set of eigenvalues, taking multiplicities into account

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11: 28.29 Definitions and Basic Properties

… ►iff ${\mathrm{e}}^{\pi \mathrm{i}\nu }$ is an eigenvalue of the matrix … ►The $\pi$-periodic or $\pi$-antiperiodic solutions are multiples of $w_{\text{I}}(z,\lambda ),w_{\text{II}}(z,\lambda )$, respectively. … ►

§28.29(iii) Discriminant and Eigenvalues in the Real Case

… ►To every equation (28.29.1), there belong two increasing infinite sequences of real eigenvalues: … ►

28.29.17 $${\mu }_n,n=1,2,3,\mathrm{\dots },\text{ with }△({\mu }_n)=-2.$$

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Defines:

${\mu }_n$: eigenvalues (locally)

Symbols:

$n$: integer

Referenced by:

§28.30(i)

Permalink:

http://dlmf.nist.gov/28.29.E17

Encodings:

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See also:

Annotations for §28.29(iii), §28.29 and Ch.28

…

12: 18.28 Askey–Wilson Class

… ►The Askey–Wilson polynomials form a system of OP’s $\{p_n(x)\}$, $n=0,1,2,\mathrm{\dots }$, that are orthogonal with respect to a weight function on a bounded interval, possibly supplemented with discrete weights on a finite set. … ►Leonard (1982) classified all (finite or infinite) discrete systems of OP’s $p_n(x)$ on a set $\{x(m)\}$ for which there is a system of discrete OP’s $q_m(y)$ on a set $\{y(n)\}$ such that $p_n(x(m))=q_m(y(n))$. … ►

18.28.26 $$\underset{\lambda \to 0}{lim}{r}_n(x/(2\lambda );\lambda ,qa{\lambda }^{-1},qc{\lambda }^{-1},b{c}^{-1}\lambda |q)=P_n(x;a,b,c;q).$$

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Symbols:

$P_n(x;a,b,c;q)$: big $q$-Jacobi polynomial, ${(a_1,a_2,\mathrm{\dots },a_r;q)}_n$: multiple $q$-Pochhammer symbol, $z$: complex variable, $q$: real variable, $\mathrm{\ell }$: nonnegative integer, $n$: nonnegative integer and $x$: real variable

Source:

Koekoek et al. (2010, (14.1.18))

Proof sketch:

In the terminating $q$-hypergeometric series which defines the ${}_4{}^{}\varphi _3^{}$ in (18.28.1_5), with ${(az{q}^{\mathrm{\ell }},a{z}^{-1}{q}^{\mathrm{\ell }};q)}_{\mathrm{\ell }}$ written as ${\prod }_{j=0}^{\mathrm{\ell }-1}(1-2a{q}^{j}x+a^2{q}^{2j})$, substitute for $x$ and the parameters according to the left-hand side of the present formula, take the limit, and obtain the ${}_3{}^{}\varphi _2^{}$ as in (18.27.5).

Referenced by:

(18.28.27), (18.28.28), §18.28(x)

Permalink:

http://dlmf.nist.gov/18.28.E26

Encodings:

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See also:

Annotations for §18.28(x), §18.28(x), §18.28 and Ch.18

… ►

18.28.27 $$\underset{\lambda \to 0}{lim}{r}_n(bqx/(2\lambda );\lambda ,qb{\lambda }^{-1},q,a|q)={(-b)}^n{q}^{n(n+1)/2}\frac{{(qa;q)}_n}{{(qb;q)}_n}{p}_n(x;a,b;q).$$

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Symbols:

$p_n(x;a,b;q)$: little $q$-Jacobi polynomial, ${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), $q$: real variable, $n$: nonnegative integer and $x$: real variable

Proved:

Koornwinder (1993, (6.4)); The limit formula there is equivalent to the present formula(proved)

Proof sketch:

Proceed in a similar way as in the proof of (18.28.26). After taking the limit one now obtains a ${}_3{}^{}\varphi _2^{}$ which equals (18.27.14_1) by (18.27.13).

