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Edit detail for SandBoxCombfuncDiscussion revision 7 of 7

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Editor: Bill Page Time: 2017/04/07 22:41:53 GMT+0
Note: better display for binomials

added:
Nicer display from binomials:
\begin{axiom}
--)set output tex off
--)set output algebra on
binomial(n,m)
opbinom := operator(operator 'binomial)$CombinatorialFunction(INT,EXPR INT)
setProperty(opbinom, '%specialDisp::Symbol,
  ((l:List EXPR INT):OutputForm +-> paren vconcat(first(l)::OutputForm,second(l)::OutputForm)) pretend None)
binomial(n,m)
\end{axiom}


removed:
-binomial(3,x)-binomial(3,3-x)

changed:
-ex1 := binomial(x+y,y)*binomial(y,y+z) 
-ex2 := BX( binomial(x+y,y)*binomial(y,y+z) )
ex1 := binomial(x+y,y)*binomial(y,y-z) 
ex2 := BX ex1

Nicer display from binomials:

fricas

(1) -> --)set output tex off
--)set output algebra on
binomial(n,m)

$\label{eq1}\hbox{\axiomType{BINOMIAL}\ } \left({n , \: m}\right)$

(1)

Type: Expression(Integer)

fricas

opbinom := operator(operator 'binomial)$CombinatorialFunction(INT,EXPR INT)

$\label{eq2}binomial$

(2)

Type: BasicOperator?

fricas

setProperty(opbinom, '%specialDisp::Symbol,
  ((l:List EXPR INT):OutputForm +-> paren vconcat(first(l)::OutputForm,second(l)::OutputForm)) pretend None)

$\label{eq3}binomial$

(3)

Type: BasicOperator?

fricas

binomial(n,m)

$\label{eq4}\left( \begin{array}{c} n \ m \$

(4)

Type: Expression(Integer)

Tests from GouldBk?.pdf --rrogers, Fri, 07 Apr 2017 16:22:25 +0000 reply

http://www.dsi.unifi.it/~resp/GouldBK.pdf

fricas

binomial(5,2)-binomial(5,5-2)

$\label{eq5}0$

(5)

Type: NonNegativeInteger?

fricas

binomial(a,b)-binomial(a,a-b)

$\label{eq6}\begin{array}{@{}l} \displaystyle {\left( \begin{array}{c} a \ b \$

(6)

Type: Expression(Integer)

Ah well no joy in Mudville I guess I need a special function of some sort, any comments?

simplification of binomials --Bill Page, Fri, 07 Apr 2017 21:38:48 +0000 reply

The simplifications provided by Combfunc apply only to specific integer values of the 2nd argument of a single kernel. What you want to do usually involves combinations of more than one kernel. For this you might expect something like simplify or expand to work but these commands are not aware of binomial. The next best thing might be to use some custom rules. E.g.

fricas

BS := rule
  binomial(a,a-b) == binomial(a,b)

$\label{eq7}= = \left({{\left( \begin{array}{c} a \ {- b + a} \$

(7)

Type: RewriteRule?(Integer,Integer,Expression(Integer))

then we have

fricas

Is(binomial(3,3-x),lhs BS)

$\label{eq8}\left[{b = x}, \:{a = 3}\right]$

(8)

Type: List(Equation(Expression(Integer)))

fricas

BS( binomial(3,x)-binomial(3,3-x) )

$\label{eq9}0$

(9)

Type: Expression(Integer)

This rule is already built-in

fricas

BN := rule
  binomial(-n+k-1,k | even? k) == binomial(n,k)
  binomial(-n+k-1,k | odd? k) == -binomial(n,k)

$\label{eq10}\begin{array}{@{}l} \displaystyle \left\{{= = \left({{\left( \begin{array}{c} {- n + k - 1} \ k \$

(10)

Type: Ruleset(Integer,Integer,Expression(Integer))

fricas

binomial(-n+1,2)

$\label{eq11}\frac{{{n}^{2}}- n}{2}$

(11)

Type: Expression(Integer)

fricas

binomial(-n+2,3)

$\label{eq12}\frac{-{{n}^{3}}+{3 \ {{n}^{2}}}-{2 \ n}}{6}$

(12)

Type: Expression(Integer)

Cross product. Note that because pattern matching is syntactic we need to take care about the lexical ordering of the variables.

fricas

BX := rule
  binomial(k,n)*binomial(n,j) == binomial(k,j)*binomial(k-j,n-j)
  binomial(k,j)*binomial(n,k) == binomial(n,j)*binomial(n-j,k-j)

$\label{eq13}\begin{array}{@{}l} \displaystyle \left\{{= = \left({{\%E \ {\left( \begin{array}{c} k \ n \$

(13)

Type: Ruleset(Integer,Integer,Expression(Integer))

fricas

ex1 := binomial(x+y,y)*binomial(y,y-z)

$\label{eq14}{\left({ \begin{array}{c} y \ {- z + y} \$

(14)

Type: Expression(Integer)

fricas

ex2 := BX ex1

$\label{eq15}{\left({ \begin{array}{c} {y + x} \ {- z + y} \$

(15)

Type: Expression(Integer)

fricas

eval(ex1-ex2,[x=3,y=5,z=7])

$\label{eq16}0$

(16)

Type: Expression(Integer)