MathAction SandBoxHyperGeometric

Topics

FrontPage
- SandBox
  - SandBoxHyperGeometric <-- You are here.

spad

)abbrev package SUMMPACK SummPack
SummPack(): Exports == Implementation where
  FPI ==> Fraction Polynomial Integer
  X ==> Expression Integer
  NNI ==> NonNegativeInteger
  PI ==> PositiveInteger
  POLINT ==> Polynomial Integer 
  BOP ==> BasicOperator

  Exports == with
    coeffMatrix : (PI,PI) -> Matrix X
    funcMatrix  : ((X,X)->X,PI,PI) -> Matrix X
    sumAndNormalize : (Matrix X,Matrix X) -> X
    retractNumeratorToPolyInt : X -> Polynomial Integer
    findCoeffs : (POLINT,Matrix X, Symbol) -> List Equation FPI
    formalExpr : (BOP,Matrix X,PI,PI) -> X
    celine : ((X,X)->X,PI,PI) -> X
    celine2 : ((X,X)->X,BOP,PI,PI) -> X

  Implementation == add

    coeffMatrix(I:PI,J:PI):Matrix X ==
      matrix [[new()$Symbol::X for i in 0..I] for j in 0..J]

    funcMatrix(F:(X,X)->X,I:PI,J:PI):Matrix X ==
      n:X:='n::X
      k:X:='k::X
      matrix [[F(n-j::X,k-i::X)/F(n,k)  for i in 0..I] for j in 0..J]

    sumAndNormalize(CM:Matrix X, FM:Matrix X):X ==
      n:=nrows(CM)
      t:X:=trace(squareMatrix(CM*transpose(FM))$SquareMatrix(n,X))
      normalize(t)$ElementaryFunctionStructurePackage(Integer,X)

    retractNumeratorToPolyInt(san:X):Polynomial Integer ==
      p:Polynomial Integer:=retract(numerator san)
      return p

    findCoeffs(p:POLINT,m:Matrix X,k:Symbol):List Equation FPI ==
      d:=degree(p,k)
      eqs:List Equation FPI:=[coefficient(p,k,l)::FPI=0 for l in 0..d]
      v:=members m
      x:=variables v
      sol:=solve(eqs,x)$SystemSolvePackage(Integer) --$TransSolvePackage(Integer)
      first sol

    convToFPI(M:Matrix X):Matrix FPI ==
      m:=copy(M)
      nr:=nrows(m)
      nc:=ncols(m)
      r:=zero(nr,nc)$Matrix(FPI)
      for i in 1..nr repeat
        for j in 1..nc repeat
          r(i,j):=retract m(i,j)
      return r

    formalExpr(op:BOP,cm:Matrix X,I:PI,J:PI):X ==
      d:=nrows(cm)
      n:X:='n::X
      k:X:='k::X
      g:Matrix(X):=matrix [[op(n-j::X,k-i::X) for i in 0..I] for j in 0..J]
      t:X:=trace(squareMatrix(cm*transpose(g))$SquareMatrix(d,X))
      return t

    celine(F:(X,X)->X,I:PI,J:PI):X ==
      cm:Matrix X:=coeffMatrix(I,J)
      fm:=funcMatrix(F,I,J)
      san:=sumAndNormalize(cm,fm)
      p:=retractNumeratorToPolyInt(san)
      --
      seq:=findCoeffs(p,cm,'k)
      e:List Equation X:=[lhs(x)::X=rhs(x)::X for x in seq]
      G:=operator 'G
      fex:X:=formalExpr(G,cm,I,J)
      subst(fex,e)

    celine2(F:(X,X)->X,op:BOP,I:PI,J:PI):X ==
      cm:Matrix X:=coeffMatrix(I,J)
      fm:=funcMatrix(F,I,J)
      san:=sumAndNormalize(cm,fm)
      p:=retractNumeratorToPolyInt(san)
      --
      seq:=findCoeffs(p,cm,'k)
      e:List Equation X:=[lhs(x)::X=rhs(x)::X for x in seq]
      fex:X:=formalExpr(op,cm,I,J)
      subst(fex,e)

