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HyperG

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Editor: pagani Time: 2022/09/14 20:05:57 GMT+0
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changed:
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\begin{spad}
)abbrev package HYPGEOM HyperGeometric
HyperGeometric(): Exports == Implementation where
  R ==> Fraction Integer  -- better than Integer for this purpose
  X ==> Expression R 
  NNI ==> NonNegativeInteger
  SMP ==> SparseMultivariatePolynomial(R, Kernel X) -- if R has IntegralDomain
  SUP ==> SparseUnivariatePolynomial SMP 
  
  EFSP   ==> ElementaryFunctionStructurePackage(R,X)
  FACSUP ==> Factored SUP
  FACREC ==> Record(factor:SUP, exponent:NNI)
  
  HYPER  ==> Record(ap:List X, bq:List X, fac:X)
  PARAM  ==> Record(ab:List SUP, const:List SUP)

  Exports == with
    factoredForm : (X->X) -> Fraction(FACSUP)
    getParameters : FACSUP -> PARAM
    convert : SUP -> X
    isPoly? :(X->X) -> Boolean
    isQuoRat? :(X->X) -> Boolean
    pFq : (X->X) -> HYPER
    pFq0 : (X->X) -> HYPER
    construct: (HYPER,Symbol) -> X
    display : HYPER -> OutputForm
    hyperLookup : HYPER -> X

  Implementation == add
  
    factoredForm(f:X->X):Fraction(FACSUP) ==
      ns:X:=new()$Symbol::X
      q:=f(ns+1)/f(ns)
      nq:=normalize(q)$EFSP
      dnq:=denom nq
      nnq:=numer nq
      unnq:=univariate(nnq,first tower(ns))
      udnq:=univariate(dnq,first tower(ns))
      funnq:=factor unnq
      fudnq:=factor udnq
      funnq/fudnq
      

    getParameters x ==
      qm:=create()$SingletonAsOrderedSet
      r:List FACREC:=factors x
      l1:List SUP:=[]
      l2:List SUP:=[]
      for s in r repeat
        p:SUP:=s.factor
        if variables p = [qm]
        then
          q1:=p-qm::SUP
          for i in 1..s.exponent repeat
            l1:=cons(q1,l1)
        else
          q2:=p 
          for i in 1..s.exponent repeat
            l2:=cons(q2,l2)
      return [l1,l2]$PARAM


    convert(x:SUP):X ==
      k:Kernel(X):=first tower('_%::X)
      y:SMP:=multivariate(x,k)$SMP
      coerce(y)$X
      
    pFq(f:X->X):HYPER ==
      ff:=factoredForm(f)
      gpn:=getParameters(numer ff).ab
      gpd:=getParameters(denom ff).ab
      [[convert t for t in gpn],[convert t for t in gpd],1$X]$HYPER
      -- 1 in bq => remove it otherwise add 1 to aq
      -- coeff and f(0) todo
      -- pFq(k+->cos(k)) , pre-filter f(n+1)/f(n) rational?

    isPoly?(f:X->X):Boolean ==
      nsym:Symbol:=new()$Symbol
      ns:X:=nsym::X
      q:=f(ns)
      D(q,nsym)$X=0 => true
      nq:=normalize(q)$EFSP
      dnq:=denom nq
      nnq:=numer nq
      unnq:=univariate(nnq,first tower(ns))
      udnq:=univariate(dnq,first tower(ns))
      empty?(variables unnq) => false
      not empty?(variables udnq) => false 
      true       
      
    isQuoRat?(f:X->X):Boolean ==
      isPoly?(f) => true  -- accelerates e.g. k^1234567
      nsym:Symbol:=new()$Symbol
      ns:X:=nsym::X
      q:=f(ns+1)/f(ns)
      D(q,nsym)$X=0 => true
      nq:=normalize(q)$EFSP
      dnq:=denom nq
      nnq:=numer nq
      unnq:=univariate(nnq,first tower(ns))
      udnq:=univariate(dnq,first tower(ns))
      empty?(variables unnq) and empty?(variables udnq) => false
      true 

    buildConst(x:List SUP):X ==
      empty?(x) => 1$X
      p:SUP:=reduce(_*,x)
      convert p

    pFq0(f:X->X):HYPER ==
      ff:=factoredForm(f)
      gpn:=getParameters(numer ff)
      gpd:=getParameters(denom ff)
      a:=[convert t for t in gpn.ab]
      b:=[convert t for t in gpd.ab]
      if member?(1$X,b) then
        b:=delete(b,position(1$X,b))
      else
        a:=cons(1$X,a)
      c:=convert unit(numer ff)
      ca:=buildConst(gpn.const)
      cb:=buildConst(gpd.const)
      [a,b,c*ca*cb]$HYPER
 
    construct(x,s) ==
      v:=s::X
      a:=x.ap
      b:=x.bq
      c:=x.fac
      av:=[paren(v+t) for t in a]
      bv:=[paren(v+t) for t in b]
      num:=1$X
      den:=1$X
      if not empty? av then
        num:=reduce(_*,av)
      if not empty? bv then
        den:=reduce(_*,bv)    
      c*num/den/(v+1$X)

    display(x:HYPER):OutputForm ==
      OF ==> OutputForm
      a:=[s::OF for s in x.ap]
      b:=[s::OF for s in x.bq]
      c:=(x.fac)::OF
      p:=(#a)::OF
      q:=(#b)::OF
      A:=sub(presub('F::OF,p),q)
      if #a < #b then
        a:=append(a,['*::OF for i in 1..#b- #a])
      else
        b:=append(b,['*::OF for i in 1..#a- #b])
      --B:=binomial(blankSeparate a, blankSeparate b)
      B:=blankSeparate [A,matrix [a,b],bracket c]
      --hconcat(A,B)

    hyperLookup(x:HYPER):X ==
      a:=x.ap
      b:=x.bq
      c:=x.fac
      p:= #a
      q:= #b
      if p=1 then
        if q=0 then
          return (1-c)^(-a.1)
      0$X

\end{spad}
\begin{axiom}
)set output algebra on
R ==> Fraction Integer  -- better than Integer for this purpose
X ==> Expression R 
NNI ==> NonNegativeInteger
SMP ==> SparseMultivariatePolynomial(R, Kernel X) -- if R has IntegralDomain
SUP ==> SparseUnivariatePolynomial SMP 
  
EFSP   ==> ElementaryFunctionStructurePackage(R,X)
FACSUP ==> Factored SUP
FACREC ==> Record(factor:SUP, exponent:NNI)
HYPER  ==> Record(ap:List X, bq:List X, fac:X)


