# on finite point sets

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##### 1: 18.2 General Orthogonal Polynomials
Let $X$ be a finite set of distinct points on $\mathbb{R}$, or a countable infinite set of distinct points on $\mathbb{R}$, and $w_{x}$, $x\in X$, be a set of positive constants. …when $X$ is a finite set of $N+1$ distinct points. … If the polynomials $p_{n}(x)$ ($n=0,1,\ldots,N$) are orthogonal on a finite set $X$ of $N+1$ distinct points as in (18.2.3), then the polynomial $p_{N+1}(x)$ of degree $N+1$, up to a constant factor defined by (18.2.8) or (18.2.10), vanishes on $X$. …
##### 2: Errata

• §18.2

In Subsection 18.2(i), Equation (18.2.1_5); the paragraph title “Orthogonality on Finite Point Sets” has been changed to “Orthogonality on Countable Sets”, and there are minor changes in the presentation of the final paragraph, including a new equation (18.2.4_5). The presentation of Subsection 18.2(iii) has changed, Equation (18.2.5_5) was added and an extra paragraph on standardizations has been included. The presentation of Subsection 18.2(iv) has changed and it has been expanded with two extra paragraphs and several new equations, (18.2.9_5), (18.2.11_1)–(18.2.11_9). Subsections 18.2(v) (with (18.2.12_5), (18.2.14)–(18.2.17)) and 18.2(vi) (with (18.2.17)–(18.2.20)) have been expanded. New subsections, 18.2(vii)18.2(xii), with Equations (18.2.21)–(18.2.46),

• §18.3

A new introduction, minor changes in the presentation, and three new paragraphs.

• §18.5

Extra details for Chebyshev polynomials, and Equations (18.5.4_5), (18.5.11_1)–(18.5.11_4), (18.5.17_5).

• §18.8

Line numbers and two extra rows were added to Table 18.8.1.

• §18.9

Subsection 18.9(i) has been expanded, and 18.9(iii) has some additional explanation. Equations (18.9.2_1), (18.9.2_2), (18.9.18_5) and Table 18.9.2 were added.

• Three extra generating functions, (18.12.2_5), (18.12.3_5), (18.12.17).

• Equation (18.14.3_5). New subsection, 18.14(iv), with Equations (18.14.25)–(18.14.27).

• Equation (18.15.4_5).

• The title of Subsection 18.16(iii) was changed from “Ultraspherical and Legendre” to “Ultraspherical, Legendre and Chebyshev”. New subsection, 18.16(vii) Discriminants, with Equations (18.16.19)–(18.16.21).

• Extra explanatory text at many places and seven extra integrals (18.17.16_5), (18.17.21_1)–(18.17.21_3), (18.17.28_5), (18.17.34_5), (18.17.41_5).

• Extra explanatory text at several places and the title of Subsection 18.18(iv) was changed from “Connection Formulas” to “Connection and Inversion Formulas”.

• A new introduction.

• Equation (18.21.13).

• Extra explanatory text in Subsection 18.25(i) and the title of Subsection 18.25(ii) was changed from “Weights and Normalizations: Continuous Cases” to “Weights and Standardizations: Continuous Cases”.

• In Subsection 18.26(i) an extra paragraph on dualities has been included, with Equations (18.26.4_1), (18.26.4_2).

• Extra text at the start of this section and twenty seven extra formulas, (18.27.4_1), (18.27.4_2), (18.27.6_5), (18.27.9_5), (18.27.12_5), (18.27.14_1)–(18.27.14_6), (18.27.17_1)–(18.27.17_3), (18.27.20_5), (18.27.25), (18.27.26), (18.28.1_5).

• A big expansion. Six extra formulas in Subsection 18.28(ii) ((18.28.6_1)–(18.28.6_5)) and three extra formulas in Subsection 18.28(viii) ((18.28.21)–(18.28.23)). New subsections, 18.28(ix)18.28(xi), with Equations (18.28.23)–(18.28.34).

• Originally this section did not have subsections. The original seven formulas have now more explanatory text and are split over two subsections. New subsections 18.30(iii)18.30(viii), with Equations (18.30.8)–(18.30.31).

• This short section has been expanded, with Equation (18.32.2).

