.18.19) follows from (18.18.12) by dividing both sides by $\lambda^{n}$ and letting $\lambda\to\infty$ . (18.18.20) is the case $\beta=\pm\frac{1}{2}$ of (18.18.19) in view of (18.7.19), (18.7.20).
Keywords:: classical orthogonal polynomials
Permalink:: http://dlmf.nist.gov/18.18.iv

¶ Jacobi

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

Symbols:: $\mathop{P^{{(\alpha,\beta)}}_{{n}}\/}\nolimits\!\left(x\right)$ : Jacobi polynomial, $\left(a\right)_{{n}}$ : Pochhammer’s symbol, : : factorial, $\ell$ : nonnegative integer, : nonnegative integer and : real variable
Referenced by:: §18.17(vi), §18.18(iv)
Permalink:: http://dlmf.nist.gov/18.18.E14
Encodings:: TeX, pMML, png

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

Symbols:: $\mathop{P^{{(\alpha,\beta)}}_{{n}}\/}\nolimits\!\left(x\right)$ : Jacobi polynomial, $\left(a\right)_{{n}}$ : Pochhammer’s symbol, $\ell$ : nonnegative integer, : nonnegative integer and : real variable
Referenced by:: §18.18(iv)
Permalink:: http://dlmf.nist.gov/18.18.E15
Encodings:: TeX, pMML, png

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

¶ Ultraspherical

Keywords:: ultraspherical polynomials

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

Symbols:: $\left(a\right)_{{n}}$ : Pochhammer’s symbol, : : factorial, $\left\lfloor x\right\rfloor$ : floor of , $\mathop{C^{{(\lambda)}}_{{n}}\/}\nolimits\!\left(x\right)$ : ultraspherical (or Gegenbauer) polynomial, $\ell$ : nonnegative integer, : nonnegative integer and : real variable
Referenced by:: §18.18(iv)
Permalink:: http://dlmf.nist.gov/18.18.E16
Encodings:: TeX, pMML, png

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

Symbols:: $\left(a\right)_{{n}}$ : Pochhammer’s symbol, : : factorial, $\left\lfloor x\right\rfloor$ : floor of , $\mathop{C^{{(\lambda)}}_{{n}}\/}\nolimits\!\left(x\right)$ : ultraspherical (or Gegenbauer) polynomial, $\ell$ : nonnegative integer, : nonnegative integer and : real variable
Referenced by:: §18.18(iv)
Permalink:: http://dlmf.nist.gov/18.18.E17
Encodings:: TeX, pMML, png

¶ Laguerre

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

Symbols:: $\left(a\right)_{{n}}$ : Pochhammer’s symbol, : : factorial, $\mathop{L^{{(\alpha)}}_{{n}}\/}\nolimits\!\left(x\right)$ : Laguerre (or generalized Laguerre) polynomial, $\ell$ : nonnegative integer, : nonnegative integer and : real variable
Referenced by:: §18.18(iv)
Permalink:: http://dlmf.nist.gov/18.18.E18
Encodings:: TeX, pMML, png

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

Symbols:: $\left(a\right)_{{n}}$ : Pochhammer’s symbol, $\mathop{L^{{(\alpha)}}_{{n}}\/}\nolimits\!\left(x\right)$ : Laguerre (or generalized Laguerre) polynomial, $\ell$ : nonnegative integer, : nonnegative integer and : real variable
Referenced by:: §18.18(iv)
Permalink:: http://dlmf.nist.gov/18.18.E19
Encodings:: TeX, pMML, png

¶ Hermite

Keywords:: Hermite polynomials, classical orthogonal polynomials

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

Symbols:: $\mathop{H_{{n}}\/}\nolimits\!\left(x\right)$ : Hermite polynomial, $\left(a\right)_{{n}}$ : Pochhammer’s symbol, : : factorial, $\left\lfloor x\right\rfloor$ : floor of , $\ell$ : nonnegative integer, : nonnegative integer and : real variable
Referenced by:: §18.18(iv)
Permalink:: http://dlmf.nist.gov/18.18.E20
Encodings:: TeX, pMML, png

§18.18(v) Linearization Formulas

Notes:: See Andrews et al. (1999, (5.1.6), Theorems 6.8.2 and 6.8.1 and Remarks 6.8.2 and 6.8.1).
Keywords:: classical orthogonal polynomials
Permalink:: http://dlmf.nist.gov/18.18.v

¶ Chebyshev

Keywords:: Chebyshev polynomials

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

Symbols:: $\mathop{T_{{n}}\/}\nolimits\!\left(x\right)$ : Chebyshev polynomial of the first kind, : nonnegative integer, : nonnegative integer and : real variable
Permalink:: http://dlmf.nist.gov/18.18.E21
Encodings:: TeX, pMML, png