T1(n) == 1/(2*n-1)/factorial(2*n+1)
T2(k) == binomial(n,k)*(-1)^k/factorial(k)
T3(k) == (-1)^k*(x/2)^(2*k+p)/factorial(k)/factorial(k+p)
T4(k) == binomial(n,k)^2*binomial(n+k,k)^2 
T5(n) == binomial(n,k)^2*binomial(n+k,k)^2  
T6(k) == n^k
T7(k) == factorial(k)
T8(k) == factorial(2*k+7)/factorial(k-3)
T9(k) == (k^2-1)*factorial(3*k+1)/(2*k+7)/factorial(k+3)
T10(k) == z^k/factorial(k)
T11(k) == 2^k/factorial(k)^2
T12(k) == binomial(n,k)

ff1:=factoredForm(T1)$HYPGEOM
ff2:=factoredForm(T2)$HYPGEOM
ff3:=factoredForm(T3)$HYPGEOM
ff4:=factoredForm(T4)$HYPGEOM
ff5:=factoredForm(T5)$HYPGEOM
ff6:=factoredForm(T6)$HYPGEOM
ff7:=factoredForm(T7)$HYPGEOM
ff8:=factoredForm(T8)$HYPGEOM
ff9:=factoredForm(T9)$HYPGEOM
ff10:=factoredForm(T10)$HYPGEOM
ff11:=factoredForm(T11)$HYPGEOM
ff12:=factoredForm(T12)$HYPGEOM


for v in [ff1,ff2,ff3,ff4,ff5,ff6,ff7,ff8,ff9,ff10,ff11,ff12] repeat
  output getParameters(numer v)
  output getParameters(denom v)
  

gp2:=getParameters(numer ff2).ab
gp3:=getParameters(denom ff3).ab

----
mp3:=multivariate(gp3.1,tower(new()$Symbol).1)$SMP::X
mp2:=multivariate(gp2.1,tower(new()$Symbol).1)$SMP::X
--- List(Expression(Fraction(Integer)))
[convert(v) for v in gp3]
[convert(v) for v in gp2]

-- near the goal ;)
pFq(T1)
pFq(T2)
pFq(T3)
pFq(T4)
pFq(T5)
pFq(T6)
pFq(T7)
pFq(T8)
pFq(T9)
pFq(k+->sin(k))

--------------


-- simple test
isQuoRat?(k+->sin k)
isQuoRat?(k+->sin asin k^4)
isQuoRat?(k+->log exp  k^4)
isQuoRat?(k+->exp  k^4)
isQuoRat?(k+->x^k) 

un:=unit numer ff1 
ud:=unit denom ff1
unud:=un/ud

pFq0(T6) 
T3(k)
T3C:=(k+x)^2/((k+p+1)*(k+1))
pFq0(T3)
ff3 
--
pFq0(T10) 
pFq0(T11) 

hy:=pFq0(T3)
hyc:=construct(hy,z)$HYPGEOM 
ff3
display(hy)$HYPGEOM
display(pFq0(T1))$HYPGEOM
display(pFq0(T2))$HYPGEOM
display(pFq0(T3))$HYPGEOM
display(pFq0(T4))$HYPGEOM
hyperLookup(pFq0 T12)


Q:=construct(pFq0(T1),Z)$HYPGEOM

T1(Z+1)/T1(Z)- distribute Q
normalize $

display(pFq0(T1))$HYPGEOM
display(pFq0(T2))$HYPGEOM
display(pFq0(T3))$HYPGEOM
display(pFq0(T4))$HYPGEOM
display(pFq0(T5))$HYPGEOM
display(pFq0(T6))$HYPGEOM
display(pFq0(T7))$HYPGEOM
display(pFq0(T8))$HYPGEOM
display(pFq0(T9))$HYPGEOM
display(pFq0(T10))$HYPGEOM
display(pFq0(T11))$HYPGEOM
display(pFq0(T12))$HYPGEOM


S:=sum(binomial(n,k),k=0..n)
hyperLookup(pFq0 T12)

\end{axiom}

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(1) -> <spad>

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)abbrev package HYPGEOM HyperGeometric
HyperGeometric(): Exports == Implementation where
  R ==> Fraction Integer  -- better than Integer for this purpose
  X ==> Expression R 
  NNI ==> NonNegativeInteger
  SMP ==> SparseMultivariatePolynomial(R, Kernel X) -- if R has IntegralDomain
  SUP ==> SparseUnivariatePolynomial SMP 

  EFSP   ==> ElementaryFunctionStructurePackage(R,X)
  FACSUP ==> Factored SUP
  FACREC ==> Record(factor:SUP, exponent:NNI)

  HYPER  ==> Record(ap:List X, bq:List X, fac:X)
  PARAM  ==> Record(ab:List SUP, const:List SUP)

  Exports == with
    factoredForm : (X->X) -> Fraction(FACSUP)
    getParameters : FACSUP -> PARAM
    convert : SUP -> X
    isPoly? :(X->X) -> Boolean
    isQuoRat? :(X->X) -> Boolean
    pFq : (X->X) -> HYPER
    pFq0 : (X->X) -> HYPER
    construct: (HYPER,Symbol) -> X
    display : HYPER -> OutputForm
    hyperLookup : HYPER -> X

  Implementation == add

    factoredForm(f:X->X):Fraction(FACSUP) ==
      ns:X:=new()$Symbol::X
      q:=f(ns+1)/f(ns)
      nq:=normalize(q)$EFSP
      dnq:=denom nq
      nnq:=numer nq
      unnq:=univariate(nnq,first tower(ns))
      udnq:=univariate(dnq,first tower(ns))
      funnq:=factor unnq
      fudnq:=factor udnq
      funnq/fudnq

    getParameters x ==
      qm:=create()$SingletonAsOrderedSet
      r:List FACREC:=factors x
      l1:List SUP:=[]
      l2:List SUP:=[]
      for s in r repeat
        p:SUP:=s.factor
        if variables p = [qm]
        then
          q1:=p-qm::SUP
          for i in 1..s.exponent repeat
            l1:=cons(q1,l1)
        else
          q2:=p 
          for i in 1..s.exponent repeat
            l2:=cons(q2,l2)
      return [l1,l2]$PARAM

    convert(x:SUP):X ==
      k:Kernel(X):=first tower('_%::X)
      y:SMP:=multivariate(x,k)$SMP
      coerce(y)$X

    pFq(f:X->X):HYPER ==
      ff:=factoredForm(f)
      gpn:=getParameters(numer ff).ab
      gpd:=getParameters(denom ff).ab
      [[convert t for t in gpn],[convert t for t in gpd],1$X]$HYPER
      -- 1 in bq => remove it otherwise add 1 to aq
      -- coeff and f(0) todo
      -- pFq(k+->cos(k)) , pre-filter f(n+1)/f(n) rational?

