# polynomial solutions

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##### 1: 29.17 Other Solutions
If (29.2.1) admits a Lamé polynomial solution $E$, then a second linearly independent solution $F$ is given by …
##### 3: 31.15 Stieltjes Polynomials
###### §31.15(i) Definitions
Stieltjes polynomials are polynomial solutions of the Fuchsian equation (31.14.1). …There exist at most $\genfrac{(}{)}{0.0pt}{}{n+N-2}{N-2}$ polynomials $V(z)$ of degree not exceeding $N-2$ such that for $\Phi(z)=V(z)$, (31.15.1) has a polynomial solution $w=S(z)$ of degree $n$. … If $z_{1},z_{2},\dots,z_{n}$ 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 $z_{1}^{\prime},z_{2}^{\prime},\dots,z_{n-1}^{\prime}$ 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 …
##### 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
###### §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
###### §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
Sometimes the parameters are suppressed.
##### 9: Bibliography S
• R. Shail (1978) Lamé polynomial solutions to some elliptic crack and punch problems. Internat. J. Engrg. Sci. 16 (8), pp. 551–563.