¶ Ultraspherical

Keywords:: ultraspherical polynomials

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

Symbols:: $\left(a\right)_{{n}}$ : Pochhammer’s symbol, : : factorial, : open interval, $\mathop{C^{{(\lambda)}}_{{n}}\/}\nolimits\!\left(x\right)$ : ultraspherical (or Gegenbauer) polynomial, $\ell$ : nonnegative integer, : nonnegative integer, : nonnegative integer and : real variable
Referenced by:: ¶ ‣ §18.18(v)
Permalink:: http://dlmf.nist.gov/18.18.E22
Encodings:: TeX, pMML, png

¶ Hermite

Keywords:: Hermite polynomials

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

Symbols:: $\mathop{H_{{n}}\/}\nolimits\!\left(x\right)$ : Hermite polynomial, $\binom{m}{n}$ : binomial coefficient, : : factorial, : open interval, $\ell$ : nonnegative integer, : nonnegative integer, : nonnegative integer and : real variable
Referenced by:: §18.17(v), ¶ ‣ §18.18(v)
Permalink:: http://dlmf.nist.gov/18.18.E23
Encodings:: TeX, pMML, png

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

Notes:: See Koornwinder (1974).
Keywords:: classical orthogonal polynomials
Permalink:: http://dlmf.nist.gov/18.18.vi

¶ Jacobi

Keywords:: Jacobi polynomials

With

18.18.24 $b_{{n,\ell}}=\binom{n}{\ell}\frac{\left(n+\alpha+\beta+1\right)_{{\ell}}\left(% -\beta-n\right)_{{n-\ell}}}{2^{{\ell}}\left(\alpha+1\right)_{{n}}},$

Symbols:: $\left(a\right)_{{n}}$ : Pochhammer’s symbol, $\binom{m}{n}$ : binomial coefficient, $\ell$ : nonnegative integer and : nonnegative integer
Permalink:: http://dlmf.nist.gov/18.18.E24
Encodings:: TeX, pMML, png

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

Symbols:: $\mathop{P^{{(\alpha,\beta)}}_{{n}}\/}\nolimits\!\left(x\right)$ : Jacobi polynomial, : real variable, $\ell$ : nonnegative integer, : nonnegative integer and : real variable
Permalink:: http://dlmf.nist.gov/18.18.E25
Encodings:: TeX, pMML, png

18.18.26 $\frac{\mathop{P^{{(\alpha,\beta)}}_{{n}}\/}\nolimits\!\left(x\right)}{\mathop{% P^{{(\alpha,\beta)}}_{{n}}\/}\nolimits\!\left(1\right)}=\sum_{{\ell=0}}^{n}b_{% {n,\ell}}(x+1)^{{\ell}}.$

Symbols:: $\mathop{P^{{(\alpha,\beta)}}_{{n}}\/}\nolimits\!\left(x\right)$ : Jacobi polynomial, $\ell$ : nonnegative integer, : nonnegative integer and : real variable
Permalink:: http://dlmf.nist.gov/18.18.E26
Encodings:: TeX, pMML, png

§18.18(vii) Poisson Kernels

Notes:: See Andrews et al. (1999, (6.2.25), (6.1.13)). For the positivity of the Poisson kernels see Askey (1975, p. 16).
Keywords:: classical orthogonal polynomials
Permalink:: http://dlmf.nist.gov/18.18.vii

¶ Laguerre

Keywords:: Laguerre polynomials

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

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\left(a\right)_{{n}}$ : Pochhammer’s symbol, $\mathop{\exp\/}\nolimits z$ : exponential function, : : factorial, $\mathop{L^{{(\alpha)}}_{{n}}\/}\nolimits\!\left(x\right)$ : Laguerre (or generalized Laguerre) polynomial, $\mathop{I_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified Bessel function, : real variable, : complex variable, : nonnegative integer and : real variable
Referenced by:: ¶ ‣ §18.18(vii)
Permalink:: http://dlmf.nist.gov/18.18.E27
Encodings:: TeX, pMML, png

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

¶ Hermite

Keywords:: Hermite polynomials

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

Symbols:: $\mathop{H_{{n}}\/}\nolimits\!\left(x\right)$ : Hermite polynomial, $\mathop{\exp\/}\nolimits z$ : exponential function, : : factorial, : real variable, : complex variable, : nonnegative integer and : real variable
Permalink:: http://dlmf.nist.gov/18.18.E28
Encodings:: TeX, pMML, png