• Additional references and a new large subsection, 18.33(vi), including Equations (18.33.17)–(18.33.32).

• This section has been expanded, including an extra orthogonality relations (18.34.5_5), (18.34.7_1)–(18.34.7_3).

• This section on Pollaczek polynomials has been significantly updated with much more explanations and as well to include the Pollaczek polynomials of type 3 which are the most general with three free parameters. The Pollaczek polynomials which were previously treated, namely those of type 1 and type 2 are special cases of the type 3 Pollaczek polynomials. In the first paragraph of this section an extensive description of the relations between the three types of Pollaczek polynomials is given which was lacking previously. Equations (18.35.0_5), (18.35.2_1)–(18.35.2_5), (18.35.4_5), (18.35.6_1)–(18.35.6_6), (18.35.10).

• This section on miscellaneous polynomials has been expanded with new subsections, 18.36(v) on non-classical Laguerre polynomials and 18.36(vi) with examples of exceptional orthogonal polynomials, with Equations (18.36.1)–(18.36.10). In the titles of Subsections 18.36(ii) and 18.36(iii) we replaced “OP’s” by “Orthogonal Polynomials”.

• The paragraphs of Subsection 18.38(i) have been re-ordered and one paragraph was added. The title of Subsection 18.38(ii) was changed from “Classical OP’s: Other Applications” to “Classical OP’s: Mathematical Developments and Applications”. Subsection 18.38(iii) has been expanded with seven new paragraphs, and Equations (18.38.4)–(18.38.11).

• This section was completely rewritten. The previous 18.39(i) Quantum Mechanics has been replaced by Subsections 18.39(i) Quantum Mechanics and 18.39(ii) A 3D Separable Quantum System, the Hydrogen Atom, containing the same essential information; the original content of the subsection is reproduced below for reference. Subsection 18.39(ii) was moved to 18.39(v) Other Applications. New subsections, 18.39(iii) Non Classical Weight Functions of Utility in DVR Method in the Physical Sciences, 18.39(iv) Coulomb–Pollaczek Polynomials and J-Matrix Methods; Equations (18.39.7)–(18.39.48); and Figures 18.39.1, 18.39.2.

The original text of 18.39(i) Quantum Mechanics was:

“Classical OP’s appear when the time-dependent Schrödinger equation is solved by separation of variables. Consider, for example, the one-dimensional form of this equation for a particle of mass $m$ with potential energy $V(x)$:

errata.1 $\left(\frac{-\hbar^{2}}{2m}\frac{{\partial}^{2}}{{\partial x}^{2}}+V(x)\right)% \psi(x,t)=i\hbar\frac{\partial}{\partial t}\psi(x,t),$

where $\hbar$ is the reduced Planck’s constant. On substituting $\psi(x,t)=\eta(x)\zeta(t)$, we obtain two ordinary differential equations, each of which involve the same constant $E$. The equation for $\eta(x)$ is

errata.2 $\frac{{\mathrm{d}}^{2}\eta}{{\mathrm{d}x}^{2}}+\frac{2m}{\hbar^{2}}\left(E-V(x% )\right)\eta=0.$

For a harmonic oscillator, the potential energy is given by

errata.3 $V(x)=\tfrac{1}{2}m\omega^{2}x^{2},$

where $\omega$ is the angular frequency. For (18.39.2) to have a nontrivial bounded solution in the interval $-\infty, the constant $E$ (the total energy of the particle) must satisfy

errata.4 $E=E_{n}=\left(n+\tfrac{1}{2}\right)\hbar\omega,$ $n=0,1,2,\dots$.

The corresponding eigenfunctions are

errata.5 $\eta_{n}(x)=\pi^{-\frac{1}{4}}2^{-\frac{1}{2}n}(n!\,b)^{-\frac{1}{2}}H_{n}% \left(x/b\right){\mathrm{e}}^{-x^{2}/2b^{2}},$

where $b=(\hbar/m\omega)^{1/2}$, and $H_{n}$ is the Hermite polynomial. For further details, see Seaborn (1991, p. 224) or Nikiforov and Uvarov (1988, pp. 71-72).

