relation to permutations

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1: 26.14 Permutations: Order Notation

… ► …

2: Bille C. Carlson

… ►He then went to Oxford as a Rhodes Scholar and completed a doctoral degree in physics. … ►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. In Permutation symmetry for theta functions (2011) he found an analogous hidden symmetry between theta functions. …

3: 34.7 Basic Properties: $\mathit{9j}$ Symbol

… ►The

\mathit{9j}

symbol has symmetry properties with respect to permutation of columns, permutation of rows, and transposition of rows and columns; these relate 72 independent

\mathit{9j}

symbols. …

4: 26.13 Permutations: Cycle Notation

§26.13 Permutations: Cycle Notation

… ►

\sigma\in\mathfrak{S}_{n}

is a one-to-one and onto mapping from

\{1,2,\ldots,n\}

to itself. … ►The permutation … ►See §26.8 for generating functions, recurrence relations, identities, and asymptotic approximations. … ►A permutation is even or odd according to the parity of the number of transpositions. …

5: 19.25 Relations to Other Functions

§19.25 Relations to Other Functions

… ►

§19.25(ii) Bulirsch’s Integrals as Symmetric Integrals

… ► … ►

§19.25(vii) Hypergeometric Function

… ►

6: 19.18 Derivatives and Differential Equations

… ►The next two equations apply to (19.16.14)–(19.16.18) and (19.16.20)–(19.16.23). … ►

19.18.6

\left(\frac{\partial}{\partial x}+\frac{\partial}{\partial y}+\frac{\partial}{% \partial z}\right)R_{F}\left(x,y,z\right)=\frac{-1}{2\sqrt{x}\sqrt{y}\sqrt{z}},

ⓘ

Symbols:: $R_{F}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : symmetric elliptic integral of first kind, $\frac{\partial\NVar{f}}{\partial\NVar{x}}$ : partial derivative of $f$ with respect to $x$ and $\,\partial\NVar{x}$ : partial differential of $x$
Referenced by:: §19.18(ii), §19.32, Erratum (V1.1.3) for Chapter 19
Permalink:: http://dlmf.nist.gov/19.18.E6
Encodings:: pMML, png, TeX
Correction (effective with 1.1.3):: The factors inside the square root on the right-hand side were written as products to ensure the correct multivalued behavior.
Suggested 2021-06-07 by Luc Maisonobe
See also:: Annotations for §19.18(ii), §19.18 and Ch.19

… ►and two similar equations obtained by permuting

x, y, z

in (19.18.10). ►More concisely, if

v=R_{-a}\left(\mathbf{b};\mathbf{z}\right)

, then each of (19.16.14)–(19.16.18) and (19.16.20)–(19.16.23) satisfies Euler’s homogeneity relation: … ►The next four differential equations apply to the complete case of

R_{F}

and

R_{G}

in the form

R_{-a}\left(\tfrac{1}{2},\tfrac{1}{2};z_{1},z_{2}\right)

(see (19.16.20) and (19.16.23)). …

7: 19.21 Connection Formulas

… ►Let

y

z

, and

p

be positive and distinct, and permute

y

and

z

to ensure that

y

does not lie between

z

and

p

. …If

0 <mrow> <mn>0</mn> <mo><</mo> <mi>p</mi> <mo><</mo> <mi>z</mi> </mrow> </math> and <math class=

, then as

p\to 0

(19.21.6) reduces to Legendre’s relation (19.21.1). … ►where both summations extend over the three cyclic permutations of

x, y, z

. … ►Change-of-parameter relations can be used to shift the parameter

p

R_{J}

from either circular region to the other, or from either hyperbolic region to the other (§19.20(iii)). …and

x, y, z

may be permuted. …

8: 1.2 Elementary Algebra

… ►and as

p\to\infty

… ►

Special Properties and Definitions Relating to Square Matrices

… ►

\mathfrak{S}_{n}

is the set of all permutations of the set

\{1,2,...,n\}

. … ►Numerical methods and issues for solution of (1.2.61) appear in §§3.2(i) to 3.2(iii). … ►

Relation of Eigenvalues to the Determinant and Trace

…

9: 19.29 Reduction of General Elliptic Integrals

… ►and

\alpha,\beta,\gamma,\delta

is any permutation of the numbers

1,2,3,4

, then … ►The reduction of

I(\mathbf{m})

is carried out by a relation derived from partial fractions and by use of two recurrence relations. …Partial fractions provide a reduction to integrals in which

\mathbf{m}

has at most one nonzero component, and these are then reduced to basic integrals by the recurrence relations. … ►where

\alpha,\beta,\gamma

is any permutation of the numbers

1,2,3

, and … ►The other recurrence relation is …

10: 20.11 Generalizations and Analogs

… ►However, in this case

q

is no longer regarded as an independent complex variable within the unit circle, because

k

is related to the variable

\tau=\tau(k)

of the theta functions via (20.9.2). … ►For specialization to the one-dimensional theta functions treated in the present chapter, see Rauch and Lebowitz (1973) and §21.7(iii). ►

§20.11(v) Permutation Symmetry

… ►The importance of these combined theta functions is that sets of twelve equations for the theta functions often can be replaced by corresponding sets of three equations of the combined theta functions, plus permutation symmetry. Such sets of twelve equations include derivatives, differential equations, bisection relations, duplication relations, addition formulas (including new ones for theta functions), and pseudo-addition formulas. …