orthogonal%20polynomials

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21: 18.39 Applications in the Physical Sciences

… ►This is not the orthogonality of Table 18.8.1, as the co-ordinate arguments depend, independently on $p$ and $q$. … ►Derivations of (18.39.42) appear in Bethe and Salpeter (1957, pp. 12–20), and Pauling and Wilson (1985, Chapter V and Appendix VII), where the derivations are based on (18.39.36), and is also the notation of Piela (2014, §4.7), typifying the common use of the associated Coulomb–Laguerre polynomials in theoretical quantum chemistry. … ►

The Coulomb–Pollaczek Polynomials

… ►These cases correspond to the two distinct orthogonality conditions of (18.35.6) and (18.35.6_3). … ►For interpretations of zeros of classical OP’s as equilibrium positions of charges in electrostatic problems (assuming logarithmic interaction), see Ismail (2000a, b).

22: 3.8 Nonlinear Equations

… ►For the computation of zeros of orthogonal polynomials as eigenvalues of finite tridiagonal matrices (§3.5(vi)), see Gil et al. (2007a, pp. 205–207). … ►The polynomial … ►

Example. Wilkinson’s Polynomial

… ►Consider $x=20$ and $j=19$. We have $p^{\prime }(20)=19!$ and $a_{19}=1+2+\mathrm{\cdots }+20=210$. …