-- C:=celine2((k,n)+->binomial(n,k),operator 'T,1,1)
-- variables(%)
-- C=0 ; %/?%
-- sum(C,k=1..n), tower(C), kernels(C), variables C, mainKernel C

spad

   Compiling FriCAS source code from file 
      /var/lib/zope2.10/instance/axiom-wiki/var/LatexWiki/4749327676615086707-25px001.spad
      using old system compiler.
   SUMMPACK abbreviates package SummPack 
------------------------------------------------------------------------
   initializing NRLIB SUMMPACK for SummPack 
   compiling into NRLIB SUMMPACK 
   compiling exported coeffMatrix : (PositiveInteger,PositiveInteger) -> Matrix Expression Integer
Time: 0.06 SEC.

   compiling exported funcMatrix : ((Expression Integer,Expression Integer) -> Expression Integer,PositiveInteger,PositiveInteger) -> Matrix Expression Integer
Time: 0.02 SEC.

   compiling exported sumAndNormalize : (Matrix Expression Integer,Matrix Expression Integer) -> Expression Integer
Time: 0.05 SEC.

   compiling exported retractNumeratorToPolyInt : Expression Integer -> Polynomial Integer
Time: 0.01 SEC.

   compiling exported findCoeffs : (Polynomial Integer,Matrix Expression Integer,Symbol) -> List Equation Fraction Polynomial Integer
Time: 0.03 SEC.

   compiling local convToFPI : Matrix Expression Integer -> Matrix Fraction Polynomial Integer
Time: 0.02 SEC.

   compiling exported formalExpr : (BasicOperator,Matrix Expression Integer,PositiveInteger,PositiveInteger) -> Expression Integer
Time: 0.04 SEC.

   compiling exported celine : ((Expression Integer,Expression Integer) -> Expression Integer,PositiveInteger,PositiveInteger) -> Expression Integer
Time: 0.02 SEC.

   compiling exported celine2 : ((Expression Integer,Expression Integer) -> Expression Integer,BasicOperator,PositiveInteger,PositiveInteger) -> Expression Integer
Time: 0.01 SEC.

(time taken in buildFunctor:  0)

;;;     ***       |SummPack| REDEFINED

;;;     ***       |SummPack| REDEFINED
Time: 0 SEC.

   Warnings: 
      [1] sumAndNormalize: not known that (AlgebraicallyClosedField) is of mode (CATEGORY domain (IF (has (Integer) (IntegralDomain)) (PROGN (ATTRIBUTE (AlgebraicallyClosedFunctionSpace (Integer))) (ATTRIBUTE (TranscendentalFunctionCategory)) (ATTRIBUTE (CombinatorialOpsCategory)) (ATTRIBUTE (LiouvillianFunctionCategory)) (ATTRIBUTE (SpecialFunctionCategory)) (SIGNATURE reduce ($ $)) (SIGNATURE number? ((Boolean) $)) (IF (has (Integer) (PolynomialFactorizationExplicit)) (ATTRIBUTE (PolynomialFactorizationExplicit)) noBranch) (SIGNATURE setSimplifyDenomsFlag ((Boolean) (Boolean))) (SIGNATURE getSimplifyDenomsFlag ((Boolean)))) noBranch))
      [2] sumAndNormalize: not known that (TranscendentalFunctionCategory) is of mode (CATEGORY domain (IF (has (Integer) (IntegralDomain)) (PROGN (ATTRIBUTE (AlgebraicallyClosedFunctionSpace (Integer))) (ATTRIBUTE (TranscendentalFunctionCategory)) (ATTRIBUTE (CombinatorialOpsCategory)) (ATTRIBUTE (LiouvillianFunctionCategory)) (ATTRIBUTE (SpecialFunctionCategory)) (SIGNATURE reduce ($ $)) (SIGNATURE number? ((Boolean) $)) (IF (has (Integer) (PolynomialFactorizationExplicit)) (ATTRIBUTE (PolynomialFactorizationExplicit)) noBranch) (SIGNATURE setSimplifyDenomsFlag ((Boolean) (Boolean))) (SIGNATURE getSimplifyDenomsFlag ((Boolean)))) noBranch))