    isPoly?(f:X->X):Boolean ==
      nsym:Symbol:=new()$Symbol
      ns:X:=nsym::X
      q:=f(ns)
      D(q,nsym)$X=0 => true
      nq:=normalize(q)$EFSP
      dnq:=denom nq
      nnq:=numer nq
      unnq:=univariate(nnq,first tower(ns))
      udnq:=univariate(dnq,first tower(ns))
      empty?(variables unnq) => false
      not empty?(variables udnq) => false 
      true       

    isQuoRat?(f:X->X):Boolean ==
      isPoly?(f) => true  -- accelerates e.g. k^1234567
      nsym:Symbol:=new()$Symbol
      ns:X:=nsym::X
      q:=f(ns+1)/f(ns)
      D(q,nsym)$X=0 => true
      nq:=normalize(q)$EFSP
      dnq:=denom nq
      nnq:=numer nq
      unnq:=univariate(nnq,first tower(ns))
      udnq:=univariate(dnq,first tower(ns))
      empty?(variables unnq) and empty?(variables udnq) => false
      true 

    buildConst(x:List SUP):X ==
      empty?(x) => 1$X
      p:SUP:=reduce(_*,x)
      convert p

    pFq0(f:X->X):HYPER ==
      ff:=factoredForm(f)
      gpn:=getParameters(numer ff)
      gpd:=getParameters(denom ff)
      a:=[convert t for t in gpn.ab]
      b:=[convert t for t in gpd.ab]
      if member?(1$X,b) then
        b:=delete(b,position(1$X,b))
      else
        a:=cons(1$X,a)
      c:=convert unit(numer ff)
      ca:=buildConst(gpn.const)
      cb:=buildConst(gpd.const)
      [a,b,c*ca*cb]$HYPER

    construct(x,s) ==
      v:=s::X
      a:=x.ap
      b:=x.bq
      c:=x.fac
      av:=[paren(v+t) for t in a]
      bv:=[paren(v+t) for t in b]
      num:=1$X
      den:=1$X
      if not empty? av then
        num:=reduce(_*,av)
      if not empty? bv then
        den:=reduce(_*,bv)    
      c*num/den/(v+1$X)

    display(x:HYPER):OutputForm ==
      OF ==> OutputForm
      a:=[s::OF for s in x.ap]
      b:=[s::OF for s in x.bq]
      c:=(x.fac)::OF
      p:=(#a)::OF
      q:=(#b)::OF
      A:=sub(presub('F::OF,p),q)
      if #a < #b then
        a:=append(a,['*::OF for i in 1..#b- #a])
      else
        b:=append(b,['*::OF for i in 1..#a- #b])
      --B:=binomial(blankSeparate a, blankSeparate b)
      B:=blankSeparate [A,matrix [a,b],bracket c]
      --hconcat(A,B)

    hyperLookup(x:HYPER):X ==
      a:=x.ap
      b:=x.bq
      c:=x.fac
      p:= #a
      q:= #b
      if p=1 then
        if q=0 then
          return (1-c)^(-a.1)
      0$X</spad>

fricas

Compiling FriCAS source code from file 
      /var/lib/zope2.10/instance/axiom-wiki/var/LatexWiki/5800620949499819823-25px001.spad
      using old system compiler.
   HYPGEOM abbreviates package HyperGeometric 
------------------------------------------------------------------------
   initializing NRLIB HYPGEOM for HyperGeometric 
   compiling into NRLIB HYPGEOM 
   compiling exported factoredForm : Expression Fraction Integer -> Expression Fraction Integer -> Fraction Factored SparseUnivariatePolynomial SparseMultivariatePolynomial(Fraction Integer,Kernel Expression Fraction Integer)
Time: 0.13 SEC.

   compiling exported getParameters : Factored SparseUnivariatePolynomial SparseMultivariatePolynomial(Fraction Integer,Kernel Expression Fraction Integer) -> Record(ab: List SparseUnivariatePolynomial SparseMultivariatePolynomial(Fraction Integer,Kernel Expression Fraction Integer),const: List SparseUnivariatePolynomial SparseMultivariatePolynomial(Fraction Integer,Kernel Expression Fraction Integer))
Time: 0.03 SEC.

   compiling exported convert : SparseUnivariatePolynomial SparseMultivariatePolynomial(Fraction Integer,Kernel Expression Fraction Integer) -> Expression Fraction Integer
Time: 0.02 SEC.

   compiling exported pFq : Expression Fraction Integer -> Expression Fraction Integer -> Record(ap: List Expression Fraction Integer,bq: List Expression Fraction Integer,fac: Expression Fraction Integer)
Time: 0 SEC.

   compiling exported isPoly? : Expression Fraction Integer -> Expression Fraction Integer -> Boolean
Time: 0.01 SEC.

   compiling exported isQuoRat? : Expression Fraction Integer -> Expression Fraction Integer -> Boolean
Time: 0.04 SEC.

   compiling local buildConst : List SparseUnivariatePolynomial SparseMultivariatePolynomial(Fraction Integer,Kernel Expression Fraction Integer) -> Expression Fraction Integer
Time: 0.01 SEC.

   compiling exported pFq0 : Expression Fraction Integer -> Expression Fraction Integer -> Record(ap: List Expression Fraction Integer,bq: List Expression Fraction Integer,fac: Expression Fraction Integer)
Time: 0.06 SEC.

   compiling exported construct : (Record(ap: List Expression Fraction Integer,bq: List Expression Fraction Integer,fac: Expression Fraction Integer),Symbol) -> Expression Fraction Integer
Time: 0.02 SEC.

   compiling exported display : Record(ap: List Expression Fraction Integer,bq: List Expression Fraction Integer,fac: Expression Fraction Integer) -> OutputForm
   processing macro definition OF ==> OutputForm 
Time: 0.02 SEC.

   compiling exported hyperLookup : Record(ap: List Expression Fraction Integer,bq: List Expression Fraction Integer,fac: Expression Fraction Integer) -> Expression Fraction Integer
Time: 0.01 SEC.

(time taken in buildFunctor:  0)

;;;     ***       |HyperGeometric| REDEFINED

;;;     ***       |HyperGeometric| REDEFINED
Time: 0.01 SEC.