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

§18.18(viii) Other Sums

Notes:: (18.18.29) follows from (18.12.4). (18.18.30) follows from (18.12.4) and (18.12.5). (18.18.31) is the limiting case of (18.18.30) as $\lambda\to 0$ . Each of the formulas (18.18.32)–(18.18.35) is equivalent to a difference formula together with a trivial case, and each difference formula can be rewritten via (18.5.1), (18.5.2) as a well-known trigonometric identity. (18.18.36) is the special case $\lambda=\mu=\frac{1}{2}$ of (18.18.29). For (18.18.37) see Szegö (1975, (5.1.13)). (18.18.38) follows from (18.12.13), and is the special case of (18.18.10). For (18.18.39) see Szegö (1975, (5.5.11)). (18.18.40) is the special case $\alpha=-\frac{1}{2}$ of (18.18.38) in view of (18.7.19).
Permalink:: http://dlmf.nist.gov/18.18.viii

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

¶ Ultraspherical

Keywords:: ultraspherical polynomials

18.18.29 $\sum_{{\ell=0}}^{n}\mathop{C^{{(\lambda)}}_{{\ell}}\/}\nolimits\!\left(x\right% )\mathop{C^{{(\mu)}}_{{n-\ell}}\/}\nolimits\!\left(x\right)=\mathop{C^{{(% \lambda+\mu)}}_{{n}}\/}\nolimits\!\left(x\right).$

Symbols:: $\mathop{C^{{(\lambda)}}_{{n}}\/}\nolimits\!\left(x\right)$ : ultraspherical (or Gegenbauer) polynomial, $\ell$ : nonnegative integer, : nonnegative integer and : real variable
Referenced by:: §18.18(viii)
Permalink:: http://dlmf.nist.gov/18.18.E29
Encodings:: TeX, pMML, png

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

Symbols:: $\mathop{C^{{(\lambda)}}_{{n}}\/}\nolimits\!\left(x\right)$ : ultraspherical (or Gegenbauer) polynomial, $\ell$ : nonnegative integer, : nonnegative integer and : real variable
Referenced by:: §18.18(viii)
Permalink:: http://dlmf.nist.gov/18.18.E30
Encodings:: TeX, pMML, png

¶ Chebyshev

Keywords:: Chebyshev polynomials

18.18.31 $\sum_{{\ell=0}}^{n}\mathop{T_{{\ell}}\/}\nolimits\!\left(x\right)x^{{n-\ell}}=% \mathop{U_{{n}}\/}\nolimits\!\left(x\right).$

Symbols:: $\mathop{T_{{n}}\/}\nolimits\!\left(x\right)$ : Chebyshev polynomial of the first kind, $\mathop{U_{{n}}\/}\nolimits\!\left(x\right)$ : Chebyshev polynomial of the second kind, $\ell$ : nonnegative integer, : nonnegative integer and : real variable
Referenced by:: §18.18(viii)
Permalink:: http://dlmf.nist.gov/18.18.E31
Encodings:: TeX, pMML, png

18.18.32 $2\sum_{{\ell=0}}^{n}\mathop{T_{{2\ell}}\/}\nolimits\!\left(x\right)=1+\mathop{% U_{{2n}}\/}\nolimits\!\left(x\right),$

Symbols:: $\mathop{T_{{n}}\/}\nolimits\!\left(x\right)$ : Chebyshev polynomial of the first kind, $\mathop{U_{{n}}\/}\nolimits\!\left(x\right)$ : Chebyshev polynomial of the second kind, $\ell$ : nonnegative integer, : nonnegative integer and : 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

18.18.33 $2\sum_{{\ell=0}}^{n}\mathop{T_{{2\ell+1}}\/}\nolimits\!\left(x\right)=\mathop{% U_{{2n+1}}\/}\nolimits\!\left(x\right).$

Symbols:: $\mathop{T_{{n}}\/}\nolimits\!\left(x\right)$ : Chebyshev polynomial of the first kind, $\mathop{U_{{n}}\/}\nolimits\!\left(x\right)$ : Chebyshev polynomial of the second kind, $\ell$ : nonnegative integer, : nonnegative integer and : real variable
A&S Ref:: 22.12.3
Permalink:: http://dlmf.nist.gov/18.18.E33
Encodings:: TeX, pMML, png

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

Symbols:: $\mathop{T_{{n}}\/}\nolimits\!\left(x\right)$ : Chebyshev polynomial of the first kind, $\mathop{U_{{n}}\/}\nolimits\!\left(x\right)$ : Chebyshev polynomial of the second kind, $\ell$ : nonnegative integer, : nonnegative integer and : real variable
A&S Ref:: 22.12.4
Permalink:: http://dlmf.nist.gov/18.18.E34
Encodings:: TeX, pMML, png

