# connection with orthogonal polynomials on the line

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##### 1: 35.4 Partitions and Zonal Polynomials

###### §35.4 Partitions and Zonal Polynomials

… ►###### Normalization

… ►###### Orthogonal Invariance

… ►###### Summation

… ►###### Mean-Value

…##### 2: 31.5 Solutions Analytic at Three Singularities: Heun Polynomials

###### §31.5 Solutions Analytic at Three Singularities: Heun Polynomials

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31.5.2
$${\mathit{Hp}}_{n,m}(a,{q}_{n,m};-n,\beta ,\gamma ,\delta ;z)=H\mathrm{\ell}(a,{q}_{n,m};-n,\beta ,\gamma ,\delta ;z)$$

►is a polynomial of degree $n$, and hence a solution of (31.2.1) that is analytic at all three finite singularities $0,1,a$.
These solutions are the *Heun polynomials*. …

##### 3: 24.1 Special Notation

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###### Bernoulli Numbers and Polynomials

►The origin of the notation ${B}_{n}$, ${B}_{n}\left(x\right)$, is not clear. … ►###### Euler Numbers and Polynomials

… ►The notations ${E}_{n}$, ${E}_{n}\left(x\right)$, as defined in §24.2(ii), were used in Lucas (1891) and Nörlund (1924). …##### 4: 18.3 Definitions

###### §18.3 Definitions

►Table 18.3.1 provides the definitions of Jacobi, Laguerre, and Hermite polynomials via orthogonality and normalization (§§18.2(i) and 18.2(iii)). … ►For exact values of the coefficients of the Jacobi polynomials ${P}_{n}^{(\alpha ,\beta )}\left(x\right)$, the ultraspherical polynomials ${C}_{n}^{(\lambda )}\left(x\right)$, the Chebyshev polynomials ${T}_{n}\left(x\right)$ and ${U}_{n}\left(x\right)$, the Legendre polynomials ${P}_{n}\left(x\right)$, the Laguerre polynomials ${L}_{n}\left(x\right)$, and the Hermite polynomials ${H}_{n}\left(x\right)$, see Abramowitz and Stegun (1964, pp. 793–801). … ►In addition to the orthogonal property given by Table 18.3.1, the Chebyshev polynomials ${T}_{n}\left(x\right)$, $n=0,1,\mathrm{\dots},N$, are orthogonal on the discrete point set comprising the zeros ${x}_{N+1,n},n=1,2,\mathrm{\dots},N+1$, of ${T}_{N+1}\left(x\right)$: … ►For another version of the discrete orthogonality property of the polynomials ${T}_{n}\left(x\right)$ see (3.11.9). …##### 5: 18.33 Polynomials Orthogonal on the Unit Circle

###### §18.33 Polynomials Orthogonal on the Unit Circle

►###### §18.33(i) Definition

… ►###### §18.33(ii) Recurrence Relations

… ►###### §18.33(iii) Connection with OP’s on the Line

… ►###### §18.33(v) Biorthogonal Polynomials on the Unit Circle

…##### 6: Bibliography I

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On polynomials orthogonal with respect to certain Sobolev inner products.
J. Approx. Theory 65 (2), pp. 151–175.
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Two families of orthogonal polynomials related to Jacobi polynomials.
Rocky Mountain J. Math. 21 (1), pp. 359–375.
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An electrostatics model for zeros of general orthogonal polynomials.
Pacific J. Math. 193 (2), pp. 355–369.
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Classical and Quantum Orthogonal Polynomials in One Variable.
Encyclopedia of Mathematics and its Applications, Vol. 98, Cambridge University Press, Cambridge.
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Connection formulae for the fourth Painlevé transcendent; Clarkson-McLeod solution.
J. Phys. A 31 (17), pp. 4073–4113.
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##### 7: 18.36 Miscellaneous Polynomials

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►These are OP’s on the interval $(-1,1)$ with respect to an orthogonality measure obtained by adding constant multiples of “Dirac delta weights” at $-1$ and $1$ to the weight function for the Jacobi polynomials.
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###### §18.36(iii) Multiple OP’s

►These are polynomials in one variable that are orthogonal with respect to a number of different measures. … ►###### §18.36(iv) Orthogonal Matrix Polynomials

►These are matrix-valued polynomials that are orthogonal with respect to a square matrix of measures on the real line. …##### 8: William P. Reinhardt

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►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*. … ►This is closely connected with his interests in classical dynamical “chaos,” an area where he coauthored a book, Chaos in atomic physics with Reinhold Blümel. …##### 9: 12.16 Mathematical Applications

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►For examples see §§13.20(iii), 13.20(iv), 14.15(v), and 14.26.
►Sleeman (1968b) considers certain orthogonality properties of the PCFs and corresponding eigenvalues.
In Brazel et al. (1992) exponential asymptotics are considered in connection with an eigenvalue problem involving PCFs.
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##### 10: 18.37 Classical OP’s in Two or More Variables

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