   Warnings: 
      [1] factoredForm: not known that (AlgebraicallyClosedField) is of mode (CATEGORY domain (IF (has (Fraction (Integer)) (IntegralDomain)) (PROGN (ATTRIBUTE (AlgebraicallyClosedFunctionSpace (Fraction (Integer)))) (ATTRIBUTE (TranscendentalFunctionCategory)) (ATTRIBUTE (CombinatorialOpsCategory)) (ATTRIBUTE (LiouvillianFunctionCategory)) (ATTRIBUTE (SpecialFunctionCategory)) (SIGNATURE reduce ($ $)) (SIGNATURE number? ((Boolean) $)) (IF (has (Fraction (Integer)) (PolynomialFactorizationExplicit)) (ATTRIBUTE (PolynomialFactorizationExplicit)) noBranch) (SIGNATURE setSimplifyDenomsFlag ((Boolean) (Boolean))) (SIGNATURE getSimplifyDenomsFlag ((Boolean)))) noBranch))
      [2] factoredForm: not known that (TranscendentalFunctionCategory) is of mode (CATEGORY domain (IF (has (Fraction (Integer)) (IntegralDomain)) (PROGN (ATTRIBUTE (AlgebraicallyClosedFunctionSpace (Fraction (Integer)))) (ATTRIBUTE (TranscendentalFunctionCategory)) (ATTRIBUTE (CombinatorialOpsCategory)) (ATTRIBUTE (LiouvillianFunctionCategory)) (ATTRIBUTE (SpecialFunctionCategory)) (SIGNATURE reduce ($ $)) (SIGNATURE number? ((Boolean) $)) (IF (has (Fraction (Integer)) (PolynomialFactorizationExplicit)) (ATTRIBUTE (PolynomialFactorizationExplicit)) noBranch) (SIGNATURE setSimplifyDenomsFlag ((Boolean) (Boolean))) (SIGNATURE getSimplifyDenomsFlag ((Boolean)))) noBranch))
      [3] getParameters:  exponent has no value
      [4] getParameters:  l1 has no value
      [5] getParameters:  l2 has no value
      [6] pFq0:  ab has no value
      [7] pFq0:  const has no value
      [8] construct:  ap has no value
      [9] construct:  bq has no value
      [10] construct:  fac has no value
      [11] display:  ap has no value
      [12] display:  bq has no value
      [13] display:  fac has no value
      [14] hyperLookup:  ap has no value
      [15] hyperLookup:  bq has no value
      [16] hyperLookup:  fac has no value

   Cumulative Statistics for Constructor HyperGeometric
      Time: 0.36 seconds

   finalizing NRLIB HYPGEOM 
   Processing HyperGeometric for Browser database:
--->-->HyperGeometric(constructor): Not documented!!!!
--->-->HyperGeometric((factoredForm ((Fraction (Factored (SparseUnivariatePolynomial (SparseMultivariatePolynomial (Fraction (Integer)) (Kernel (Expression (Fraction (Integer)))))))) (Mapping (Expression (Fraction (Integer))) (Expression (Fraction (Integer))))))): Not documented!!!!
--->-->HyperGeometric((getParameters ((Record (: ab (List (SparseUnivariatePolynomial (SparseMultivariatePolynomial (Fraction (Integer)) (Kernel (Expression (Fraction (Integer)))))))) (: const (List (SparseUnivariatePolynomial (SparseMultivariatePolynomial (Fraction (Integer)) (Kernel (Expression (Fraction (Integer))))))))) (Factored (SparseUnivariatePolynomial (SparseMultivariatePolynomial (Fraction (Integer)) (Kernel (Expression (Fraction (Integer)))))))))): Not documented!!!!
--->-->HyperGeometric((convert ((Expression (Fraction (Integer))) (SparseUnivariatePolynomial (SparseMultivariatePolynomial (Fraction (Integer)) (Kernel (Expression (Fraction (Integer))))))))): Not documented!!!!
--->-->HyperGeometric((isPoly? ((Boolean) (Mapping (Expression (Fraction (Integer))) (Expression (Fraction (Integer))))))): Not documented!!!!
--->-->HyperGeometric((isQuoRat? ((Boolean) (Mapping (Expression (Fraction (Integer))) (Expression (Fraction (Integer))))))): Not documented!!!!
--->-->HyperGeometric((pFq ((Record (: ap (List (Expression (Fraction (Integer))))) (: bq (List (Expression (Fraction (Integer))))) (: fac (Expression (Fraction (Integer))))) (Mapping (Expression (Fraction (Integer))) (Expression (Fraction (Integer))))))): Not documented!!!!
--->-->HyperGeometric((pFq0 ((Record (: ap (List (Expression (Fraction (Integer))))) (: bq (List (Expression (Fraction (Integer))))) (: fac (Expression (Fraction (Integer))))) (Mapping (Expression (Fraction (Integer))) (Expression (Fraction (Integer))))))): Not documented!!!!
--->-->HyperGeometric((construct ((Expression (Fraction (Integer))) (Record (: ap (List (Expression (Fraction (Integer))))) (: bq (List (Expression (Fraction (Integer))))) (: fac (Expression (Fraction (Integer))))) (Symbol)))): Not documented!!!!
--->-->HyperGeometric((display ((OutputForm) (Record (: ap (List (Expression (Fraction (Integer))))) (: bq (List (Expression (Fraction (Integer))))) (: fac (Expression (Fraction (Integer)))))))): Not documented!!!!
--->-->HyperGeometric((hyperLookup ((Expression (Fraction (Integer))) (Record (: ap (List (Expression (Fraction (Integer))))) (: bq (List (Expression (Fraction (Integer))))) (: fac (Expression (Fraction (Integer)))))))): Not documented!!!!
--->-->HyperGeometric(): Missing Description
; compiling file "/var/aw/var/LatexWiki/HYPGEOM.NRLIB/HYPGEOM.lsp" (written 14 SEP 2022 08:05:57 PM):

; /var/aw/var/LatexWiki/HYPGEOM.NRLIB/HYPGEOM.fasl written
; compilation finished in 0:00:00.118
------------------------------------------------------------------------
   HyperGeometric is now explicitly exposed in frame initial 
   HyperGeometric will be automatically loaded when needed from 
      /var/aw/var/LatexWiki/HYPGEOM.NRLIB/HYPGEOM

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)set output algebra on

R ==> Fraction Integer  -- better than Integer for this purpose

Type: Void

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X ==> Expression R

Type: Void

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NNI ==> NonNegativeInteger

Type: Void

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SMP ==> SparseMultivariatePolynomial(R, Kernel X) -- if R has IntegralDomain

Type: Void

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SUP ==> SparseUnivariatePolynomial SMP

Type: Void

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EFSP   ==> ElementaryFunctionStructurePackage(R,X)

Type: Void

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FACSUP ==> Factored SUP

Type: Void

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FACREC ==> Record(factor:SUP, exponent:NNI)

Type: Void

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HYPER  ==> Record(ap:List X, bq:List X, fac:X)