Referenced by:

§18.28(x)

Permalink:

http://dlmf.nist.gov/18.28.E27

Encodings:

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See also:

Annotations for §18.28(x), §18.28(x), §18.28 and Ch.18

►

18.28.28 $$\underset{\mu \to 0,\lambda /\mu \to 0}{lim}{r}_n(x/(2\lambda \mu );\lambda /\mu ,qa\mu /\lambda ,1/(\lambda \mu ),qb\lambda \mu |q)=p_n(x;a,b;q).$$

ⓘ

Symbols:

$p_n(x;a,b;q)$: little $q$-Jacobi polynomial, $q$: real variable, $n$: nonnegative integer and $x$: real variable

Proof sketch:

Proceed in a similar way as in the proof of (18.28.26). After taking the limit one now obtains a ${}_2{}^{}\varphi _1^{}$ which equals (18.27.13).

Referenced by:

§18.28(x)

Permalink:

http://dlmf.nist.gov/18.28.E28

Encodings:

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See also:

Annotations for §18.28(x), §18.28(x), §18.28 and Ch.18

…

13: 1.2 Elementary Algebra

… ►

Eigenvectors and Eigenvalues of Square Matrices

… ►Eigenvalues are the roots of the polynomial equation … ►The diagonal elements are not necessarily distinct, and the number of identical (degenerate) diagonal elements is the multiplicity of that specific eigenvalue. … ►Thus $det(𝐀)$ is the product of the $n$ (counted according to their multiplicities) eigenvalues of $𝐀$. …Thus $\mathrm{tr}(𝐀)$ is the sum of the (counted according to their multiplicities) eigenvalues of $𝐀$. …

14: 28.6 Expansions for Small $q$

… ►

§28.6(i) Eigenvalues

►Leading terms of the power series for $a_m\left(q\right)$ and $b_m\left(q\right)$ for $m\le 6$ are: … ►The coefficients of the power series of $a_{2n}\left(q\right)$, $b_{2n}\left(q\right)$ and also $a_{2n+1}\left(q\right)$, $b_{2n+1}\left(q\right)$ are the same until the terms in $q^{2n-2}$ and $q^{2n}$, respectively. … ►Higher coefficients in the foregoing series can be found by equating coefficients in the following continued-fraction equations: … ►Here $j=1$ for $a_{2n}\left(q\right)$, $j=2$ for $b_{2n+2}\left(q\right)$, and $j=3$ for $a_{2n+1}\left(q\right)$ and $b_{2n+1}\left(q\right)$. …

15: 29.3 Definitions and Basic Properties

… ►

§29.3(i) Eigenvalues

►For each pair of values of $\nu$ and $k$ there are four infinite unbounded sets of real eigenvalues $h$ for which equation (29.2.1) has even or odd solutions with periods $2K$ or $4K$. … ►

§29.3(ii) Distribution

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

16: 3.11 Approximation Techniques

… ►It is denoted by ${[p/q]}_f\left(z\right)$. … ►We take $n$ complex exponentials ${\varphi }_k(x)={\mathrm{e}}^{\mathrm{i}kx}$, $k=0,1,\mathrm{\dots },n-1$, and approximate $f(x)$ by the linear combination (3.11.31). … ►requires approximately $n^2$ multiplications. … ►The set of all the polynomials defines a function, the spline, on $[a,b]$. By taking more derivatives into account, the smoothness of the spline will increase. …

17: 31.11 Expansions in Series of Hypergeometric Functions

… ►Taking $P_j=P_j^5$ or $P_j=P_j^6$ the coefficients $c_j$ satisfy the equations …where we take $c_0=1$ and where … ►The expansion (31.11.1) for a Heun function that is associated with any branch of (31.11.2)—other than a multiple of the right-hand side of (31.11.12)—is convergent inside the ellipse $ℰ$. … ►

§31.11(v) Doubly-Infinite Series

►Schmidt (1979) gives expansions of path-multiplicative solutions (§31.6) in terms of doubly-infinite series of hypergeometric functions. …

18: 18.39 Applications in the Physical Sciences

… ►The properties of $V(x)$ determine whether the spectrum, this being the set of eigenvalues of $\mathscr{H}$, is discrete, continuous, or mixed, see §1.18. Below we consider two potentials with analytically known eigenfunctions and eigenvalues where the spectrum is entirely point, or discrete, with all eigenfunctions being $L^2$ and forming a complete set. … ►and the corresponding eigenvalues are … ►The eigenvalues and radial wave functions are independent of $m_l$ and they both do depend on $l$ due to the presence of the ‘fictitious’ centrifugal potential ${\mathrm{\hslash }}^{2}l(l+1)/(2m{r}^2)$, which is a result of the choice of co-ordinate system, and not the physical potential energy of interaction $V(r)$. … ►with eigenvalues …

