View source for gei'au

==== valsi ====
gei'au
==== type ====
experimental cmavo
==== creator ====
[[personal/krtisfranks|krtisfranks]]
==== time entered ====
Sat Nov 12 21:28:56 2022
 
== English ==
=== Definition #73028 - Preferred ===
 
==== selma'o ====
VUhU3
==== definition ====

mekso 7-ary operator: for input  MATH
 $(X_1 = z, X_2 = (a_i)_i, X_3 = (b_j)_j, X_4 = p, X_5 = q, X_6 = h_1, X_7 = h_2)$
 
<span class="MATH">(<i>X</i><sub>1</sub> = <i>z</i>, <i>X</i><sub>2</sub> = (<i>a</i><sub>i</sub>)<sub>i</sub>, <i>X</i><sub>3</sub> = (<i>b</i><sub>j</sub>)<sub>j</sub>, <i>X</i><sub>4</sub> = <i>p</i>, <i>X</i><sub>5</sub> = <i>q</i>, <i>X</i><sub>6</sub> = <i>h</i><sub>1</sub>, <i>X</i><sub>7</sub> = <i>h</i><sub>2</sub>)</span>, this word/function outputs/yields  MATH
 $\sum_{n=0}^{\infty} (((\prod_{i = 1}^{p} ($
 
<span class="MATH"><img alt="$\sum_{{n=0}}^{{\infty}}$" src="/jbovlaste_export/oOYGQcM56a/img1.png" style="height: 2.66ex; vertical-align: -0.81ex; "/>(((<img alt="$\prod_{{i = 1}}^{{p}}$" src="/jbovlaste_export/oOYGQcM56a/img2.png" style="height: 2.66ex; vertical-align: -0.81ex; "/>(</span>ne'o'o MATH
 $(a_i,n,1,h_1)))/(\prod_{j = 1}^{q} ($
 
<span class="MATH">(<i>a</i><sub>i</sub>, <i>n</i>, 1, <i>h</i><sub>1</sub>)))/(<img alt="$\prod_{{j = 1}}^{{q}}$" src="/jbovlaste_export/oOYGQcM56a/img3.png" style="height: 3.01ex; vertical-align: -1.13ex; "/>(</span>ne'o'o MATH
 $(b_j,n,1,h_2)))) z^n/(n!))$
 
<span class="MATH">(<i>b</i><sub>j</sub>, <i>n</i>, 1, <i>h</i><sub>2</sub>))))<i>z</i><sup>n</sup>/(<i>n</i>!))</span>; by default,  MATH
 $X_6 = 1 = X_7$
 
<span class="MATH"><i>X</i><sub>6</sub> = 1 = <i>X</i><sub>7</sub></span> unless explicitly specified otherwise.

==== notes ====

See "[[ne'o'o|ne'o'o]]" (Pochhammer symbol); however, for the purposes of this definition and regardless of the definition(s) provided at "ne'o'o", here ne'e'o MATH
 $(Y_1, Y_2, Y_3, Y_4) = \prod_<a class="undefined" href="../dict/k%20%3d%200?bg=1;langidarg=2">k = 0</a>^<a class="undefined" href="../dict/Y_2%20-%201?bg=1;langidarg=2">Y_2 - 1</a> (Y_1 + (-1)^<a class="undefined" href="../dict/(1%20-%20Y_3)?bg=1;langidarg=2">(1 - Y_3)</a> Y_4 k)$
 
<span class="MATH">(<i>Y</i><sub>1</sub>, <i>Y</i><sub>2</sub>, <i>Y</i><sub>3</sub>, <i>Y</i><sub>4</sub>) = <img alt="$\prod_{{k = 0}}^{{Y_2 - 1}}$" src="/jbovlaste_export/blRiJYDuqd/img1.png" style="height: 3.01ex; vertical-align: -0.81ex; "/>(<i>Y</i><sub>1</sub> + (- 1)<sup>(1-Y<sub>3</sub>)</sup><i>Y</i><sub>4</sub><i>k</i>)</span> by local definition; ne'o'o MATH
 $(Y_1, Y_2, Y_3, Y_4) = 1$
 
<span class="MATH">(<i>Y</i><sub>1</sub>, <i>Y</i><sub>2</sub>, <i>Y</i><sub>3</sub>, <i>Y</i><sub>4</sub>) = 1</span> for all <span class="MATH"><i>Y</i><sub>2</sub> &lt; 1</span>, regardless of the other inputs <span class="MATH"><i>Y</i><sub>m</sub></span> (so long as they belong to the domain); by default, <span class="MATH"><i>Y</i><sub>4</sub> = 1</span> unless explicitly defined otherwise. For the main definition: <span class="MATH"><i>z</i></span> is a complex number, and <span class="MATH">(<i>a</i><sub>i</sub>)<sub>i</sub></span> and <span class="MATH">(<i>b</i><sub>j</sub>)<sub>j</sub></span> are (pre)defined sequences (id est: functions with domains being exactly the set of exactly all positive integers) such that, for all positive integer indices <span class="MATH"><i>i</i></span> and <span class="MATH"><i>j</i></span> respectively, their terms/outputs are complex numbers; both <span class="MATH">(<i>a</i><sub>i</sub>)<sub>i</sub></span> and <span class="MATH">(<i>b</i><sub>j</sub>)<sub>j</sub></span> are indexed starting at <span class="MATH">1</span>; also,  MATH
 $p, q, h_1, h_2$
 
<span class="MATH"><i>p</i>, <i>q</i>, <i>h</i><sub>1</sub>, <i>h</i><sub>2</sub></span> are positive integers; by default,  MATH
 $h_1 = 1 = h_2$
 
<span class="MATH"><i>h</i><sub>1</sub> = 1 = <i>h</i><sub>2</sub></span> unless explicitly specified otherwise. <span class="MATH"><i>X</i><sub>2</sub></span> and <span class="MATH"><i>X</i><sub>3</sub></span> may require usage of "[[mau'au|mau'au]]" and "[[zai'ai|zai'ai]]".

==== gloss words ====

<ul><li>[[natlang/en/generalized hypergeometric function|generalized hypergeometric function]]</li><li>[[natlang/en/hypergeometric function|hypergeometric function ; generalized]]</li></ul>

==== created by ====
[[personal/krtisfranks|krtisfranks]]
==== vote information ====
1
     
==== time ====
Sat Nov 12 22:55:55 2022



<br/><font size="+1">Examples</font>
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=== Etymology ===