Type: Void

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T1(n) == 1/(2*n-1)/factorial(2*n+1)

Type: Void

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T2(k) == binomial(n,k)*(-1)^k/factorial(k)

Type: Void

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T3(k) == (-1)^k*(x/2)^(2*k+p)/factorial(k)/factorial(k+p)

Type: Void

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T4(k) == binomial(n,k)^2*binomial(n+k,k)^2

Type: Void

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T5(n) == binomial(n,k)^2*binomial(n+k,k)^2

Type: Void

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T6(k) == n^k

Type: Void

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T7(k) == factorial(k)

Type: Void

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T8(k) == factorial(2*k+7)/factorial(k-3)

Type: Void

fricas

T9(k) == (k^2-1)*factorial(3*k+1)/(2*k+7)/factorial(k+3)

Type: Void

fricas

T10(k) == z^k/factorial(k)

Type: Void

fricas

T11(k) == 2^k/factorial(k)^2

Type: Void

fricas

T12(k) == binomial(n,k)

Type: Void

fricas

ff1:=factoredForm(T1)$HYPGEOM

fricas

Compiling function T1 with type Expression(Fraction(Integer)) -> 
      Expression(Fraction(Integer)) 

               1      1
               - (? - -)
               4      2
   (22)  ---------------------
              1             3
         (? + -)(? + 1)(? + -)
              2             2

$\label{eq1}\frac{{\frac{1}{4}}\ {\left(? -{\frac{1}{2}}\right)}}{{\left(? +{\frac{1}{2}}\right)}\ {\left(? + 1 \right)}\ {\left(? +{\frac{3}{2}}\right)}}$

(1)

Type: Fraction(Factored(SparseUnivariatePolynomial?(SparseMultivariatePolynomial?(Fraction(Integer),Kernel(Expression(Fraction(Integer)))))))

fricas

ff2:=factoredForm(T2)$HYPGEOM

fricas

Compiling function T2 with type Expression(Fraction(Integer)) -> 
      Expression(Fraction(Integer)) 

           ? - n
   (23)  --------
                2
         (? + 1)

$\label{eq2}\frac{? - n}{{\left(? + 1 \right)}^{2}}$

(2)

Type: Fraction(Factored(SparseUnivariatePolynomial?(SparseMultivariatePolynomial?(Fraction(Integer),Kernel(Expression(Fraction(Integer)))))))

fricas

ff3:=factoredForm(T3)$HYPGEOM

fricas

Compiling function T3 with type Expression(Fraction(Integer)) -> 
      Expression(Fraction(Integer)) 

                  1  2
                  - x
                  4
   (24)  - ------------------
           (? + 1)(? + p + 1)

$\label{eq3}-{\frac{{\frac{1}{4}}\ {{x}^{2}}}{{\left(? + 1 \right)}\ {\left(? + p + 1 \right)}}}$

(3)

Type: Fraction(Factored(SparseUnivariatePolynomial?(SparseMultivariatePolynomial?(Fraction(Integer),Kernel(Expression(Fraction(Integer)))))))

fricas

ff4:=factoredForm(T4)$HYPGEOM

fricas

Compiling function T4 with type Expression(Fraction(Integer)) -> 
      Expression(Fraction(Integer)) 

                2           2
         (? - n) (? + n + 1)
   (25)  --------------------
                      4
               (? + 1)

$\label{eq4}\frac{{{\left(? - n \right)}^{2}}\ {{\left(? + n + 1 \right)}^{2}}}{{\left(? + 1 \right)}^{4}}$

(4)

Type: Fraction(Factored(SparseUnivariatePolynomial?(SparseMultivariatePolynomial?(Fraction(Integer),Kernel(Expression(Fraction(Integer)))))))

fricas

ff5:=factoredForm(T5)$HYPGEOM

fricas

Compiling function T5 with type Expression(Fraction(Integer)) -> 
      Expression(Fraction(Integer)) 

                    2
         (? + k + 1)
   (26)  ------------
                    2
         (? - k + 1)

$\label{eq5}\frac{{\left(? + k + 1 \right)}^{2}}{{\left(? - k + 1 \right)}^{2}}$

(5)

Type: Fraction(Factored(SparseUnivariatePolynomial?(SparseMultivariatePolynomial?(Fraction(Integer),Kernel(Expression(Fraction(Integer)))))))

fricas

ff6:=factoredForm(T6)$HYPGEOM

fricas

Compiling function T6 with type Expression(Fraction(Integer)) -> 
      Expression(Fraction(Integer)) 

   (27)  n

$\label{eq6}n$

(6)

Type: Fraction(Factored(SparseUnivariatePolynomial?(SparseMultivariatePolynomial?(Fraction(Integer),Kernel(Expression(Fraction(Integer)))))))

fricas

ff7:=factoredForm(T7)$HYPGEOM

fricas

Compiling function T7 with type Expression(Fraction(Integer)) -> 
      Expression(Fraction(Integer)) 

   (28)  ? + 1

$\label{eq7}? + 1$

(7)

Type: Fraction(Factored(SparseUnivariatePolynomial?(SparseMultivariatePolynomial?(Fraction(Integer),Kernel(Expression(Fraction(Integer)))))))

fricas

ff8:=factoredForm(T8)$HYPGEOM

fricas

Compiling function T8 with type Expression(Fraction(Integer)) -> 
      Expression(Fraction(Integer)) 

                       9
         4 (? + 4)(? + -)
                       2
   (29)  ----------------
               ? - 2

$\label{eq8}\frac{4 \ {\left(? + 4 \right)}\ {\left(? +{\frac{9}{2}}\right)}}{? - 2}$

(8)

Type: Fraction(Factored(SparseUnivariatePolynomial?(SparseMultivariatePolynomial?(Fraction(Integer),Kernel(Expression(Fraction(Integer)))))))

fricas

ff9:=factoredForm(T9)$HYPGEOM

fricas

Compiling function T9 with type Expression(Fraction(Integer)) -> 
      Expression(Fraction(Integer)) 

                  2      4             7
         27 ?(? + -)(? + -)(? + 2)(? + -)
                  3      3             2
   (30)  --------------------------------
                                  9
               (? - 1)(? + 4)(? + -)
                                  2

$\label{eq9}\frac{{27}\ ? \ {\left(? +{\frac{2}{3}}\right)}\ {\left(? +{\frac{4}{3}}\right)}\ {\left(? + 2 \right)}\ {\left(? +{\frac{7}{2}}\right)}}{{\left(? - 1 \right)}\ {\left(? + 4 \right)}\ {\left(? +{\frac{9}{2}}\right)}}$

(9)