A second example is provided by the three-dimensional time-independent Schrödinger equation

errata.6 $\nabla^{2}\psi+\frac{2m}{\hbar^{2}}\left(E-V(\mathbf{x})\right)\psi=0,$

when this is solved by separation of variables in spherical coordinates (§1.5(ii)). The eigenfunctions of one of the separated ordinary differential equations are Legendre polynomials. See Seaborn (1991, pp. 69-75).

For a third example, one in which the eigenfunctions are Laguerre polynomials, see Seaborn (1991, pp. 87-93) and Nikiforov and Uvarov (1988, pp. 76-80 and 320-323).”

• Section 18.40

The old section is now Subsection 18.40(i) and a large new subsection, 18.40(ii), on the classical moment problem has been added, with formulae (18.40.1)–(18.40.10) and Figures 18.40.1, 18.40.2.

• ##### 3: 23.20 Mathematical Applications
Let $T$ denote the set of points on $C$ that are of finite order (that is, those points $P$ for which there exists a positive integer $n$ with $nP=o$), and let $I,K$ be the sets of points with integer and rational coordinates, respectively. …
##### 4: 1.9 Calculus of a Complex Variable
A domain $D$, say, is an open set in $\mathbb{C}$ that is connected, that is, any two points can be joined by a polygonal arc (a finite chain of straight-line segments) lying in the set. …
##### 5: 18.38 Mathematical Applications
The terminology DVR arises as an otherwise continuous variable, such as the co-ordinate $x$, is replaced by its values at a finite set of zeros of appropriate OP’s resulting in expansions using functions localized at these points. …
##### 6: 36.12 Uniform Approximation of Integrals
###### §36.12(i) General Theory for Cuspoids
Also, $f$ is real analytic, and $\ifrac{{\partial}^{K+2}f}{{\partial u}^{K+2}}>0$ for all $\mathbf{y}$ such that all $K+1$ critical points coincide. … In (36.12.10), both second derivatives vanish when critical points coalesce, but their ratio remains finite. … Also, $\Delta^{1/4}/\sqrt{f_{+}^{\prime\prime}}$ and $\Delta^{1/4}/\sqrt{-f_{-}^{\prime\prime}}$ are chosen to be positive real when $y$ is such that both critical points are real, and by analytic continuation otherwise. …
##### 7: 2.1 Definitions and Elementary Properties
Let $\mathbf{X}$ be a point set with a limit point $c$. … If $c$ is a finite limit point of $\mathbf{X}$, then … Suppose $u$ is a parameter (or set of parameters) ranging over a point set (or sets) $\mathbf{U}$, and for each nonnegative integer $n$Similarly for finite limit point $c$ in place of $\infty$. … where $c$ is a finite, or infinite, limit point of $\mathbf{X}$. …
##### 8: 21.7 Riemann Surfaces
Consider the set of points in ${\mathbb{C}}^{2}$ that satisfy the equation …This compact curve may have singular points, that is, points at which the gradient of $\tilde{P}$ vanishes. … On this surface, we choose $2g$ cycles (that is, closed oriented curves, each with at most a finite number of singular points) $a_{j}$, $b_{j}$, $j=1,2,\dots,g$, such that their intersection indices satisfy … The zeros $\lambda_{j}$, $j=1,2,\dots,2g+1$ of $Q(\lambda)$ specify the finite branch points $P_{j}$, that is, points at which $\mu_{j}=0$, on the Riemann surface. Denote the set of all branch points by $B=\{P_{1},P_{2},\dots,P_{2g+1},P_{\infty}\}$. …
##### 9: 1.10 Functions of a Complex Variable
If the poles are infinite in number, then the point at infinity is called an essential singularity: it is the limit point of the poles. … A cut domain is one from which the points on finitely many nonintersecting simple contours (§1.9(iii)) have been removed. … Alternatively, take $z_{0}$ to be any point in $D$ and set $F(z_{0})={\mathrm{e}}^{\alpha\ln\left(1-z_{0}\right)}{\mathrm{e}}^{\beta\ln% \left(1+z_{0}\right)}$ where the logarithms assume their principal values. … (The integer $k$ may be greater than one to allow for a finite number of zero factors.) …
##### 10: 1.4 Calculus of One Variable
where the supremum is over all sets of points $x_{0} in the closure of $(a,b)$, that is, $(a,b)$ with $a,b$ added when they are finite. …