   Cumulative Statistics for Constructor SummPack
      Time: 0.26 seconds

   finalizing NRLIB SUMMPACK 
   Processing SummPack for Browser database:
--->-->SummPack(constructor): Not documented!!!!
--->-->SummPack((coeffMatrix ((Matrix (Expression (Integer))) (PositiveInteger) (PositiveInteger)))): Not documented!!!!
--->-->SummPack((funcMatrix ((Matrix (Expression (Integer))) (Mapping (Expression (Integer)) (Expression (Integer)) (Expression (Integer))) (PositiveInteger) (PositiveInteger)))): Not documented!!!!
--->-->SummPack((sumAndNormalize ((Expression (Integer)) (Matrix (Expression (Integer))) (Matrix (Expression (Integer)))))): Not documented!!!!
--->-->SummPack((retractNumeratorToPolyInt ((Polynomial (Integer)) (Expression (Integer))))): Not documented!!!!
--->-->SummPack((findCoeffs ((List (Equation (Fraction (Polynomial (Integer))))) (Polynomial (Integer)) (Matrix (Expression (Integer))) (Symbol)))): Not documented!!!!
--->-->SummPack((formalExpr ((Expression (Integer)) (BasicOperator) (Matrix (Expression (Integer))) (PositiveInteger) (PositiveInteger)))): Not documented!!!!
--->-->SummPack((celine ((Expression (Integer)) (Mapping (Expression (Integer)) (Expression (Integer)) (Expression (Integer))) (PositiveInteger) (PositiveInteger)))): Not documented!!!!
--->-->SummPack((celine2 ((Expression (Integer)) (Mapping (Expression (Integer)) (Expression (Integer)) (Expression (Integer))) (BasicOperator) (PositiveInteger) (PositiveInteger)))): Not documented!!!!
--->-->SummPack(): Missing Description
; compiling file "/var/aw/var/LatexWiki/SUMMPACK.NRLIB/SUMMPACK.lsp" (written 04 APR 2022 08:22:36 PM):

; /var/aw/var/LatexWiki/SUMMPACK.NRLIB/SUMMPACK.fasl written
; compilation finished in 0:00:00.075
------------------------------------------------------------------------
   SummPack is now explicitly exposed in frame initial 
   SummPack will be automatically loaded when needed from 
      /var/aw/var/LatexWiki/SUMMPACK.NRLIB/SUMMPACK

Test different flavours

fricas

--)co sumpack
X==>EXPR INT

Type: Void

fricas

F:=(n:X,k:X):X+->k*binomial(n,k)

$\label{eq1}\mbox{theMap (...)}$

(1)

Type: ((Expression(Integer), Expression(Integer)) -> Expression(Integer))

fricas

cm:=coeffMatrix(1,1)

$\label{eq2}\left[ \begin{array}{cc} \%A & \%B \ \%C & \%D$

(2)

Type: Matrix(Expression(Integer))

fricas

fm:=funcMatrix(F,1,1)

$\label{eq3}\left[ \begin{array}{cc} 1 &{{{\left(k - 1 \right)}\ {\hbox{\axiomType{BINOMIAL}\ } \left({n , \:{k - 1}}\right)}}\over{k \ {\hbox{\axiomType{BINOMIAL}\ } \left({n , \: k}\right)}}} \ {{\hbox{\axiomType{BINOMIAL}\ } \left({{n - 1}, \: k}\right)}\over{\hbox{\axiomType{BINOMIAL}\ } \left({n , \: k}\right)}}&{{{\left(k - 1 \right)}\ {\hbox{\axiomType{BINOMIAL}\ } \left({{n - 1}, \:{k - 1}}\right)}}\over{k \ {\hbox{\axiomType{BINOMIAL}\ } \left({n , \: k}\right)}}}$

