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1: 1.11 Zeros of Polynomials

… ►The discriminant of $f(z)$ is defined by ►

1.11.9

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Defines:

$D$: discriminant of $f(z)$ (locally)

Symbols:

$z$: variable, $j$: integer, $k$: integer and $n$: nonnegative integer

Referenced by:

§1.11(ii)

Permalink:

http://dlmf.nist.gov/1.11.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §1.11(ii), §1.11(ii), §1.11 and Ch.1

… ►The discriminant of $g(w)$ is ►

1.11.12 $$D=-4p^3-27q^2.$$

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Defines:

$D$: discriminant (locally)

Symbols:

$p$ and $q$

A&S Ref:

3.8.1

Permalink:

http://dlmf.nist.gov/1.11.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §1.11(iii), §1.11(iii), §1.11 and Ch.1

… ►The discriminant of $g(w)$ is …

2: 27.14 Unrestricted Partitions

… ►

§27.14(vi) Ramanujan’s Tau Function

►The discriminant function $\mathrm{\Delta }\left(\tau \right)$ is defined by ►

27.14.16 $$\mathrm{\Delta }\left(\tau \right)={(2\pi )}^{12}{(\eta \left(\tau \right))}^{24},$$ $\mathrm{\Im }\tau >0$,

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Defines:

$\mathrm{\Delta }\left(\tau \right)$: discriminant function

Symbols:

$\eta \left(\tau \right)$: Dedekind’s eta function (or Dedekind modular function), $\pi$: the ratio of the circumference of a circle to its diameter and $\mathrm{\Im }$: imaginary part

Permalink:

http://dlmf.nist.gov/27.14.E16

Encodings:

pMML, png, TeX

See also:

Annotations for §27.14(vi), §27.14 and Ch.27

… ►

27.14.17 $$\mathrm{\Delta }\left(\frac{a\tau +b}{c\tau +d}\right)={(c\tau +d)}^{12}\mathrm{\Delta }\left(\tau \right),$$

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Symbols:

$\mathrm{\Delta }\left(\tau \right)$: discriminant function and $\mathrm{\Im }$: imaginary part

Permalink:

http://dlmf.nist.gov/27.14.E17

Encodings:

pMML, png, TeX

See also:

Annotations for §27.14(vi), §27.14 and Ch.27

…

3: 18.16 Zeros

… ►

§18.16(vii) Discriminants

►The discriminant (18.2.20) can be given explicitly for classical OP’s. … ►

18.16.19 $$\mathrm{Disc}\left(P_n^{(\alpha ,\beta )}\right)=2^{-n(n-1)}\prod _{j=1}^n{j}^{j-2n+2}{(j+\alpha )}^{j-1}{(j+\beta )}^{j-1}{(n+j+\alpha +\beta )}^{n-j}.$$

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Symbols:

$\mathrm{Disc}\left(x\right)$: discriminant function, $P_n^{(\alpha ,\beta )}\left(x\right)$: Jacobi polynomial and $n$: nonnegative integer

Proved:

Ismail (2009, (3.4.16))(proved)

Referenced by:

Erratum (V1.2.0) Section 18.16

Permalink:

http://dlmf.nist.gov/18.16.E19

Encodings:

pMML, png, TeX

See also:

Annotations for §18.16(vii), §18.16(vii), §18.16 and Ch.18

… ►

18.16.20 $$\mathrm{Disc}\left(L_n^{(\alpha )}\right)=\prod _{j=1}^n{j}^{j-2n+2}{(j+\alpha )}^{j-1}.$$

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Symbols:

$\mathrm{Disc}\left(x\right)$: discriminant function, $L_n^{(\alpha )}\left(x\right)$: Laguerre (or generalized Laguerre) polynomial and $n$: nonnegative integer

Proved:

Ismail (2009, (3.4.15))(proved)

Permalink:

http://dlmf.nist.gov/18.16.E20

Encodings:

pMML, png, TeX

See also:

Annotations for §18.16(vii), §18.16(vii), §18.16 and Ch.18

… ►

18.16.21 $$\mathrm{Disc}\left(H_n\right)=2^{\frac{3}{2}n(n-1)}\prod _{j=1}^n{j}^j.$$

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Symbols:

$\mathrm{Disc}\left(x\right)$: discriminant function, $H_n\left(x\right)$: Hermite polynomial and $n$: nonnegative integer

Proved:

Ismail (2009, (3.4.14))(proved)

Referenced by:

Erratum (V1.2.0) Section 18.16

Permalink:

http://dlmf.nist.gov/18.16.E21

Encodings:

pMML, png, TeX

See also:

Annotations for §18.16(vii), §18.16(vii), §18.16 and Ch.18

4: 23.19 Interrelations

… ►

23.19.4 $$\mathrm{\Delta }={(2\pi )}^{12}{\eta }^{24}\left(\tau \right).$$

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Symbols:

$\eta \left(\tau \right)$: Dedekind’s eta function (or Dedekind modular function), $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{\Delta }$: discriminant and $\tau$: complex variable

Referenced by:

§23.19

Permalink:

http://dlmf.nist.gov/23.19.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §23.19 and Ch.23

5: 23.3 Differential Equations

… ►

§23.3(i) Invariants, Roots, and Discriminant

… ►The discriminant (§1.11(ii)) is given by ►

23.3.4 $$\mathrm{\Delta }=g_2^3-27g_3^2=16{(e_2-e_3)}^2{(e_3-e_1)}^2{(e_1-e_2)}^2.$$

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Symbols:

$e_j$: Weierstrass lattice roots, $g_j$: Weierstrass lattice invariants $g_2$, $g_3$ , $𝕃$: lattice and $\mathrm{\Delta }$: discriminant

A&S Ref:

18.1.8

Referenced by:

§23.19

Permalink:

http://dlmf.nist.gov/23.3.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §23.3(i), §23.3 and Ch.23

…

6: 28.29 Definitions and Basic Properties

… ►

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

… ►

28.29.15 $$△(\lambda )=w_{\text{I}}(\pi ,\lambda )+w_{\text{II}}^{\prime }(\pi ,\lambda )$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $w_{\text{I}}(z;a,q)$: fundamental solution, $w_{\text{II}}(z;a,q)$: fundamental solution and $\lambda$: parameter

Permalink:

http://dlmf.nist.gov/28.29.E15

Encodings:

pMML, png, TeX

See also:

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

►is called the discriminant of (28.29.1). … ►For this purpose the discriminant can be expressed as an infinite determinant involving the Fourier coefficients of $Q(x)$; see Magnus and Winkler (1966, §2.3, pp. 28–36). …

7: 23.1 Special Notation

… ► ►►►

$𝕃$	lattice in $ℂ$.
…
$\mathrm{\Delta }$	discriminant $g_2^3-27g_3^2$.
…

…

8: 18.2 General Orthogonal Polynomials

… ►

Discriminants

… ►The discriminant of $p_n$ is defined by ►

18.2.20

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Defines:

$\mathrm{Disc}\left(x\right)$: discriminant function

Symbols:

$p_n(x)$: polynomial of degree $n$ , $n$: nonnegative integer, $x$: real variable and $k_n$

Source:

Ismail (2009, (3.1.8))

Referenced by:

§18.16(vii), §18.2(v), Erratum (V1.2.0) Section 18.2

Permalink:

http://dlmf.nist.gov/18.2.E20

Encodings:

pMML, png, TeX

See also:

Annotations for §18.2(vi), §18.2(vi), §18.2 and Ch.18

►See Ismail (2009, §3.4) for another expression of the discriminant in the case of a general OP. …

9: 23.22 Methods of Computation

… ►

In the general case, given by $cd\ne 0$, we compute the roots $\alpha$, $\beta$, $\gamma$, say, of the cubic equation $4t^3-ct-d=0$; see §1.11(iii). These roots are necessarily distinct and represent $e_1$, $e_2$, $e_3$ in some order.

If $c$ and $d$ are real, and the discriminant is positive, that is $c^3-27d^2>0$, then $e_1$, $e_2$, $e_3$ can be identified via (23.5.1), and $k^2$, $k_{}^{\prime }{}_{}{}^2$ obtained from (23.6.16).

If , or $c$ and $d$ are not both real, then we label $\alpha$, $\beta$, $\gamma$ so that the triangle with vertices $\alpha$, $\beta$, $\gamma$ is positively oriented and $[\alpha ,\gamma ]$ is its longest side (chosen arbitrarily if there is more than one). In particular, if $\alpha$, $\beta$, $\gamma$ are collinear, then we label them so that $\beta$ is on the line segment $(\alpha ,\gamma )$. In consequence, $k^2=(\beta -\gamma )/(\alpha -\gamma )$, $k_{}^{\prime }{}_{}{}^2=(\alpha -\beta )/(\alpha -\gamma )$ satisfy $\mathrm{\Im }{k}^2\ge 0\ge \mathrm{\Im }k_{}^{\prime }{}_{}{}^2$ (with strict inequality unless $\alpha$, $\beta$, $\gamma$ are collinear); also $|k^2|$, $|k_{}^{\prime }{}_{}{}^2|\le 1$.

Finally, on taking the principal square roots of $k^2$ and $k_{}^{\prime }{}_{}{}^2$ we obtain values for $k$ and $k^{\prime }$ that lie in the 1st and 4th quadrants, respectively, and $2{\omega }_1$, $2{\omega }_3$ are given by

23.22.1 $$2{\omega }_{1}M(1,k^{\prime })=-2\mathrm{i}{\omega }_{3}M(1,k)=\frac{\pi }{3}\sqrt{\frac{c(2+k^{2}k_{}^{\prime }{}_{}{}^2)(k_{}^{\prime }{}_{}{}^2-k^2)}{d(1-k^{2}k_{}^{\prime }{}_{}{}^2)}},$$

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Symbols:

$M(a,g)$: arithmetic-geometric mean, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{i}$: imaginary unit, ${\omega }_1$, ${\omega }_3$, ${\omega }_2=-{\omega }_1-{\omega }_3$: lattice generators, $k$: modulus and $k^{\prime }$: complementary modulus

Proof sketch:

Combine (23.6.16), (23.6.17), (22.20.6), (23.3.5)–(23.3.7), and use analytical continuation.

Referenced by:

§23.22(ii)

Permalink:

http://dlmf.nist.gov/23.22.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §23.22(ii), §23.22(ii), §23.22 and Ch.23

where $M$ denotes the arithmetic-geometric mean (see §§19.8(i) and 22.20(ii)). This process yields 2 possible pairs ($2{\omega }_1$, $2{\omega }_3$), corresponding to the 2 possible choices of the square root.

…

10: Bibliography M

… ►

Exact misclassification probabilities for plug-in normal quadratic discriminant functions. I. The equal-means case. J. Multivariate Anal. 77 (1), pp. 21–53.

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External Links:

ISSN 0047-259X, Document, MathReview (Peter A. Lachenbruch), ZentralBlatt

Referenced by:

§35.9.

Encodings:

BibTeX

►

Exact misclassification probabilities for plug-in normal quadratic discriminant functions. II. The heterogeneous case. J. Multivariate Anal. 82 (2), pp. 299–330.

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External Links:

ISSN 0047-259X, Document, MathReview (Peter A. Lachenbruch), ZentralBlatt

Referenced by:

§35.9.

Encodings:

BibTeX

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