# set of eigenvalues, taking multiplicities into account

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##### 1: 3.2 Linear Algebra
###### §3.2(iv) Eigenvalues and Eigenvectors
The multiplicity of an eigenvalue is its multiplicity as a zero of the characteristic polynomial (§3.8(i)). To an eigenvalue of multiplicity $m$, there correspond $m$ linearly independent eigenvectors provided that $\mathbf{A}$ is nondefective, that is, $\mathbf{A}$ has a complete set of $n$ linearly independent eigenvectors.
##### 2: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions
If an eigenvalue has multiplicity $>1$, the eigenfunctions may always be orthogonalized in this degenerate sub-space. … For $\mathcal{D}(T)$ we can take $C^{2}(X)$, with appropriate boundary conditions, and with compact support if $X$ is bounded, which space is dense in $L^{2}\left(X\right)$, and for $X$ unbounded require that possible non-$L^{2}$ eigenfunctions of (1.18.28), with real eigenvalues, are non-zero but bounded on open intervals, including $\pm\infty$. Stated informally, the spectrum of $T$ is the set of it’s eigenvalues, these being real as $T$ is self-adjoint. … … If an eigenvalue is of multiplicity greater than $1$ then an orthonormal basis of eigenfunctions can be given for the eigenspace. …
##### 3: 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. When $\widehat{\nu}=0$ or $1$, the notation for the two sets of eigenvalues corresponding to each $\widehat{\nu}$ is shown in Table 28.2.1, together with the boundary conditions of the associated eigenvalue problem. …
##### 4: Mathematical Introduction
The mathematical project team has endeavored to take into account the hundreds of research papers and numerous books on special functions that have appeared since 1964. … In addition, there is a comprehensive account of the great variety of analytical methods that are used for deriving and applying the extremely important asymptotic properties of the special functions, including double asymptotic properties (Chapter 2 and §§10.41(iv), 10.41(v)). …
 $(a,b]$ or $[a,b)$ half-closed intervals. … set subtraction. set of all integers. set of all integer multiples of $n$.
##### 5: 17.4 Basic Hypergeometric Functions
Here and elsewhere it is assumed that the $b_{j}$ do not take any of the values $q^{-n}$. …
17.4.3 ${{}_{r}\psi_{s}}\left({a_{1},a_{2},\dots,a_{r}\atop b_{1},b_{2},\dots,b_{s}};q% ,z\right)={{}_{r}\psi_{s}}\left(a_{1},a_{2},\dots,a_{r};b_{1},b_{2},\dots,b_{s% };q,z\right)=\sum_{n=-\infty}^{\infty}\frac{\left(a_{1},a_{2},\dots,a_{r};q% \right)_{n}(-1)^{(s-r)n}q^{(s-r)\genfrac{(}{)}{0.0pt}{}{n}{2}}z^{n}}{\left(b_{% 1},b_{2},\dots,b_{s};q\right)_{n}}=\sum_{n=0}^{\infty}\frac{\left(a_{1},a_{2},% \dots,a_{r};q\right)_{n}(-1)^{(s-r)n}q^{(s-r)\genfrac{(}{)}{0.0pt}{}{n}{2}}z^{% n}}{\left(b_{1},b_{2},\dots,b_{s};q\right)_{n}}+\sum_{n=1}^{\infty}\frac{\left% (q/b_{1},q/b_{2},\dots,q/b_{s};q\right)_{n}}{\left(q/a_{1},q/a_{2},\dots,q/a_{% r};q\right)_{n}}\left(\frac{b_{1}b_{2}\cdots b_{s}}{a_{1}a_{2}\cdots a_{r}z}% \right)^{n}.$