(3)

Type: Matrix(Expression(Integer))

fricas

san:=sumAndNormalize(cm,fm)

$\label{eq4}{\left( \begin{array}{@{}l} \displaystyle {{\left(\%C + \%A \right)}\ {{n}^{2}}}+{{\left({ \begin{array}{@{}l} \displaystyle {{\left({ \begin{array}{@{}l} \displaystyle \%D -{2 \ \%C}+ \%B - \ \ \displaystyle \%A$

(4)

Type: Expression(Integer)

fricas

p:=retractNumeratorToPolyInt(san)

$\label{eq5}\begin{array}{@{}l} \displaystyle {{\left(\%C + \%A \right)}\ {{n}^{2}}}+{{\left({ \begin{array}{@{}l} \displaystyle {{\left(\%D -{2 \ \%C}+ \%B - \%A \right)}\ k}- \%D + \ \ \displaystyle \%C - \%B + \%A$

(5)

Type: Polynomial(Integer)

fricas

--
d:=degree(p,k)

$\label{eq6}2$

(6)

Type: PositiveInteger?

fricas

eqs:=[coefficient(p,k,l)=0 for l in 0..d]

$\label{eq7}\begin{array}{@{}l} \displaystyle \left[{{{{\left(\%C + \%A \right)}\ {{n}^{2}}}+{{\left(- \%D + \%C - \%B + \%A \right)}\ n}- \%D}= 0}, \: \right. \ \ \displaystyle \left.{{{{\left(\%D -{2 \ \%C}+ \%B - \%A \right)}\ n}+{2 \ \%D}- \%C}= 0}, \: \right. \ \ \displaystyle \left.{{- \%D + \%C}= 0}\right]$

(7)

Type: List(Equation(Polynomial(Integer)))

fricas

v:=members cm

$\label{eq8}\left[ \%A , \: \%B , \: \%C , \: \%D \right]$

(8)

Type: List(Expression(Integer))

fricas

vv:=variables(v)

$\label{eq9}\left[ \%A , \: \%B , \: \%C , \: \%D \right]$

(9)

Type: List(Symbol)

fricas

x:=[s::Symbol for s in v]

$\label{eq10}\left[ \%A , \: \%B , \: \%C , \: \%D \right]$

(10)

Type: List(Symbol)

fricas

sol:=solve(eqs,x) -- check #sol=1

$\label{eq11}\left[{\left[{\%A ={{-{\%F \ n}+ \%F}\over n}}, \:{\%B = 0}, \:{\%C = \%F}, \:{\%D = \%F}\right]}\right]$

(11)

Type: List(List(Equation(Fraction(Polynomial(Integer)))))

fricas

e:=findCoeffs(p,cm,k)

$\label{eq12}\left[{\%A ={{-{\%G \ n}+ \%G}\over n}}, \:{\%B = 0}, \:{\%C = \%G}, \:{\%D = \%G}\right]$

(12)

Type: List(Equation(Fraction(Polynomial(Integer))))

fricas

ss:=eval(cm,sol.1)

$\label{eq13}\left[ \begin{array}{cc} {{-{\%F \ n}+ \%F}\over n}& 0 \ \%F & \%F$

(13)

Type: Matrix(Expression(Integer))

fricas

--
G:=operator 'G

$\label{eq14}G$

(14)

Type: BasicOperator?

fricas

I:=J:=1

$\label{eq15}1$

(15)

Type: PositiveInteger?

fricas

g:Matrix(X):=matrix [[G(n-j,k-i) for i in 0..I] for j in 0..J]

