# polynomial solutions

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##### 1: 29.17 Other Solutions

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►If (29.2.1) admits a Lamé polynomial solution
$E$, then a second linearly independent solution
$F$ is given by
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##### 2: 31.15 Stieltjes Polynomials

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###### §31.15(i) Definitions

►Stieltjes polynomials are polynomial solutions of the Fuchsian equation (31.14.1). …There exist at most $\left(\genfrac{}{}{0.0pt}{}{n+N-2}{N-2}\right)$ polynomials $V(z)$ of degree not exceeding $N-2$ such that for $\mathrm{\Phi}(z)=V(z)$, (31.15.1) has a polynomial solution $w=S(z)$ of degree $n$. … ►If ${z}_{1},{z}_{2},\mathrm{\dots},{z}_{n}$ are the zeros of an $n$th degree Stieltjes polynomial $S(z)$, then every zero ${z}_{k}$ is either one of the parameters ${a}_{j}$ or a solution of the system of equations … ►If ${t}_{k}$ is a zero of the Van Vleck polynomial $V(z)$, corresponding to an $n$th degree Stieltjes polynomial $S(z)$, and ${z}_{1}^{\prime},{z}_{2}^{\prime},\mathrm{\dots},{z}_{n-1}^{\prime}$ are the zeros of ${S}^{\prime}(z)$ (the derivative of $S(z)$), then ${t}_{k}$ is either a zero of ${S}^{\prime}(z)$ or a solution of the equation …##### 3: 31.5 Solutions Analytic at Three Singularities: Heun Polynomials

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

… ►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*. …

##### 4: 28.31 Equations of Whittaker–Hill and Ince

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###### §28.31(ii) Equation of Ince; Ince Polynomials

… ►When $p$ is a nonnegative integer, the parameter $\eta $ can be chosen so that solutions of (28.31.3) are trigonometric polynomials, called*Ince polynomials*. …##### 5: 32.10 Special Function Solutions

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###### §32.10(iv) Fourth Painlevé Equation

…##### 6: 31.1 Special Notation

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►Sometimes the parameters are suppressed.

##### 7: Bibliography S

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Lamé polynomial solutions to some elliptic crack and punch problems.
Internat. J. Engrg. Sci. 16 (8), pp. 551–563.
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##### 8: 18.38 Mathematical Applications

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###### Differential Equations: Spectral Methods

… ►###### Quadrature “Extended” to Pseudo-Spectral (DVR) Representations of Operators in One and Many Dimensions

…##### 9: 18.40 Methods of Computation

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##### 10: 14.31 Other Applications

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►Many additional physical applications of Legendre polynomials and associated Legendre functions include solution of the Helmholtz equation, as well as the Laplace equation, in spherical coordinates (Temme (1996b)), quantum mechanics (Edmonds (1974)), and high-frequency scattering by a sphere (Nussenzveig (1965)).
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