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11: 30.9 Asymptotic Approximations and Expansions

… ►For the eigenfunctions see Meixner and Schäfke (1954, §3.251) and Müller (1963). … ►For the eigenfunctions see Meixner and Schäfke (1954, §3.252) and Müller (1962). …

12: 3.7 Ordinary Differential Equations

… ►The values ${\lambda }_k$ are the eigenvalues and the corresponding solutions $w_k$ of the differential equation are the eigenfunctions. The eigenvalues ${\lambda }_k$ are simple, that is, there is only one corresponding eigenfunction (apart from a normalization factor), and when ordered increasingly the eigenvalues satisfy …

13: 18.38 Mathematical Applications

… ►Eigenvalue equations involving Dunkl type operators have as eigenfunctions nonsymmetric analogues of multivariable special functions associated with root systems. … ► … ►The Dunkl type operator is a $q$-difference-reflection operator acting on Laurent polynomials and its eigenfunctions, the nonsymmetric Askey–Wilson polynomials, are linear combinations of the symmetric Laurent polynomial $R_n(z;a,b,c,d|q)$ and the ‘anti-symmetric’ Laurent polynomial $z^{-1}(1-az)(1-bz)R_{n-1}(z;qa,qb,c,d|q)$, where $R_n(z)$ is given in (18.28.1_5). …

14: 29.12 Definitions

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Table 29.12.1: Lamé polynomials.

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$\nu$	eigenvalue $h$	eigenfunction $w(z)$	polynomial form	real period	imag. period	parity of $w(z)$	parity of $w(z-K)$	parity of $w(z-K-\mathrm{i}{K}^{\prime })$
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15: Bibliography V

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Error estimates for Rayleigh-Ritz approximations of eigenvalues and eigenfunctions of the Mathieu and spheroidal wave equation. Constr. Approx. 20 (1), pp. 39–54.

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External Links:

ISSN 0176-4276, Document, MathReview, ZentralBlatt

Referenced by:

§30.16(i), §30.16(ii), §30.16(ii).

Encodings:

BibTeX

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16: 30.4 Functions of the First Kind

… ►The eigenfunctions of (30.2.1) that correspond to the eigenvalues ${\lambda }_n^m\left({\gamma }^2\right)$ are denoted by ${\mathrm{𝖯𝗌}}_n^m(x,{\gamma }^2)$, $n=m,m+1,m+2,\mathrm{\dots }$. …

17: 30.13 Wave Equation in Prolate Spheroidal Coordinates

… ►The corresponding eigenfunctions are given by (30.13.8), (30.13.14), (30.13.13), (30.13.12), with $b_1=b_2=0$. …The corresponding eigenfunctions are given as before with $b_2=0$. …

18: Bibliography R

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On the computation of Lamé functions, of eigenvalues and eigenfunctions of some potential operators. Z. Angew. Math. Mech. 78 (1), pp. 66–72.

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External Links:

ISSN 0044-2267, Document, MathReview Entry, ZentralBlatt

Referenced by:

§29.20(ii).

Encodings:

BibTeX

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19: 18.28 Askey–Wilson Class

… ► $y$) such that $P_n(z)=p_n(\frac{1}{2}(z+z^{-1}))$ in the Askey–Wilson case, and $P_n(y)=p_n(q^{-y}+c{q}^{y+1})$ in the $q$-Racah case, and both are eigenfunctions of a second order $q$-difference operator similar to (18.27.1). … ►In Tsujimoto et al. (2012) an extension of the Bannai–Ito polynomials occurs as eigenfunctions of a Dunkl type operator. …

20: 30.14 Wave Equation in Oblate Spheroidal Coordinates

… ►The corresponding eigenfunctions are then given by (30.13.8), (30.14.8), (30.13.13), (30.13.12), with $b_1=b_2=0$. …