$\label{eq16}\left[ \begin{array}{cc} {G \left({n , \: k}\right)}&{G \left({n , \:{k - 1}}\right)} \ {G \left({{n - 1}, \: k}\right)}&{G \left({{n - 1}, \:{k - 1}}\right)}$

(16)

Type: Matrix(Expression(Integer))

fricas

sf:Matrix X:=ss*transpose(g)

$\label{eq17}\left[ \begin{array}{cc} {{{\left(-{\%F \ n}+ \%F \right)}\ {G \left({n , \: k}\right)}}\over n}&{{{\left(-{\%F \ n}+ \%F \right)}\ {G \left({{n - 1}, \: k}\right)}}\over n} \ {{\%F \ {G \left({n , \: k}\right)}}+{\%F \ {G \left({n , \:{k - 1}}\right)}}}&{{\%F \ {G \left({{n - 1}, \: k}\right)}}+{\%F \ {G \left({{n - 1}, \:{k - 1}}\right)}}}$

(17)

Type: Matrix(Expression(Integer))

fricas

res:X:=reduce(_+,[sf(i,i) for i in 1..I+1])

$\label{eq18}{\left( \begin{array}{@{}l} \displaystyle {{\left(-{\%F \ n}+ \%F \right)}\ {G \left({n , \: k}\right)}}+{\%F \ n \ {G \left({{n - 1}, \: k}\right)}}+ \ \ \displaystyle {\%F \ n \ {G \left({{n - 1}, \:{k - 1}}\right)}}$

(18)

Type: Expression(Integer)

fricas

fex:=formalExpr(G,cm,I,J)

$\label{eq19}\begin{array}{@{}l} \displaystyle {\%A \ {G \left({n , \: k}\right)}}+{\%B \ {G \left({n , \:{k - 1}}\right)}}+{\%C \ {G \left({{n - 1}, \: k}\right)}}+ \ \ \displaystyle {\%D \ {G \left({{n - 1}, \:{k - 1}}\right)}}$

(19)

Type: Expression(Integer)

fricas

H:=(n:X,k:X):X+->binomial(n,k)

$\label{eq20}\mbox{theMap (...)}$

(20)

Type: ((Expression(Integer), Expression(Integer)) -> Expression(Integer))

fricas

celine(H,1,1)

$\label{eq21}-{\%L \ {G \left({n , \: k}\right)}}+{\%L \ {G \left({{n - 1}, \: k}\right)}}+{\%L \ {G \left({{n - 1}, \:{k - 1}}\right)}}$

(21)

Type: Expression(Integer)

fricas

Q:=(n:X,k:X):X+->binomial(n,k)^2

$\label{eq22}\mbox{theMap (...)}$

(22)

Type: ((Expression(Integer), Expression(Integer)) -> Expression(Integer))

fricas

celine(Q,2,2)

$\label{eq23}{\left( \begin{array}{@{}l} \displaystyle {\%V \ n \ {G \left({n , \: k}\right)}}+{{\left(-{2 \ \%V \ n}+ \%V \right)}\ {G \left({{n - 1}, \: k}\right)}}+ \ \ \displaystyle {{\left(-{2 \ \%V \ n}+ \%V \right)}\ {G \left({{n - 1}, \:{k - 1}}\right)}}+ \ \ \displaystyle {{\left({\%V \ n}- \%V \right)}\ {G \left({{n - 2}, \: k}\right)}}+ \ \ \displaystyle {{\left(-{2 \ \%V \ n}+{2 \ \%V}\right)}\ {G \left({{n - 2}, \:{k - 1}}\right)}}+ \ \ \displaystyle {{\left({\%V \ n}- \%V \right)}\ {G \left({{n - 2}, \:{k - 2}}\right)}}$

(23)

Type: Expression(Integer)

fricas

R:=(n:X,k:X):X+->(-1)^k*factorial(n)*'x::X^k/(factorial(k)^2*factorial(n-k))

