# polynomial solutions

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## 1—10 of 66 matching pages

##### 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: 18.40 Methods of Computation

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##### 3: 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 …##### 4: 18.38 Mathematical Applications

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###### Differential Equations

…##### 5: 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*. …

##### 6: 18.39 Physical Applications

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###### §18.39(i) Quantum Mechanics

… ►The corresponding eigenfunctions are … ►A second example is provided by the three-dimensional time-independent Schrödinger equation … ► …##### 7: 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*. …##### 8: 31.1 Special Notation

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►Sometimes the parameters are suppressed.

##### 9: 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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##### 10: 32.10 Special Function Solutions

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