19: 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 $a={\lambda }_{\nu }\left(q\right)$. …

20: 17.8 Special Cases of ${}_r{}^{}\psi _r^{}$ Functions

… ►

17.8.1 $$\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}{(-z)}^n{q}^{n(n-1)/2}={(q,z,q/z;q)}_{\mathrm{\infty }};$$

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Symbols:

${(a_1,a_2,\mathrm{\dots },a_r;q)}_n$: multiple $q$-Pochhammer symbol, $q$: complex base, $n$: nonnegative integer and $z$: complex variable

Permalink:

http://dlmf.nist.gov/17.8.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §17.8, §17.8 and Ch.17

… ►

17.8.2 $${}_1{}^{}\psi _1^{}(\genfrac{}{}{0pt}{}{a}{b};q,z)=\frac{{(q,b/a,az,q/(az);q)}_{\mathrm{\infty }}}{{(b,q/a,z,b/(az);q)}_{\mathrm{\infty }}}.$$

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Symbols:

${}_r{}^{}\psi _s^{}(a_1,\mathrm{\dots },a_r;b_1,\mathrm{\dots },b_s;q,z)$ or ${}_r{}^{}\psi _s^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_r}{b_1,\mathrm{\dots },b_s};q,z)$: bilateral basic hypergeometric (or bilateral $q$-hypergeometric) function, ${(a_1,a_2,\mathrm{\dots },a_r;q)}_n$: multiple $q$-Pochhammer symbol, $q$: complex base and $z$: complex variable

Permalink:

http://dlmf.nist.gov/17.8.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §17.8, §17.8 and Ch.17

… ►

17.8.3 $$\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}{(-1)}^n{q}^{n(3n-1)/2}{z}^{3n}(1+z{q}^n)={(q,-z,-q/z;q)}_{\mathrm{\infty }}{(q{z}^2,q/z^2;q^2)}_{\mathrm{\infty }}.$$

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Symbols:

${(a_1,a_2,\mathrm{\dots },a_r;q)}_n$: multiple $q$-Pochhammer symbol, $q$: complex base, $n$: nonnegative integer and $z$: complex variable

Permalink:

http://dlmf.nist.gov/17.8.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §17.8, §17.8 and Ch.17

… ►

17.8.4 $${}_2{}^{}\psi _2^{}(b,c;aq/b,aq/c;q,-aq/(bc))=\frac{{(aq/(bc);q)}_{\mathrm{\infty }}{(a{q}^2/b^2,a{q}^2/c^2,q^2,aq,q/a;q^2)}_{\mathrm{\infty }}}{{(aq/b,aq/c,q/b,q/c,-aq/(bc);q)}_{\mathrm{\infty }}},$$

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Symbols:

${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), ${}_r{}^{}\psi _s^{}(a_1,\mathrm{\dots },a_r;b_1,\mathrm{\dots },b_s;q,z)$ or ${}_r{}^{}\psi _s^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_r}{b_1,\mathrm{\dots },b_s};q,z)$: bilateral basic hypergeometric (or bilateral $q$-hypergeometric) function, ${(a_1,a_2,\mathrm{\dots },a_r;q)}_n$: multiple $q$-Pochhammer symbol and $q$: complex base

Permalink:

http://dlmf.nist.gov/17.8.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §17.8, §17.8 and Ch.17

►

17.8.5 $${}_3{}^{}\psi _3^{}(\genfrac{}{}{0pt}{}{b,c,d}{q/b,q/c,q/d};q,\frac{q}{bcd})=\frac{{(q,q/(bc),q/(bd),q/(cd);q)}_{\mathrm{\infty }}}{{(q/b,q/c,q/d,q/(bcd);q)}_{\mathrm{\infty }}},$$

ⓘ

Symbols:

${}_r{}^{}\psi _s^{}(a_1,\mathrm{\dots },a_r;b_1,\mathrm{\dots },b_s;q,z)$ or ${}_r{}^{}\psi _s^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_r}{b_1,\mathrm{\dots },b_s};q,z)$: bilateral basic hypergeometric (or bilateral $q$-hypergeometric) function, ${(a_1,a_2,\mathrm{\dots },a_r;q)}_n$: multiple $q$-Pochhammer symbol and $q$: complex base

Permalink:

http://dlmf.nist.gov/17.8.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §17.8, §17.8 and Ch.17

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