$\label{eq24}\mbox{theMap (...)}$

(24)

Type: ((Expression(Integer), Expression(Integer)) -> Expression(Integer))

fricas

RC:=celine(R,2,2)

$\label{eq25}{\left( \begin{array}{@{}l} \displaystyle -{\%BH \ {{n}^{2}}\ {G \left({n , \: k}\right)}}+ \ \ \displaystyle {{\left({2 \ \%BG \ {{n}^{2}}}-{\%BG \ n}\right)}\ {G \left({n , \:{k - 1}}\right)}}+ \ \ \displaystyle {{\left({2 \ \%BH \ {{n}^{2}}}-{\%BH \ n}\right)}\ {G \left({{n - 1}, \: k}\right)}}+ \ \ \displaystyle {{\left({ \begin{array}{@{}l} \displaystyle -{\%BH \ n \ x}-{4 \ \%BG \ {{n}^{2}}}+ \ \ \displaystyle {4 \ \%BG \ n}- \%BG$

(25)

Type: Expression(Integer)

fricas

ch:=n*R(n,k-1)+(-2*n+1)*R(n-1,k-1)+'x::X*R(n-1,k-2)+(n-1)*R(n-2,k-1)

$\label{eq26}{\left( \begin{array}{@{}l} \displaystyle { \begin{array}{@{}l} \displaystyle {\left({ \begin{array}{@{}l} \displaystyle {n \ {{{\left(k - 2 \right)}!}^{2}}\ {{\left(n - k - 1 \right)}!}\ {{\left(n - k \right)}!}\ {n !}}+ \ \ \displaystyle {{\left(-{2 \ n}+ 1 \right)}\ {{{\left(k - 2 \right)}!}^{2}}\ {{\left(n - k - 1 \right)}!}\ {{\left(n - k + 1 \right)}!}\ {{\left(n - 1 \right)}!}}+ \ \ \displaystyle {{\left(n - 1 \right)}\ {{{\left(k - 2 \right)}!}^{2}}\ {{\left(n - k \right)}!}\ {{\left(n - k + 1 \right)}!}\ {{\left(n - 2 \right)}!}}$

(26)

Type: Expression(Integer)

fricas

normalize(ch)

$\label{eq27}0$

(27)

Type: Expression(Integer)

fricas

celine((n,k)+->n*k,1,1)

$\label{eq28}{\left( \begin{array}{@{}l} \displaystyle {{\left(-{\%BM \ n}+ \%BM \right)}\ {G \left({n , \: k}\right)}}+ \ \ \displaystyle {{\left(-{\%BN \ n}+ \%BN \right)}\ {G \left({n , \:{k - 1}}\right)}}+ \ \ \displaystyle {\%BM \ n \ {G \left({{n - 1}, \: k}\right)}}+{\%BN \ n \ {G \left({{n - 1}, \:{k - 1}}\right)}}$

(28)

Type: Expression(Integer)

fricas

-- [13] (D,List(Equation(D2))) -> D from D
--            if D has EVALAB(D2) and D2 has SETCAT
-- [32] (Fraction(Polynomial(D3)),List(Equation(Fraction(Polynomial(D3)
--            )))) -> Fraction(Polynomial(D3))
--            from RationalFunction(D3) if D3 has INTDOM

SC:=celine((n,k)+->binomial(n,k)*binomial(2*k,k)*(-1/2)^k,2,2)

$\label{eq29}{\left( \begin{array}{@{}l} \displaystyle {{\left(- \%BY -{2 \ \%BX}\right)}\ n \ {G \left({n , \: k}\right)}}- \ \ \displaystyle {2 \ \%BY \ n \ {G \left({n , \:{k - 1}}\right)}}+ \ \ \displaystyle {{\left({{\left({2 \ \%BY}+{4 \ \%BX}\right)}\ n}- \%BY -{2 \ \%BX}\right)}\ {G \left({{n - 1}, \: k}\right)}}+ \ \ \displaystyle {{\left({ \begin{array}{@{}l} \displaystyle {{\left({2 \ \%BY}-{4 \ \%BX}\right)}\ n}- \%BY + \ \ \displaystyle {2 \ \%BX}$