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

Symbols:: $\mathop{T_{{n}}\/}\nolimits\!\left(x\right)$ : Chebyshev polynomial of the first kind, $\mathop{U_{{n}}\/}\nolimits\!\left(x\right)$ : Chebyshev polynomial of the second kind, $\ell$ : nonnegative integer, : nonnegative integer and : 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

¶ Legendre and Chebyshev

Keywords:: Legendre polynomials

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

Symbols:: $\mathop{U_{{n}}\/}\nolimits\!\left(x\right)$ : Chebyshev polynomial of the second kind, $\mathop{P_{{n}}\/}\nolimits\!\left(x\right)$ : Legendre polynomial, $\ell$ : nonnegative integer, : nonnegative integer and : real variable
Referenced by:: §18.18(viii)
Permalink:: http://dlmf.nist.gov/18.18.E36
Encodings:: TeX, pMML, png

¶ Laguerre

18.18.37 $\sum_{{\ell=0}}^{n}\mathop{L^{{(\alpha)}}_{{\ell}}\/}\nolimits\!\left(x\right)% =\mathop{L^{{(\alpha+1)}}_{{n}}\/}\nolimits\!\left(x\right),$

Symbols:: $\mathop{L^{{(\alpha)}}_{{n}}\/}\nolimits\!\left(x\right)$ : Laguerre (or generalized Laguerre) polynomial, $\ell$ : nonnegative integer, : nonnegative integer and : real variable
Referenced by:: §18.18(viii)
Permalink:: http://dlmf.nist.gov/18.18.E37
Encodings:: TeX, pMML, png

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:: $\mathop{L^{{(\alpha)}}_{{n}}\/}\nolimits\!\left(x\right)$ : Laguerre (or generalized Laguerre) polynomial, : real variable, $\ell$ : nonnegative integer, : nonnegative integer and : 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

¶ Hermite and Laguerre

Keywords:: Hermite polynomials

18.18.39 $\sum_{{\ell=0}}^{n}\binomial{n}{\ell}\mathop{H_{{\ell}}\/}\nolimits\!\left(2^{% {\frac{1}{2}}}x\right)\mathop{H_{{n-\ell}}\/}\nolimits\!\left(2^{{\frac{1}{2}}% }y\right)=2^{{\frac{1}{2}n}}\mathop{H_{{n}}\/}\nolimits\!\left(x+y\right),$

Symbols:: $\mathop{H_{{n}}\/}\nolimits\!\left(x\right)$ : Hermite polynomial, $\binom{m}{n}$ : binomial coefficient, : real variable, $\ell$ : nonnegative integer, : nonnegative integer and : real variable
A&S Ref:: 22.12.8
Referenced by:: §18.18(viii)
Permalink:: http://dlmf.nist.gov/18.18.E39
Encodings:: TeX, pMML, png

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

Symbols:: $\mathop{H_{{n}}\/}\nolimits\!\left(x\right)$ : Hermite polynomial, $\binom{m}{n}$ : binomial coefficient, : : factorial, $\mathop{L^{{(\alpha)}}_{{n}}\/}\nolimits\!\left(x\right)$ : Laguerre (or generalized Laguerre) polynomial, : real variable, $\ell$ : nonnegative integer, : nonnegative integer and : real variable
Referenced by:: §18.18(viii)
Permalink:: http://dlmf.nist.gov/18.18.E40
Encodings:: TeX, pMML, png

§18.18(ix) Compendia

Keywords:: classical orthogonal polynomials
Permalink:: http://dlmf.nist.gov/18.18.ix

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).

© 2010–2013 NIST / Privacy Policy / Disclaimer / Feedback; Version 1.0.6; Release date 2013-05-06 Math-Image version (See MathML) . A Printed Companionis available. 18.17 Integrals 18.19 Hahn Class: Definitions

§18.18 Sums

Contents

§18.18(i) Series Expansions of Arbitrary Functions

¶ Jacobi

¶ Chebyshev

¶ Legendre

¶ Laguerre

¶ Hermite

§18.18(ii) Addition Theorems

¶ Ultraspherical

¶ Legendre

¶ Laguerre

¶ Hermite

§18.18(iii) Multiplication Theorems

¶ Laguerre

¶ Hermite

§18.18(iv) Connection Formulas

¶ Jacobi

¶ Ultraspherical

¶ Laguerre

¶ Hermite

§18.18(v) Linearization Formulas

¶ Chebyshev

¶ Ultraspherical

¶ Hermite

§18.18(vi) Bateman-Type Sums

¶ Jacobi

§18.18(vii) Poisson Kernels

¶ Laguerre

¶ Hermite

§18.18(viii) Other Sums

¶ Ultraspherical

¶ Chebyshev

¶ Legendre and Chebyshev

¶ Laguerre

¶ Hermite and Laguerre

§18.18(ix) Compendia