Type: Fraction(Factored(SparseUnivariatePolynomial?(SparseMultivariatePolynomial?(Fraction(Integer),Kernel(Expression(Fraction(Integer)))))))

fricas

ff10:=factoredForm(T10)$HYPGEOM

fricas

Compiling function T10 with type Expression(Fraction(Integer)) -> 
      Expression(Fraction(Integer)) 

           z
   (31)  -----
         ? + 1

$\label{eq10}\frac{z}{? + 1}$

(10)

Type: Fraction(Factored(SparseUnivariatePolynomial?(SparseMultivariatePolynomial?(Fraction(Integer),Kernel(Expression(Fraction(Integer)))))))

fricas

ff11:=factoredForm(T11)$HYPGEOM

fricas

Compiling function T11 with type Expression(Fraction(Integer)) -> 
      Expression(Fraction(Integer)) 

             2
   (32)  --------
                2
         (? + 1)

$\label{eq11}\frac{2}{{\left(? + 1 \right)}^{2}}$

(11)

Type: Fraction(Factored(SparseUnivariatePolynomial?(SparseMultivariatePolynomial?(Fraction(Integer),Kernel(Expression(Fraction(Integer)))))))

fricas

ff12:=factoredForm(T12)$HYPGEOM

fricas

Compiling function T12 with type Expression(Fraction(Integer)) -> 
      Expression(Fraction(Integer)) 

           ? - n
   (33)  - -----
           ? + 1

$\label{eq12}-{\frac{? - n}{? + 1}}$

(12)

Type: Fraction(Factored(SparseUnivariatePolynomial?(SparseMultivariatePolynomial?(Fraction(Integer),Kernel(Expression(Fraction(Integer)))))))

fricas

for v in [ff1,ff2,ff3,ff4,ff5,ff6,ff7,ff8,ff9,ff10,ff11,ff12] repeat
  output getParameters(numer v)
  output getParameters(denom v)

            1
   [ab = [- -], const = []]
            2
          3     1
   [ab = [-, 1, -], const = []]
          2     2
   [ab = [- n], const = []]
   [ab = [1, 1], const = []]
   [ab = [], const = [x, x]]
   [ab = [p + 1, 1], const = []]
   [ab = [n + 1, n + 1, - n, - n], const = []]
   [ab = [1, 1, 1, 1], const = []]
   [ab = [k + 1, k + 1], const = []]
   [ab = [- k + 1, - k + 1], const = []]
   [ab = [], const = [n]]
   [ab = [], const = []]
   [ab = [1], const = []]
   [ab = [], const = []]
          9
   [ab = [-, 4], const = []]
          2
   [ab = [- 2], const = []]
          7     4  2
   [ab = [-, 2, -, -, 0], const = []]
          2     3  3
          9
   [ab = [-, 4, - 1], const = []]
          2
   [ab = [], const = [z]]
   [ab = [1], const = []]
   [ab = [], const = []]
   [ab = [1, 1], const = []]
   [ab = [- n], const = []]
   [ab = [1], const = []]

Type: Void

fricas

gp2:=getParameters(numer ff2).ab

   (35)  [- n]

$\label{eq13}\left[ - n \right]$

(13)

Type: List(SparseUnivariatePolynomial?(SparseMultivariatePolynomial?(Fraction(Integer),Kernel(Expression(Fraction(Integer))))))

fricas

gp3:=getParameters(denom ff3).ab

   (36)  [p + 1, 1]

$\label{eq14}\left[{p + 1}, \: 1 \right]$

(14)

Type: List(SparseUnivariatePolynomial?(SparseMultivariatePolynomial?(Fraction(Integer),Kernel(Expression(Fraction(Integer))))))

fricas

----
mp3:=multivariate(gp3.1,tower(new()$Symbol).1)$SMP::X

   (37)  p + 1

$\label{eq15}p + 1$

(15)

Type: Expression(Fraction(Integer))

fricas

mp2:=multivariate(gp2.1,tower(new()$Symbol).1)$SMP::X

   (38)  - n

$\label{eq16}- n$

(16)

Type: Expression(Fraction(Integer))

fricas

--- List(Expression(Fraction(Integer)))
[convert(v) for v in gp3]

   (39)  [p + 1, 1]

$\label{eq17}\left[{p + 1}, \: 1 \right]$

(17)

Type: List(Expression(Fraction(Integer)))

fricas

[convert(v) for v in gp2]

   (40)  [- n]

$\label{eq18}\left[ - n \right]$

(18)

Type: List(Expression(Fraction(Integer)))

fricas

-- near the goal ;)
pFq(T1)

                  1         3     1
   (41)  [ap = [- -], bq = [-, 1, -], fac = 1]
                  2         2     2

$\label{eq19}\left[{ap ={\left[ -{\frac{1}{2}}\right]}}, \:{bq ={\left[{\frac{3}{2}}, \: 1, \:{\frac{1}{2}}\right]}}, \:{fac = 1}\right]$

(19)

Type: Record(ap: List(Expression(Fraction(Integer))),bq: List(Expression(Fraction(Integer))),fac: Expression(Fraction(Integer)))

fricas

pFq(T2)

   (42)  [ap = [- n], bq = [1, 1], fac = 1]

$\label{eq20}\left[{ap ={\left[ - n \right]}}, \:{bq ={\left[ 1, \: 1 \right]}}, \:{fac = 1}\right]$

(20)

Type: Record(ap: List(Expression(Fraction(Integer))),bq: List(Expression(Fraction(Integer))),fac: Expression(Fraction(Integer)))

fricas

pFq(T3)

   (43)  [ap = [], bq = [p + 1, 1], fac = 1]

$\label{eq21}\left[{ap ={\left[ \right]}}, \:{bq ={\left[{p + 1}, \: 1 \right]}}, \:{fac = 1}\right]$

(21)

Type: Record(ap: List(Expression(Fraction(Integer))),bq: List(Expression(Fraction(Integer))),fac: Expression(Fraction(Integer)))

fricas

pFq(T4)

   (44)  [ap = [n + 1, n + 1, - n, - n], bq = [1, 1, 1, 1], fac = 1]

$\label{eq22}\left[{ap ={\left[{n + 1}, \:{n + 1}, \: - n , \: - n \right]}}, \:{bq ={\left[ 1, \: 1, \: 1, \: 1 \right]}}, \:{fac = 1}\right]$

(22)

Type: Record(ap: List(Expression(Fraction(Integer))),bq: List(Expression(Fraction(Integer))),fac: Expression(Fraction(Integer)))

fricas

pFq(T5)

   (45)  [ap = [k + 1, k + 1], bq = [- k + 1, - k + 1], fac = 1]