(29)

Type: Expression(Integer)

fricas

SCA:=subst(SC,[%DA=a,%CZ=b])*4*(n-1)

$\label{eq30}\begin{array}{@{}l} \displaystyle {{\left(- \%BY -{2 \ \%BX}\right)}\ n \ {G \left({n , \: k}\right)}}-{2 \ \%BY \ n \ {G \left({n , \:{k - 1}}\right)}}+ \ \ \displaystyle {{\left({{\left({2 \ \%BY}+{4 \ \%BX}\right)}\ n}- \%BY -{2 \ \%BX}\right)}\ {G \left({{n - 1}, \: k}\right)}}+ \ \ \displaystyle {{\left({{\left({2 \ \%BY}-{4 \ \%BX}\right)}\ n}- \%BY +{2 \ \%BX}\right)}\ {G \left({{n - 1}, \:{k - 1}}\right)}}+ \ \ \displaystyle {{\left(-{4 \ \%BY \ n}+{2 \ \%BY}\right)}\ {G \left({{n - 1}, \:{k - 2}}\right)}}+ \ \ \displaystyle {{\left({{\left(- \%BY -{2 \ \%BX}\right)}\ n}+ \%BY +{2 \ \%BX}\right)}\ {G \left({{n - 2}, \: k}\right)}}+ \ \ \displaystyle {{\left({4 \ \%BX \ n}-{4 \ \%BX}\right)}\ {G \left({{n - 2}, \:{k - 1}}\right)}}+ \ \ \displaystyle {{\left({4 \ \%BY \ n}-{4 \ \%BY}\right)}\ {G \left({{n - 2}, \:{k - 2}}\right)}}$

(30)

Type: Expression(Integer)

fricas

--

SC2:=celine((n,k)+->binomial(n,k)*binomial(2*k,k)*(-2)^(n-k),2,2)

$\label{eq31}{\left( \begin{array}{@{}l} \displaystyle {{\left(- \%CJ -{2 \ \%CI}\right)}\ n \ {G \left({n , \: k}\right)}}-{2 \ \%CJ \ n \ {G \left({n , \:{k - 1}}\right)}}+ \ \ \displaystyle {{\left({ \begin{array}{@{}l} \displaystyle {{\left(-{4 \ \%CJ}-{8 \ \%CI}\right)}\ n}+{2 \ \%CJ}+ \ \ \displaystyle {4 \ \%CI}$

(31)

Type: Expression(Integer)

fricas

SCA2:=subst(SC2,[%CJ=a,%CI=b])*16*(n-1)

$\label{eq32}\begin{array}{@{}l} \displaystyle {{\left(-{2 \ b}- a \right)}\ n \ {G \left({n , \: k}\right)}}-{2 \ a \ n \ {G \left({n , \:{k - 1}}\right)}}+ \ \ \displaystyle {{\left({{\left(-{8 \ b}-{4 \ a}\right)}\ n}+{4 \ b}+{2 \ a}\right)}\ {G \left({{n - 1}, \: k}\right)}}+ \ \ \displaystyle {{\left({{\left({8 \ b}-{4 \ a}\right)}\ n}-{4 \ b}+{2 \ a}\right)}\ {G \left({{n - 1}, \:{k - 1}}\right)}}+ \ \ \displaystyle {{\left({8 \ a \ n}-{4 \ a}\right)}\ {G \left({{n - 1}, \:{k - 2}}\right)}}+ \ \ \displaystyle {{\left({{\left(-{8 \ b}-{4 \ a}\right)}\ n}+{8 \ b}+{4 \ a}\right)}\ {G \left({{n - 2}, \: k}\right)}}+ \ \ \displaystyle {{\left({{16}\ b \ n}-{{16}\ b}\right)}\ {G \left({{n - 2}, \:{k - 1}}\right)}}+ \ \ \displaystyle {{\left({{16}\ a \ n}-{{16}\ a}\right)}\ {G \left({{n - 2}, \:{k - 2}}\right)}}$