Here and elsewhere the $b_{j}$ must not take any of the values $q^{-n}$, and the $a_{j}$ must not take any of the values $q^{n+1}$. The infinite series converge when $s\geq r$ provided that $|(b_{1}\cdots b_{s})/(a_{1}\cdots a_{r}z)|<1$ and also, in the case $s=r$, $|z|<1$. …
17.4.6 $\Phi^{(2)}\left(a;b,b^{\prime};c,c^{\prime};q;x,y\right)=\sum_{m,n\geq 0}\frac% {\left(a;q\right)_{m+n}\left(b;q\right)_{m}\left(b^{\prime};q\right)_{n}x^{m}y% ^{n}}{\left(q,c;q\right)_{m}\left(q,c^{\prime};q\right)_{n}},$
##### 6: 17.9 Further Transformations of ${{}_{r+1}\phi_{r}}$ Functions
17.9.3_5 ${{}_{2}\phi_{1}}\left({a,b\atop c};q,z\right)=\frac{\left(c/a,c/b;q\right)_{% \infty}}{\left(c,c/(ab);q\right)_{\infty}}{{}_{3}\phi_{2}}\left({a,b,abz/c% \atop qab/c,0};q,q\right)+\frac{\left(a,b,abz/c;q\right)_{\infty}}{\left(c,ab/% c,z;q\right)_{\infty}}{{}_{3}\phi_{2}}\left({c/a,c/b,z\atop qc/(ab),0};q,q% \right),$
17.9.7 ${{}_{3}\phi_{2}}\left({a,b,c\atop d,e};q,de/(abc)\right)=\frac{\left(b,de/(ab)% ,de/(bc);q\right)_{\infty}}{\left(d,e,de/(abc);q\right)_{\infty}}\*{{}_{3}\phi% _{2}}\left({d/b,e/b,de/(abc)\atop de/(ab),de/(bc)};q,b\right),$
17.9.13 ${{}_{3}\phi_{2}}\left({a,b,c\atop d,e};q,\frac{de}{abc}\right)=\frac{\left(e/b% ,e/c;q\right)_{\infty}}{\left(e,e/(bc);q\right)_{\infty}}{{}_{3}\phi_{2}}\left% ({d/a,b,c\atop d,bcq/e};q,q\right)+\frac{\left(d/a,b,c,de/(bc);q\right)_{% \infty}}{\left(d,e,bc/e,de/(abc);q\right)_{\infty}}{{}_{3}\phi_{2}}\left({e/b,% e/c,de/(abc)\atop de/(bc),eq/(bc)};q,q\right).$
17.9.14 ${{}_{4}\phi_{3}}\left({q^{-n},a,b,c\atop d,e,f};q,q\right)=\frac{\left(e/a,f/a% ;q\right)_{n}}{\left(e,f;q\right)_{n}}a^{n}{{}_{4}\phi_{3}}\left({q^{-n},a,d/b% ,d/c\atop d,aq^{1-n}/e,aq^{1-n}/f};q,q\right)=\frac{\left(a,ef/(ab),ef/(ac);q% \right)_{n}}{\left(e,f,ef/(abc);q\right)_{n}}{{}_{4}\phi_{3}}\left({q^{-n},e/a% ,f/a,ef/(abc)\atop ef/(ab),ef/(ac),q^{1-n}/a};q,q\right).$
17.9.16 ${{}_{8}\phi_{7}}\left({a,qa^{\frac{1}{2}},-qa^{\frac{1}{2}},b,c,d,e,f\atop a^{% \frac{1}{2}},-a^{\frac{1}{2}},aq/b,aq/c,aq/d,aq/e,aq/f};q,\frac{a^{2}q^{2}}{% bcdef}\right)=\frac{\left(aq,aq/(de),aq/(df),aq/(ef);q\right)_{\infty}}{\left(% aq/d,aq/e,aq/f,aq/(def);q\right)_{\infty}}{{}_{4}\phi_{3}}\left({aq/(bc),d,e,f% \atop aq/b,aq/c,def/a};q,q\right)+\frac{\left(aq,aq/(bc),d,e,f,a^{2}q^{2}/(% bdef),a^{2}q^{2}/(cdef);q\right)_{\infty}}{\left(aq/b,aq/c,aq/d,aq/e,aq/f,a^{2% }q^{2}/(bcdef),def/(aq);q\right)_{\infty}}\*{{}_{4}\phi_{3}}\left({aq/(de),aq/% (df),aq/(ef),a^{2}q^{2}/(bcdef)\atop a^{2}q^{2}/(bdef),a^{2}q^{2}/(cdef),aq^{2% }/(def)};q,q\right).$