$\label{eq23}\left[{ap ={\left[{k + 1}, \:{k + 1}\right]}}, \:{bq ={\left[{- k + 1}, \:{- k + 1}\right]}}, \:{fac = 1}\right]$

(23)

Type: Record(ap: List(Expression(Fraction(Integer))),bq: List(Expression(Fraction(Integer))),fac: Expression(Fraction(Integer)))

fricas

pFq(T6)

   (46)  [ap = [], bq = [], fac = 1]

$\label{eq24}\left[{ap ={\left[ \right]}}, \:{bq ={\left[ \right]}}, \:{fac = 1}\right]$

(24)

Type: Record(ap: List(Expression(Fraction(Integer))),bq: List(Expression(Fraction(Integer))),fac: Expression(Fraction(Integer)))

fricas

pFq(T7)

   (47)  [ap = [1], bq = [], fac = 1]

$\label{eq25}\left[{ap ={\left[ 1 \right]}}, \:{bq ={\left[ \right]}}, \:{fac = 1}\right]$

(25)

Type: Record(ap: List(Expression(Fraction(Integer))),bq: List(Expression(Fraction(Integer))),fac: Expression(Fraction(Integer)))

fricas

pFq(T8)

                9
   (48)  [ap = [-, 4], bq = [- 2], fac = 1]
                2

$\label{eq26}\left[{ap ={\left[{\frac{9}{2}}, \: 4 \right]}}, \:{bq ={\left[ - 2 \right]}}, \:{fac = 1}\right]$

(26)

Type: Record(ap: List(Expression(Fraction(Integer))),bq: List(Expression(Fraction(Integer))),fac: Expression(Fraction(Integer)))

fricas

pFq(T9)

                7     4  2            9
   (49)  [ap = [-, 2, -, -, 0], bq = [-, 4, - 1], fac = 1]
                2     3  3            2

$\label{eq27}\begin{array}{@{}l} \displaystyle \left[{ap ={\left[{\frac{7}{2}}, \: 2, \:{\frac{4}{3}}, \:{\frac{2}{3}}, \: 0 \right]}}, \:{bq ={\left[{\frac{9}{2}}, \: 4, \: - 1 \right]}}, \: \right. \ \ \displaystyle \left.{fac = 1}\right]$

(27)

Type: Record(ap: List(Expression(Fraction(Integer))),bq: List(Expression(Fraction(Integer))),fac: Expression(Fraction(Integer)))

fricas

pFq(k+->sin(k))

   (50)  [ap = [], bq = [], fac = 1]

$\label{eq28}\left[{ap ={\left[ \right]}}, \:{bq ={\left[ \right]}}, \:{fac = 1}\right]$

(28)

Type: Record(ap: List(Expression(Fraction(Integer))),bq: List(Expression(Fraction(Integer))),fac: Expression(Fraction(Integer)))

fricas

--------------

-- simple test
isQuoRat?(k+->sin k)

   (51)  false

$\label{eq29} \mbox{\rm false}$

(29)

Type: Boolean

fricas

isQuoRat?(k+->sin asin k^4)

   (52)  true

$\label{eq30} \mbox{\rm true}$

(30)

Type: Boolean

fricas

isQuoRat?(k+->log exp  k^4)

   (53)  true

$\label{eq31} \mbox{\rm true}$

(31)

Type: Boolean

fricas

isQuoRat?(k+->exp  k^4)

   (54)  true

$\label{eq32} \mbox{\rm true}$

(32)

Type: Boolean

fricas

isQuoRat?(k+->x^k) 

   (55)  true

$\label{eq33} \mbox{\rm true}$

(33)

Type: Boolean

fricas

un:=unit numer ff1 

         1
   (56)  -
         4

$\label{eq34}\frac{1}{4}$

(34)

Type: SparseUnivariatePolynomial?(SparseMultivariatePolynomial?(Fraction(Integer),Kernel(Expression(Fraction(Integer)))))

fricas

ud:=unit denom ff1

   (57)  1

$\label{eq35}1$

(35)

Type: SparseUnivariatePolynomial?(SparseMultivariatePolynomial?(Fraction(Integer),Kernel(Expression(Fraction(Integer)))))

fricas

unud:=un/ud

         1
   (58)  -
         4

$\label{eq36}\frac{1}{4}$

(36)

Type: Fraction(SparseUnivariatePolynomial?(SparseMultivariatePolynomial?(Fraction(Integer),Kernel(Expression(Fraction(Integer))))))

fricas

pFq0(T6) 

   (59)  [ap = [1], bq = [], fac = n]

$\label{eq37}\left[{ap ={\left[ 1 \right]}}, \:{bq ={\left[ \right]}}, \:{fac = n}\right]$

(37)

Type: Record(ap: List(Expression(Fraction(Integer))),bq: List(Expression(Fraction(Integer))),fac: Expression(Fraction(Integer)))

fricas

T3(k)

fricas

Compiling function T3 with type Variable(k) -> Expression(Integer) 

              k x p + 2 k
         (- 1) (-)
                2
   (60)  ----------------
            k!(p + k)!

$\label{eq38}\frac{{{\left(- 1 \right)}^{k}}\ {{\frac{x}{2}}^{p +{2 \ k}}}}{{k !}\ {{\left(p + k \right)}!}}$

(38)

Type: Expression(Integer)

fricas

T3C:=(k+x)^2/((k+p+1)*(k+1))

              2            2
             x  + 2 k x + k
   (61)  -----------------------
                     2
         (k + 1)p + k  + 2 k + 1

$\label{eq39}\frac{{{x}^{2}}+{2 \ k \ x}+{{k}^{2}}}{{{\left(k + 1 \right)}\ p}+{{k}^{2}}+{2 \ k}+ 1}$

(39)

Type: Fraction(Polynomial(Integer))

fricas

pFq0(T3)

                                         1  2
   (62)  [ap = [], bq = [p + 1], fac = - - x ]
                                         4

$\label{eq40}\left[{ap ={\left[ \right]}}, \:{bq ={\left[{p + 1}\right]}}, \:{fac = -{{\frac{1}{4}}\ {{x}^{2}}}}\right]$

(40)

Type: Record(ap: List(Expression(Fraction(Integer))),bq: List(Expression(Fraction(Integer))),fac: Expression(Fraction(Integer)))

fricas

ff3 

                  1  2
                  - x
                  4
   (63)  - ------------------
           (? + 1)(? + p + 1)

$\label{eq41}-{\frac{{\frac{1}{4}}\ {{x}^{2}}}{{\left(? + 1 \right)}\ {\left(? + p + 1 \right)}}}$

(41)