(32)

Type: Expression(Integer)

fricas

SCB2:=subst(SCA2,[a=1,b=0])

$\label{eq33}\begin{array}{@{}l} \displaystyle -{n \ {G \left({n , \: k}\right)}}-{2 \ n \ {G \left({n , \:{k - 1}}\right)}}+{{\left(-{4 \ n}+ 2 \right)}\ {G \left({{n - 1}, \: k}\right)}}+ \ \ \displaystyle {{\left(-{4 \ n}+ 2 \right)}\ {G \left({{n - 1}, \:{k - 1}}\right)}}+{{\left({8 \ n}- 4 \right)}\ {G \left({{n - 1}, \:{k - 2}}\right)}}+ \ \ \displaystyle {{\left(-{4 \ n}+ 4 \right)}\ {G \left({{n - 2}, \: k}\right)}}+{{\left({{16}\ n}-{16}\right)}\ {G \left({{n - 2}, \:{k - 2}}\right)}}$

(33)

Type: Expression(Integer)

fricas

-- Example 4.3.3 of A=B (subst to get same result)

SCB3:=subst(SCA2,[a=-2*b])

$\label{eq34}\begin{array}{@{}l} \displaystyle {4 \ b \ n \ {G \left({n , \:{k - 1}}\right)}}+{{\left({{16}\ b \ n}-{8 \ b}\right)}\ {G \left({{n - 1}, \:{k - 1}}\right)}}+ \ \ \displaystyle {{\left(-{{16}\ b \ n}+{8 \ b}\right)}\ {G \left({{n - 1}, \:{k - 2}}\right)}}+ \ \ \displaystyle {{\left({{16}\ b \ n}-{{16}\ b}\right)}\ {G \left({{n - 2}, \:{k - 1}}\right)}}+ \ \ \displaystyle {{\left(-{{32}\ b \ n}+{{32}\ b}\right)}\ {G \left({{n - 2}, \:{k - 2}}\right)}}$

(34)

Type: Expression(Integer)

fricas

SCB3:=subst(%,[b=1/4])

$\label{eq35}\begin{array}{@{}l} \displaystyle {n \ {G \left({n , \:{k - 1}}\right)}}+{{\left({4 \ n}- 2 \right)}\ {G \left({{n - 1}, \:{k - 1}}\right)}}+ \ \ \displaystyle {{\left(-{4 \ n}+ 2 \right)}\ {G \left({{n - 1}, \:{k - 2}}\right)}}+{{\left({4 \ n}- 4 \right)}\ {G \left({{n - 2}, \:{k - 1}}\right)}}+ \ \ \displaystyle {{\left(-{8 \ n}+ 8 \right)}\ {G \left({{n - 2}, \:{k - 2}}\right)}}$

(35)

Type: Expression(Integer)

fricas

SCB3:=subst(%,[k=k+1])

$\label{eq36}\begin{array}{@{}l} \displaystyle {n \ {G \left({n , \: k}\right)}}+{{\left({4 \ n}- 2 \right)}\ {G \left({{n - 1}, \: k}\right)}}+ \ \ \displaystyle {{\left(-{4 \ n}+ 2 \right)}\ {G \left({{n - 1}, \:{k - 1}}\right)}}+{{\left({4 \ n}- 4 \right)}\ {G \left({{n - 2}, \: k}\right)}}+ \ \ \displaystyle {{\left(-{8 \ n}+ 8 \right)}\ {G \left({{n - 2}, \:{k - 1}}\right)}}$

(36)

Type: Expression(Integer)