# Ince polynomials

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##### 1: 28.31 Equations of Whittaker–Hill and Ince
###### §28.31(ii) Equation of Ince; IncePolynomials
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. They are denoted by … … The normalization is given by …
##### 3: 29.21 Tables
• Ince (1940a) tabulates the eigenvalues $a^{m}_{\nu}\left(k^{2}\right)$, $b^{m+1}_{\nu}\left(k^{2}\right)$ (with $a^{2m+1}_{\nu}$ and $b^{2m+1}_{\nu}$ interchanged) for $k^{2}=0.1,0.5,0.9$, $\nu=-\frac{1}{2},0(1)25$, and $m=0,1,2,3$. Precision is 4D.

• ##### 4: 29.6 Fourier Series
In addition, if $H$ satisfies (29.6.2), then (29.6.3) applies. … Consequently, $\mathit{Ec}^{2m}_{\nu}\left(z,k^{2}\right)$ reduces to a Lamé polynomial; compare §§29.12(i) and 29.15(i). …
##### 6: 29.1 Special Notation
All derivatives are denoted by differentials, not by primes. The main functions treated in this chapter are the eigenvalues $a^{2m}_{\nu}\left(k^{2}\right)$, $a^{2m+1}_{\nu}\left(k^{2}\right)$, $b^{2m+1}_{\nu}\left(k^{2}\right)$, $b^{2m+2}_{\nu}\left(k^{2}\right)$, the Lamé functions $\mathit{Ec}^{2m}_{\nu}\left(z,k^{2}\right)$, $\mathit{Ec}^{2m+1}_{\nu}\left(z,k^{2}\right)$, $\mathit{Es}^{2m+1}_{\nu}\left(z,k^{2}\right)$, $\mathit{Es}^{2m+2}_{\nu}\left(z,k^{2}\right)$, and the Lamé polynomials $\mathit{uE}^{m}_{2n}\left(z,k^{2}\right)$, $\mathit{sE}^{m}_{2n+1}\left(z,k^{2}\right)$, $\mathit{cE}^{m}_{2n+1}\left(z,k^{2}\right)$, $\mathit{dE}^{m}_{2n+1}\left(z,k^{2}\right)$, $\mathit{scE}^{m}_{2n+2}\left(z,k^{2}\right)$, $\mathit{sdE}^{m}_{2n+2}\left(z,k^{2}\right)$, $\mathit{cdE}^{m}_{2n+2}\left(z,k^{2}\right)$, $\mathit{scdE}^{m}_{2n+3}\left(z,k^{2}\right)$. The notation for the eigenvalues and functions is due to Erdélyi et al. (1955, §15.5.1) and that for the polynomials is due to Arscott (1964b, §9.3.2). … Other notations that have been used are as follows: Ince (1940a) interchanges $a^{2m+1}_{\nu}\left(k^{2}\right)$ with $b^{2m+1}_{\nu}\left(k^{2}\right)$. …The relation to the Lamé functions ${\rm Ec}^{m}_{\nu}$, ${\rm Es}^{m}_{\nu}$ of Ince (1940b) is given by …
##### 7: Bibliography I
• E. L. Ince (1926) Ordinary Differential Equations. Longmans, Green and Co., London.
• E. L. Ince (1932) Tables of the elliptic cylinder functions. Proc. Roy. Soc. Edinburgh Sect. A 52, pp. 355–433.
• E. L. Ince (1940a) The periodic Lamé functions. Proc. Roy. Soc. Edinburgh 60, pp. 47–63.
• E. L. Ince (1940b) Further investigations into the periodic Lamé functions. Proc. Roy. Soc. Edinburgh 60, pp. 83–99.
• M. E. H. Ismail and D. R. Masson (1991) Two families of orthogonal polynomials related to Jacobi polynomials. Rocky Mountain J. Math. 21 (1), pp. 359–375.
• ##### 8: 29.17 Other Solutions
If (29.2.1) admits a Lamé polynomial solution $E$, then a second linearly independent solution $F$ is given by … See Erdélyi (1941c), Ince (1940b), and Lambe (1952). …
##### 9: 29.7 Asymptotic Expansions
Müller (1966a, b) found three formal asymptotic expansions for a fundamental system of solutions of (29.2.1) (and (29.11.1)) as $\nu\to\infty$, one in terms of Jacobian elliptic functions and two in terms of Hermite polynomials. …
##### 10: 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. We have also completely expanded our discussion on applications of orthogonal polynomials in the physical sciences, and also methods of computation for orthogonal polynomials. …
• Section 1.13

In Equation (1.13.4), the determinant form of the two-argument Wronskian

1.13.4 $\mathscr{W}\left\{w_{1}(z),w_{2}(z)\right\}=\det\begin{bmatrix}w_{1}(z)&w_{2}(% z)\\ w_{1}^{\prime}(z)&w_{2}^{\prime}(z)\end{bmatrix}=w_{1}(z)w_{2}^{\prime}(z)-w_{% 2}(z)w_{1}^{\prime}(z)$

was added as an equality. In ¶Wronskian (in §1.13(i)), immediately below Equation (1.13.4), a sentence was added indicating that in general the $n$-argument Wronskian is given by $\mathscr{W}\left\{w_{1}(z),\ldots,w_{n}(z)\right\}=\det\left[w_{k}^{(j-1)}(z)\right]$, where $1\leq j,k\leq n$. Immediately below Equation (1.13.4), a sentence was added giving the definition of the $n$-argument Wronskian. It is explained just above (1.13.5) that this equation is often referred to as Abel’s identity. Immediately below Equation (1.13.5), a sentence was added explaining how it generalizes for $n$th-order differential equations. A reference to Ince (1926, §5.2) was added.