Type: Fraction(Factored(SparseUnivariatePolynomial?(SparseMultivariatePolynomial?(Fraction(Integer),Kernel(Expression(Fraction(Integer)))))))

fricas

--
pFq0(T10) 

   (64)  [ap = [], bq = [], fac = z]

$\label{eq42}\left[{ap ={\left[ \right]}}, \:{bq ={\left[ \right]}}, \:{fac = z}\right]$

(42)

Type: Record(ap: List(Expression(Fraction(Integer))),bq: List(Expression(Fraction(Integer))),fac: Expression(Fraction(Integer)))

fricas

pFq0(T11) 

   (65)  [ap = [], bq = [1], fac = 2]

$\label{eq43}\left[{ap ={\left[ \right]}}, \:{bq ={\left[ 1 \right]}}, \:{fac = 2}\right]$

(43)

Type: Record(ap: List(Expression(Fraction(Integer))),bq: List(Expression(Fraction(Integer))),fac: Expression(Fraction(Integer)))

fricas

hy:=pFq0(T3)

                                         1  2
   (66)  [ap = [], bq = [p + 1], fac = - - x ]
                                         4

$\label{eq44}\left[{ap ={\left[ \right]}}, \:{bq ={\left[{p + 1}\right]}}, \:{fac = -{{\frac{1}{4}}\ {{x}^{2}}}}\right]$

(44)

Type: Record(ap: List(Expression(Fraction(Integer))),bq: List(Expression(Fraction(Integer))),fac: Expression(Fraction(Integer)))

fricas

hyc:=construct(hy,z)$HYPGEOM 

                  1  2
                  - x
                  4
   (67)  - ------------------
           (z + 1)(z + p + 1)

$\label{eq45}-{\frac{{\frac{1}{4}}\ {{x}^{2}}}{{\left(z + 1 \right)}\ {\left(z + p + 1 \right)}}}$

(45)

Type: Expression(Fraction(Integer))

fricas

ff3

                  1  2
                  - x
                  4
   (68)  - ------------------
           (? + 1)(? + p + 1)

$\label{eq46}-{\frac{{\frac{1}{4}}\ {{x}^{2}}}{{\left(? + 1 \right)}\ {\left(? + p + 1 \right)}}}$

(46)

Type: Fraction(Factored(SparseUnivariatePolynomial?(SparseMultivariatePolynomial?(Fraction(Integer),Kernel(Expression(Fraction(Integer)))))))

fricas

display(hy)$HYPGEOM

              +  *  +    1  2
   (69)   F   |     | [- - x ]
         0  1 +p + 1+    4

$\label{eq47}{{{}_{0}^{\ }F_{\ }^{\ }}_{1}}\ {\left[ \begin{array}{c} * \ {p + 1}$

(47)

Type: OutputForm?

fricas

display(pFq0(T1))$HYPGEOM

              +  1   +
              |- -  *|
              |  2   |  1
   (70)   F   |      | [-]
         1  2 | 3   1|  4
              | -   -|
              + 2   2+

$\label{eq48}\begin{array}{@{}l} \displaystyle {{{}_{1}^{\ }F_{\ }^{\ }}_{2}}\ \cdot \ \ \displaystyle {\left[ \begin{array}{cc} -{\frac{1}{2}}& * \ {\frac{3}{2}}&{\frac{1}{2}}$

(48)

Type: OutputForm?

fricas

display(pFq0(T2))$HYPGEOM

              +- n+
   (71)   F   |   | [1]
         1  1 + 1 +

$\label{eq49}{{{}_{1}^{\ }F_{\ }^{\ }}_{1}}\ {\left[ \begin{array}{c} - n \ 1$

(49)

Type: OutputForm?

fricas

display(pFq0(T3))$HYPGEOM

              +  *  +    1  2
   (72)   F   |     | [- - x ]
         0  1 +p + 1+    4

$\label{eq50}{{{}_{0}^{\ }F_{\ }^{\ }}_{1}}\ {\left[ \begin{array}{c} * \ {p + 1}$

(50)

Type: OutputForm?

fricas

display(pFq0(T4))$HYPGEOM

              +n + 1  n + 1  - n  - n+
   (73)   F   |                      | [1]
         4  3 +  1      1     1    * +

$\label{eq51}\begin{array}{@{}l} \displaystyle {{{}_{4}^{\ }F_{\ }^{\ }}_{3}}\ \cdot \ \ \displaystyle {\left[ \begin{array}{cccc} {n + 1}&{n + 1}& - n & - n \ 1 & 1 & 1 & *$

(51)

Type: OutputForm?

fricas

hyperLookup(pFq0 T12)

          n
   (74)  2

$\label{eq52}{2}^{n}$

(52)

Type: Expression(Fraction(Integer))

fricas

Q:=construct(pFq0(T1),Z)$HYPGEOM

               1      1
               - (Z - -)
               4      2
   (75)  ---------------------
                     1      3
         (Z + 1)(Z + -)(Z + -)
                     2      2

$\label{eq53}\frac{{\frac{1}{4}}\ {\left(Z -{\frac{1}{2}}\right)}}{{\left(Z + 1 \right)}\ {\left(Z +{\frac{1}{2}}\right)}\ {\left(Z +{\frac{3}{2}}\right)}}$

(53)

Type: Expression(Fraction(Integer))

fricas

T1(Z+1)/T1(Z)- distribute Q

fricas

Compiling function T1 with type Polynomial(Integer) -> Expression(
      Integer)

fricas

Compiling function T1 with type Variable(Z) -> Expression(Integer) 

            1     1                3      2   1     3
         (- - Z + -)(2 Z + 3)! + (Z  + 2 Z  + - Z - -)(2 Z + 1)!
            4     8                           4     4
   (76)  -------------------------------------------------------
                       3      2   11     3
                     (Z  + 3 Z  + -- Z + -)(2 Z + 3)!
                                   4     4

$\label{eq54}\frac{{{\left(-{{\frac{1}{4}}\ Z}+{\frac{1}{8}}\right)}\ {{\left({2 \ Z}+ 3 \right)}!}}+{{\left({{Z}^{3}}+{2 \ {{Z}^{2}}}+{{\frac{1}{4}}\ Z}-{\frac{3}{4}}\right)}\ {{\left({2 \ Z}+ 1 \right)}!}}}{{\left({{Z}^{3}}+{3 \ {{Z}^{2}}}+{{\frac{11}{4}}\ Z}+{\frac{3}{4}}\right)}\ {{\left({2 \ Z}+ 3 \right)}!}}$

(54)

Type: Expression(Fraction(Integer))

fricas

normalize $

  Line 105: normalize $
           ..........A
  Error  A: syntax error at top level
  Error  A: Improper syntax.
   2 error(s) parsing