as eigenvalues of q-difference operator

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11: 18.38 Mathematical Applications

… ►A further operator, the so-called Casimir operator … ►

Dunkl Type Operators and Nonsymmetric Orthogonal Polynomials

… ►Eigenvalue equations involving Dunkl type operators have as eigenfunctions nonsymmetric analogues of multivariable special functions associated with root systems. … ►In the one-variable case the Dunkl operator eigenvalue equation … …

12: 29.3 Definitions and Basic Properties

… ►

§29.3(i) Eigenvalues

… ►

§29.3(ii) Distribution

►The eigenvalues interlace according to …The eigenvalues coalesce according to … ►

§29.3(vii) Power Series

…

13: 18.39 Applications in the Physical Sciences

… ►The nature of, and notations and common vocabulary for, the eigenvalues and eigenfunctions of self-adjoint second order differential operators is overviewed in §1.18. … ►and the corresponding eigenvalues are … ►with eigenvalues … ►The radial operator (18.39.28) … ►The Schrödinger operator essential singularity, seen in the accumulation of discrete eigenvalues for the attractive Coulomb problem, is mirrored in the accumulation of jumps in the discrete Pollaczek–Stieltjes measure as $x\to -1-$. …

14: 28.15 Expansions for Small $q$

… ►

§28.15(i) Eigenvalues ${\lambda }_{\nu }\left(q\right)$

►

28.15.1 $${\lambda }_{\nu }\left(q\right)={\nu }^2+\frac{1}{2({\nu }^2-1)}{q}^2+\frac{5{\nu }^2+7}{32{({\nu }^2-1)}^3({\nu }^2-4)}{q}^4+\frac{9{\nu }^4+58{\nu }^2+29}{64{({\nu }^2-1)}^5({\nu }^2-4)({\nu }^2-9)}{q}^6+\mathrm{\cdots }.$$

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Symbols:

${\lambda }_{\nu +2n}\left(q\right)$: eigenvalues of Mathieu equation, $q=h^2$: parameter and $\nu$: complex parameter

A&S Ref:

20.3.15 (in slightly different form)

Permalink:

http://dlmf.nist.gov/28.15.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §28.15(i), §28.15 and Ch.28

►Higher coefficients can be found by equating powers of $q$ in the following continued-fraction equation, with $a={\lambda }_{\nu }\left(q\right)$: ►

28.15.2 $$a-{\nu }^2-\frac{q^2}{a-{(\nu +2)}^2-}\frac{q^2}{a-{(\nu +4)}^2-}\mathrm{\cdots }=\frac{q^2}{a-{(\nu -2)}^2-}\frac{q^2}{a-{(\nu -4)}^2-}\mathrm{\cdots }.$$

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Symbols:

$q=h^2$: parameter, $\nu$: complex parameter and $a$: parameter

Referenced by:

item (e)

Permalink:

http://dlmf.nist.gov/28.15.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §28.15(i), §28.15 and Ch.28

…

15: 29.22 Software

… ►

LA1: Eigenvalues for Lamé functions.

… ►

LA5: Coefficients ${\tau }_j$ of the asymptotic expansions for the eigenvalues of the Lamé functions; see §29.7(i).

… ►

LA3: Eigenvalues for Lamé polynomials.

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16: 29.16 Asymptotic Expansions

§29.16 Asymptotic Expansions

►Hargrave and Sleeman (1977) give asymptotic approximations for Lamé polynomials and their eigenvalues, including error bounds. …

17: 31.13 Asymptotic Approximations

§31.13 Asymptotic Approximations

►For asymptotic approximations for the accessory parameter eigenvalues $q_m$, see Fedoryuk (1991) and Slavyanov (1996). …

18: 28.16 Asymptotic Expansions for Large $q$

§28.16 Asymptotic Expansions for Large $q$

… ►

28.16.1 $${\lambda }_{\nu }\left(h^2\right)\sim -2h^2+2sh-\frac{1}{8}(s^2+1)-\frac{1}{2^{7}h}(s^3+3s)-\frac{1}{2^{12}{h}^2}(5s^4+34s^2+9)-\frac{1}{2^{17}{h}^3}(33s^5+410s^3+405s)-\frac{1}{2^{20}{h}^4}(63s^6+1260s^4+2943s^2+486)-\frac{1}{2^{25}{h}^5}(527s^7+15617s^5+69001s^3+41607s)+\mathrm{\cdots }.$$

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Symbols:

${\lambda }_{\nu +2n}\left(q\right)$: eigenvalues of Mathieu equation, $\sim$: Poincaré asymptotic expansion, $h$: parameter and $\nu$: complex parameter

Permalink:

http://dlmf.nist.gov/28.16.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §28.16 and Ch.28

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19: 28.2 Definitions and Basic Properties

… ►

§28.2(v) Eigenvalues $a_n$, $b_n$

►For given $\nu$ and $q$, equation (28.2.16) determines an infinite discrete set of values of $a$, the eigenvalues or characteristic values, of Mathieu’s equation. … ►

Distribution

… ►

Change of Sign of $q$

… ►Table 28.2.2 gives the notation for the eigenfunctions corresponding to the eigenvalues in Table 28.2.1. …

20: 28.12 Definitions and Basic Properties

… ►

§28.12(i) Eigenvalues ${\lambda }_{\nu +2n}\left(q\right)$

… ►For given $\nu$ (or $\cos \left(\nu \pi \right)$) and $q$, equation (28.2.16) determines an infinite discrete set of values of $a$, denoted by ${\lambda }_{\nu +2n}\left(q\right)$, $n=0,\pm 1,\pm 2,\mathrm{\dots }$. …For other values of $q$, ${\lambda }_{\nu +2n}\left(q\right)$ is determined by analytic continuation. … … ►Two eigenfunctions correspond to each eigenvalue $a={\lambda }_{\nu }\left(q\right)$. …