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1: 22.7 Landen Transformations

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§22.7(i) Descending Landen Transformation

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2: 22.17 Moduli Outside the Interval [0,1]

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§22.17(ii) Complex Moduli

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3: 26.12 Plane Partitions

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26.12.18

\begin{array}[]{ccccc}6&6&6&4&3\\ &3&3\\ &&2\end{array}

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Permalink:: http://dlmf.nist.gov/26.12.E18
Encodings:: pMML, png, TeX
See also:: Annotations for §26.12(i), §26.12 and Ch.26

►A descending plane partition is a strict shifted plane partition in which the number of parts in each row is strictly less than the largest part in that row and is greater than or equal to the largest part in the next row. The example of a strict shifted plane partition also satisfies the conditions of a descending plane partition. The number of descending plane partitions in

B(r,r,r)

is … ►

26.12.24

\sum_{\begin{subarray}{c}\pi\subseteq B(r,r,r)\\ \pi\mbox{\scriptsize\ descending plane partition}\end{subarray}}q^{|\pi|}=% \prod_{1\leq h<j\leq r}\frac{1-q^{r+h+j-1}}{1-q^{2h+j-1}}.

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Symbols:: $\subseteq$ : is, or is contained in, $h$ : nonnegative integer, $j$ : nonnegative integer, $\pi$ : plane partition, $r$ : positive integer, $B(r,s,t)$ : Box and $q$ : parameter
Permalink:: http://dlmf.nist.gov/26.12.E24
Encodings:: pMML, png, TeX
See also:: Annotations for §26.12(ii), §26.12 and Ch.26

…

4: 19.36 Methods of Computation

… ►The step from

n

n+1

is an ascending Landen transformation if

\theta=1

(leading ultimately to a hyperbolic case of

R_{C}

) or a descending Gauss transformation if

\theta=-1

(leading to a circular case of

R_{C}

). … ►Thompson (1997, pp. 499, 504) uses descending Landen transformations for both

F\left(\phi,k\right)

and

E\left(\phi,k\right)

. … ►Descending Gauss transformations of

\Pi\left(\phi,\alpha^{2},k\right)

(see (19.8.20)) are used in Fettis (1965) to compute a large table (see §19.37(iii)). … ►The function

\operatorname{el2}\left(x,k_{c},a,b\right)

is computed by descending Landen transformations if

x

is real, or by descending Gauss transformations if

x

is complex (Bulirsch (1965b)). …

5: 19.8 Quadratic Transformations

… ►

Descending Landen Transformation

… ►We consider only the descending Gauss transformation because its (ascending) inverse moves

F\left(\phi,k\right)

closer to the singularity at

k=\sin\phi=1

. …

6: 19.22 Quadratic Transformations

… ►If

x, y, z

are real and positive, then (19.22.18)–(19.22.21) are ascending Landen transformations when

x,y<z

(implying

a<z_{-}<z_{+}

), and descending Gauss transformations when

z<x,y

(implying

z_{+}<z_{-}<a

). …Descending Gauss transformations include, as special cases, transformations of complete integrals into complete integrals; ascending Landen transformations do not. … ►The transformations inverse to the ones just described are the descending Landen transformations and the ascending Gauss transformations. …