##### 7: Bille C. Carlson
In theoretical physics he is known for the “Carlson-Keller Orthogonalization”, published in 1957, Orthogonalization Procedures and the Localization of Wannier Functions, and the “Carlson-Keller Theorem”, published in 1961, Eigenvalues of Density Matrices. … The main theme of Carlson’s mathematical research has been to expose previously hidden permutation symmetries that can eliminate a set of transformations and thereby replace many formulas by a few. … In Symmetry in c, d, n of Jacobian elliptic functions (2004) he found a previously hidden symmetry in relations between Jacobian elliptic functions, which can now take a form that remains valid when the letters c, d, and n are permuted. This invariance usually replaces sets of twelve equations by sets of three equations and applies also to the relation between the first symmetric elliptic integral and the Jacobian functions. …
##### 8: 3.8 Nonlinear Equations
Sometimes the equation takes the form … For multiple zeros the convergence is linear, but if the multiplicity $m$ is known then quadratic convergence can be restored by multiplying the ratio $f(z_{n})/f^{\prime}(z_{n})$ in (3.8.4) by $m$. …
###### Eigenvalue Methods
For the computation of zeros of orthogonal polynomials as eigenvalues of finite tridiagonal matrices (§3.5(vi)), see Gil et al. (2007a, pp. 205–207). … It is called a Julia set. …
In doing this, however, we would like to place the mathematically significant phase values, specifically the multiples of $\pi/2$ correponding to the real and imaginary axes, at more immediately recognizable colors. … The conventional CMYK color wheel (not to be confused with the traditional Artist’s color wheel) places the additive colors (red, green, blue) and the subtractive colors (yellow, cyan, magenta) at multiples of 60 degrees. … We therefore use a piecewise linear mapping as illustrated below, that takes phase $0$ to red, $\pi/2$ to yellow, $\pi$ to cyan and $3\pi/2$ to blue. …
##### 10: 18.27 $q$-Hahn Class
The $q$-Hahn class OP’s comprise systems of OP’s $\{p_{n}(x)\}$, $n=0,1,\dots,N$, or $n=0,1,2,\dots$, that are eigenfunctions of a second order $q$-difference operator. …where $A(x)$, $B(x)$, and $C(x)$ are independent of $n$, and where the $\lambda_{n}$ are the eigenvalues. In the $q$-Hahn class OP’s the role of the operator $\ifrac{\mathrm{d}}{\mathrm{d}x}$ in the Jacobi, Laguerre, and Hermite cases is played by the $q$-derivative $\mathcal{D}_{q}$, as defined in (17.2.41). …
18.27.12 $v_{x}=\frac{\left(qx/c,-qx/d;q\right)_{\infty}}{\left(q^{\alpha+1}x/c,-q^{% \beta+1}x/d;q\right)_{\infty}},$ $\alpha,\beta>-1$, $c,d>0$.
18.27.22 $\sum_{\ell=0}^{\infty}\left(h_{n}\left(q^{\ell};q\right)h_{m}\left(q^{\ell};q% \right)+h_{n}\left(-q^{\ell};q\right)h_{m}\left(-q^{\ell};q\right)\right)\*% \left(q^{\ell+1},-q^{\ell+1};q\right)_{\infty}q^{\ell}=\left(q;q\right)_{n}% \left({q,-1,-q};q\right)_{\infty}q^{n(n-1)/2}\delta_{n,m}.$