Gauss–Laguerre formula

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11: Errata

… ►We also discuss non-classical Laguerre polynomials and give much more details and examples on exceptional orthogonal polynomials. … ►

Equation (18.34.1)

18.34.1

y_{n}\left(x;a\right)={{}_{2}F_{0}}\left({-n,n+a-1\atop-};-\frac{x}{2}\right)=% {\left(n+a-1\right)_{n}}\left(\frac{x}{2}\right)^{n}{{}_{1}F_{1}}\left({-n% \atop-2n-a+2};\frac{2}{x}\right)=n!\left(-\tfrac{1}{2}x\right)^{n}L^{(1-a-2n)}% _{n}\left(2x^{-1}\right)=\left(\tfrac{1}{2}x\right)^{1-\frac{1}{2}a}{\mathrm{e% }}^{1/x}W_{1-\frac{1}{2}a,\frac{1}{2}(a-1)+n}\left(2x^{-1}\right)

This equation was updated to include the definition of Bessel polynomials in terms of Laguerre polynomials and the Whittaker confluent hypergeometric function.

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Chapter 35 Functions of Matrix Argument

The generalized hypergeometric function of matrix argument ${{}_{p}F_{q}}\left(a_{1},\dots,a_{p};b_{1},\dots,b_{q};\mathbf{T}\right)$ , was linked inadvertently as its single variable counterpart ${{}_{p}F_{q}}\left(a_{1},\dots,a_{p};b_{1},\dots,b_{q};\mathbf{T}\right)$ . Furthermore, the Jacobi function of matrix argument $P^{(\gamma,\delta)}_{\nu}\left(\mathbf{T}\right)$ , and the Laguerre function of matrix argument $L^{(\gamma)}_{\nu}\left(\mathbf{T}\right)$ , were also linked inadvertently (and incorrectly) in terms of the single variable counterparts given by $P^{(\gamma,\delta)}_{\nu}\left(\mathbf{T}\right)$ , and $L^{(\gamma)}_{\nu}\left(\mathbf{T}\right)$ . In order to resolve these inconsistencies, these functions now link correctly to their respective definitions.

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Subsections 15.4(i), 15.4(ii)

Sentences were added specifying that some equations in these subsections require special care under certain circumstances. Also, (15.4.6) was expanded by adding the formula $F\left(a,b;a;z\right)=(1-z)^{-b}$ .

Report by Louis Klauder on 2017-01-01.

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Equation (18.15.22)

Because of the use of the $O$ order symbol on the right-hand side, the asymptotic expansion for the generalized Laguerre polynomial $L^{(\alpha)}_{n}\left(\nu x\right)$ was rewritten as an equality.

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12: 18.2 General Orthogonal Polynomials

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§18.2(v) Christoffel–Darboux Formula

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Confluent Form

… ►For usage of the zeros of an OP in Gauss quadrature see §3.5(v). … ►

Degree lowering and raising differentiation formulas and structure relations

►For a large class of OP’s

p_{n}

there exist pairs of differentiation formulas …