# permutation symmetry

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## 8 matching pages

##### 1: 19.15 Advantages of Symmetry
The function $R_{-a}\left(b_{1},b_{2},\dots,b_{n};z_{1},z_{2},\dots,z_{n}\right)$ (Carlson (1963)) reveals the full permutation symmetry that is partially hidden in $F_{D}$, and leads to symmetric standard integrals that simplify many aspects of theory, applications, and numerical computation. …
##### 2: Bille C. Carlson
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. …In Permutation symmetry for theta functions (2011) he found an analogous hidden symmetry between theta functions. …
##### 3: 20.11 Generalizations and Analogs
###### §20.11(v) PermutationSymmetry
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. …
##### 5: Bibliography C
• B. C. Carlson (2011) Permutation symmetry for theta functions. J. Math. Anal. Appl. 378 (1), pp. 42–48.
• ##### 6: 34.3 Basic Properties: $\mathit{3j}$ Symbol
###### §34.3(ii) Symmetry
Even permutations of columns of a $\mathit{3j}$ symbol leave it unchanged; odd permutations of columns produce a phase factor $(-1)^{j_{1}+j_{2}+j_{3}}$, for example, … Equations (34.3.11) and (34.3.12) are called Regge symmetries. Additional symmetries are obtained by applying (34.3.8)–(34.3.10) to (34.3.11)) and (34.3.12). …
##### 7: 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. …
##### 8: 19.25 Relations to Other Functions
The transformations in §19.7(ii) result from the symmetry and homogeneity of functions on the right-hand sides of (19.25.5), (19.25.7), and (19.25.14). …Thus the five permutations induce five transformations of Legendre’s integrals (and also of the Jacobian elliptic functions). … With $0\leq k^{2}\leq 1$ and $\mathrm{p,q,r}$ any permutation of the letters $\mathrm{c,d,n}$, define … In (19.25.38) and (19.25.39) $j,k,\ell$ is any permutation of the numbers $1,2,3$. … (${F_{1}}$ and $F_{D}$ are equivalent to the $R$-function of 3 and $n$ variables, respectively, but lack full symmetry.) …