Lax pairs

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21: 15.11 Riemann’s Differential Equation

… ►Here $\{a_1,a_2\}$, $\{b_1,b_2\}$, $\{c_1,c_2\}$ are the exponent pairs at the points $\alpha$, $\beta$, $\gamma$, respectively. Cases in which there are fewer than three singularities are included automatically by allowing the choice $\{0,1\}$ for exponent pairs. …

22: 26.15 Permutations: Matrix Notation

… ►The inversion number of $\sigma$ is a sum of products of pairs of entries in the matrix representation of $\sigma$: … ► $N_k(B)$ is the number of permutations in ${𝔖}_n$ for which exactly $k$ of the pairs $(j,\sigma (j))$ are elements of $B$. …

23: 36.7 Zeros

… ►Inside the cusp, that is, for , the zeros form pairs lying in curved rows. … ►Away from the $z$-axis and approaching the cusp lines (ribs) (36.4.11), the lattice becomes distorted and the rings are deformed, eventually joining to form “hairpins” whose arms become the pairs of zeros (36.7.1) of the cusp canonical integral. …

24: 8.13 Zeros

… ►When $x>x_n^{\ast }$ a pair of conjugate trajectories emanate from the point $a=a_n^{\ast }$ in the complex $a$-plane. …

25: 15.19 Methods of Computation

… ►This is because the linear transformations map the pair $\{{\mathrm{e}}^{\pi \mathrm{i}/3},{\mathrm{e}}^{-\pi \mathrm{i}/3}\}$ onto itself. …

26: 23.20 Mathematical Applications

… ►The two pairs of edges $[0,{\omega }_1]\cup [{\omega }_1,2{\omega }_3]$ and $[2{\omega }_3,2{\omega }_3-{\omega }_1]\cup [2{\omega }_3-{\omega }_1,0]$ of $R$ are each mapped strictly monotonically by $\mathrm{\wp }$ onto the real line, with $0\to \mathrm{\infty }$, ${\omega }_1\to e_1$, $2{\omega }_3\to -\mathrm{\infty }$; similarly for the other pair of edges. For each pair of edges there is a unique point $z_0$ such that $\mathrm{\wp }\left(z_0\right)=0$. …

27: 2.7 Differential Equations

… ►One pair of independent solutions of the equation …In theory either pair may be used to construct any other solution …This kind of cancellation cannot take place with $w_1(z)$ and $w_2(z)$, and for this reason, and following Miller (1950), we call $w_1(z)$ and $w_2(z)$ a numerically satisfactory pair of solutions. … ►This is characteristic of numerically satisfactory pairs. … ►In oscillatory intervals, and again following Miller (1950), we call a pair of solutions numerically satisfactory if asymptotically they have the same amplitude and are $\frac{1}{2}\pi$ out of phase.

28: 12.2 Differential Equations

… ►For real values of $z$ $(=x)$, numerically satisfactory pairs of solutions (§2.7(iv)) of (12.2.2) are $U(a,x)$ and $V(a,x)$ when $x$ is positive, or $U(a,-x)$ and $V(a,-x)$ when $x$ is negative. For (12.2.3) $W(a,x)$ and $W(a,-x)$ comprise a numerically satisfactory pair, for all $x\in ℝ$. … ►In $ℂ$, for $j=0,1,2,3$, $U({(-1)}^{j-1}a,{(-\mathrm{i})}^{j-1}z)$ and $U({(-1)}^{j}a,{(-\mathrm{i})}^{j}z)$ comprise a numerically satisfactory pair of solutions in the half-plane $\frac{1}{4}(2j-3)\pi \le \mathrm{ph}z\le \frac{1}{4}(2j+1)\pi$. …

29: 18.25 Wilson Class: Definitions

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Table 18.25.1: Wilson class OP’s: transformations of variable, orthogonality ranges, and parameter constraints.

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OP	$p_n(x)$	$x=\lambda (y)$	Orthogonality range for $y$	Constraints
Wilson	$W_n(x;a,b,c,d)$	$y^2$	$(0,\mathrm{\infty })$	$\mathrm{\Re }(a,b,c,d)>0$; nonreal parameters in conjugate pairs
continuous dual Hahn	$S_n(x;a,b,c)$	$y^2$	$(0,\mathrm{\infty })$	$\mathrm{\Re }(a,b,c)>0$; nonreal parameters in conjugate pairs
…

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… ►, for the case and $a+b$, $a+c$, $a+d$ are positive or a pair of complex conjugates with positive real parts, see Wilson (1980, (3.3)) or Koekoek et al. (2010, (9.1.3)). …

30: 4.43 Cubic Equations

… ►Note that in Case (a) all the roots are real, whereas in Cases (b) and (c) there is one real root and a conjugate pair of complex roots. …