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11: 18.38 Mathematical Applications

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Approximation Theory

►The monic Chebyshev polynomial $2^{1-n}{T}_n\left(x\right)$, $n\ge 1$, enjoys the ‘minimax’ property on the interval $[-1,1]$, that is, $|2^{1-n}{T}_n\left(x\right)|$ has the least maximum value among all monic polynomials of degree $n$. … ►

Integrable Systems

… ►Ultraspherical polynomials are zonal spherical harmonics. … ►

Group Representations

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12: 32.8 Rational Solutions

… ►where the $Q_n(z)$ are monic polynomials (coefficient of highest power of $z$ is $1$) satisfying … ►Next, let $p_m(z)$ be the polynomials defined by $p_m(z)=0$ for , and … ►where $P_m(z)$ and $Q_m(z)$ are polynomials of degree $m$, with no common zeros. … ►where $P_{j,n-1}(z)$ and $Q_{j,n}(z)$ are polynomials of degrees $n-1$ and $n$, respectively, with no common zeros. … ►where $\lambda$, $\mu$ are constants, and $P_{n-1}(z)$, $Q_n(z)$ are polynomials of degrees $n-1$ and $n$, respectively, with no common zeros. …

13: 18.33 Polynomials Orthogonal on the Unit Circle

… ►Simon (2005a, b) gives the general theory of these OP’s in terms of monic OP’s ${\mathrm{\Phi }}_n(x)$, see §18.33(vi). … ►Instead of (18.33.9) one might take monic OP’s $\{q_n(x)\}$ with weight function $(1+x)w_1(x)$, and then express $q_n\left(\frac{1}{2}(z+z^{-1})\right)$ in terms of ${\varphi }_{2n}(z^{\pm 1})$ or ${\varphi }_{2n+1}(z^{\pm 1})$. … ►

§18.33(vi) Alternative Set-up with Monic Polynomials

►Instead of orthonormal polynomials $\{{\varphi }_n(z)\}$ Simon (2005a, b) uses monic polynomials ${\mathrm{\Phi }}_n(z)$. …A system of monic polynomials $\{{\mathrm{\Phi }}_n(z)\}$, $n=0,1,\mathrm{\dots }$, where ${\mathrm{\Phi }}_n(x)$ is of proper degree $n$, is orthogonal on the unit circle with respect to the measure $\mu$ if …

14: 18.35 Pollaczek Polynomials

§18.35 Pollaczek Polynomials

… ►There are 3 types of Pollaczek polynomials: … ►For the monic polynomials … ► … ►More generally, the $P_n^{(\lambda )}(x;a,b)$ are OP’s if and only if one of the following three conditions holds (in case (iii) work with the monic polynomials (18.35.2_2)). …

15: 18.9 Recurrence Relations and Derivatives

… ►For the monic versions of the classical OP’s the recurrence coefficients $b_n$ and $c_n$ (there written as ${\alpha }_n$ and ${\beta }_n$, respectively) are given in §3.5(vi). … ►

Jacobi

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Ultraspherical

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Laguerre

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Hermite

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16: Wolter Groenevelt

… ►Groenevelt’s research interests is in special functions and orthogonal polynomials and their relations with representation theory and interacting particle systems. ►As of September 20, 2022, Groenevelt performed a complete analysis and acted as main consultant for the update of the source citation and proof metadata for every formula in Chapter 18 Orthogonal Polynomials. …

17: William P. Reinhardt

… ►Reinhardt is a theoretical chemist and atomic physicist, who has always been interested in orthogonal polynomials and in the analyticity properties of the functions of mathematical physics. …Older work on the scattering theory of the atomic Coulomb problem led to the discovery of new classes of orthogonal polynomials relating to the spectral theory of Schrödinger operators, and new uses of old ones: this work was strongly motivated by his original ownership of a 1964 hard copy printing of the original AMS 55 NBS Handbook of Mathematical Functions. … ►

20 Theta Functions

… ►In November 2015, Reinhardt was named Senior Associate Editor of the DLMF and Associate Editor for Chapters 20, 22, and 23.

18: 20 Theta Functions

Chapter 20 Theta Functions

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19: 18.5 Explicit Representations

§18.5 Explicit Representations

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Laguerre

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Hermite

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20: Errata

… ►We have significantly expanded the section on associated orthogonal polynomials, including expanded properties of associated Laguerre, Hermite, Meixner–Pollaczek, and corecursive orthogonal and numerator and denominator orthogonal polynomials. …In regard to orthogonal polynomials on the unit circle, we now discuss monic polynomials, Verblunsky’s Theorem, and Szegő’s theorem. We also discuss non-classical Laguerre polynomials and give much more details and examples on exceptional orthogonal polynomials. … ►

Several new equations have been added. See (8.17.24), (20.7.34), §20.11(v), (26.12.27), (36.2.28), and (36.2.29).

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Bibliographic citations were added in §§1.13(v), 10.14, 10.21(ii), 18.15(v), 18.32, 30.16(iii), 32.13(ii), and as general references in Chapters 19, 20, 22, and 23.

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