MathAction HYPGEOM

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Sat 9 May 22:36:23 CEST 2026 --------------------------------------

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)abbrev package HYPGEOM HyperGeometric
--------------------------------------
HyperGeometric(): Exports == Implementation where
  R ==> Fraction Integer  -- must be FRAC INT not INT!!
  X ==> Expression R
  NNI ==> NonNegativeInteger
  SMP ==> SparseMultivariatePolynomial(R, Kernel X) -- if R has IntegralDomain
  SUP ==> SparseUnivariatePolynomial SMP 
  FSF ==> FunctionalSpecialFunction(R,X)

  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
    rational? : (X,Symbol) -> Boolean
      ++ rational?(r,x) returns true if r is of the form p/q where p and q
      ++ are univariate polynomials in x.
    factoredForm : (X->X) -> Fraction(FACSUP)
      ++ factoredForm(f) returns the fraction f(?+1)/f(?) in the variable "?", 
      ++ where numerator and denominator are factored, provided that this 
      ++ fraction is a rational function; otherwise an error message is
      ++ produced. 
    factoredForm : X -> Fraction(FACSUP) 
    admissible? : Fraction(FACSUP) -> Boolean
      ++ admissible?(ff), where ff is the output of factoredForm, checks if
      ++ the factorization is complete (i.e. linear) and the b coefficients
      ++ are positive.      
    getParameters : FACSUP -> PARAM
      ++ getParameters(p) gives the parameters of sparse univariate polynomial
      ++ p, that is the list of zeroes and possibly a constant factor when
      ++ different from 1.
    convert : SUP -> X
      ++ convert(p) converts the SparseUnivariatePolynomial p to Expression
      ++ Integer.
    pFq : (X->X) -> HYPER
      ++ pFq(f) provides the hypergeometric parameters as a record of the
      ++ form Record(ap:List X, bq:List X, fac:X), where X=EXPR(INT).
    pFq : X -> HYPER
    construct: (HYPER,Symbol) -> X
      ++ construct(pfq,x) constructs the hypergeometric expression with 
      ++ the output of pFq in the symbol x.
    display : HYPER -> OutputForm
      ++ Fancy display the output of pFq.
    hyperLookup : HYPER -> X
      ++ Lookup the output of pFq in a table.
    createHyperF : HYPER -> X
      ++
    hgFtoHYPER : X -> HYPER
      ++ hgFtoHYPER(f) converts f:=hypergeometricF([a..],[b..],x) to
      ++ a HYPER record [[a..],[b..],x]$HYPER.
    sum : (X,Symbol) -> X

  Implementation == add

    rational0?(p:X,xs:Symbol):Boolean ==
      x:=xs::X
      up:=univariate(p,kernel xs)
      dup:=denom up
      nup:=numer up
      cdup:= coefficients dup
      ndup:= coefficients nup
      fox:=(u:X):Boolean+->freeOf?(u,x)
      every?(fox,cdup) and every?(fox,ndup)

    rational?(f:X, xs:Symbol):Boolean ==
      k:Kernel X:=kernel xs
      for kk in tower f repeat
        a:Boolean:=member?(k, tower(kk::X)) 
        b:Boolean:=(symbolIfCan(kk) case "failed")
        a and b => return false
      true

    factoredForm(f:X->X):Fraction(FACSUP) ==
      ns:X:=(nsym:=new()$Symbol)::X
      q:=f(ns+1)/f(ns)
      nq:=normalize(q)$EFSP
      not rational?(nq,retract ns) => error "Not rational"
      dnq:=denom nq
      nnq:=numer nq
      unnq:=univariate(nnq,kernel nsym)
      udnq:=univariate(dnq,kernel nsym)
      funnq:=factor unnq
      fudnq:=factor udnq
      funnq/fudnq

    factoredForm(S:X):Fraction(FACSUP) ==
      mk:=mainKernel S
      mk case "failed" => error "mainKernel failed."
      not (name mk = 'summation) => error "Sum expected."
      args:=argument mk
      not (#args = 3) => error "Unexpected number of kernel arguments."
      summand:=first args
      var:=second args
      not freeOf?(summand,third args) => error "Strange summand."
      q:=eval(summand,var=var+1)/summand
      nq:=normalize(q)$EFSP
      not rational?(nq,retract var) => error "Not rational"
      dnq:=denom nq
      nnq:=numer nq
      unnq:=univariate(nnq,kernel retract var)
      udnq:=univariate(dnq,kernel retract var)
      funnq:=factor unnq
      fudnq:=factor udnq
      funnq/fudnq

    admissible? ff ==
      fn:=factors(numer ff)
      fd:=factors(denom ff)
      not empty? fn and max [degree(u.factor) for u in fn] > 1 => false
      not empty? fd and max [degree(u.factor) for u in fd] > 1 => false
      --todo: check fd: b's>0
      --qm:=create()$SingletonAsOrderedSet
      empty? fd => true
      --[(s.factor-qm) ??? for s in fd]
      true

    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

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

    pFq(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

    pFq(S:X):HYPER ==
      ff:=factoredForm(S)
      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)

    checkGauss2F1(x:HYPER):Boolean ==
      not(#(x.ap)=2 and #(x.bq)=1) => false
      not(x.fac=1$X) => false
      a:Union(Integer,"failed"):=retractIfCan(x.ap.1)
      b:Union(Integer,"failed"):=retractIfCan(x.ap.2)
      if a case Integer then 
        a:=a::Integer
        if a < 0 then return true        
      if b case Integer then
        b:=b::Integer 
        if b < 0 then return true
      r:X:=x.bq.1 - x.ap.1 - x.ap.2 
      not smaller?(r,0::X) => true
      false

    checkKummer2F1(x:HYPER):Boolean ==
      not(#(x.ap)=2 and #(x.bq)=1) => false
      not(x.fac=-1$X) => false 
      r:X:=x.ap.1 - x.ap.2 + x.bq.1       
      r=1$X => true
      false

    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)
      checkGauss2F1(x) => Gamma(b.1-a.1-a.2)*Gamma(b.1)/ _
                          (Gamma(b.1-a.1)*Gamma(b.1-a.2))   
      checkKummer2F1(x) => Gamma((1/2)*a.2+1)*Gamma(a.2-a.1+1)/ _
                          (Gamma(a.2+1)*Gamma((1/2)*a.2-a.1+1))                             
      0$X

    -- better: mainKernel hypergeometricF
    --         argument mainKernel hypergeometricF([a1..ap],[b1..bq],x) -->
    --         [a1,..,ap,b1,..,bq,x,p,q] as List(X)
    -- name mainKernel hyper... -->  hypergeometricF @ Symbol

    hgFtoHYPER(f:X):HYPER ==
      k:=mainKernel(f) 
      k case "failed" => error "Not a kernel?" 
      not(name k = 'hypergeometricF) => error "Not hypergeometricF."
      args:List X:=argument k
      narg:=#args
      q:Union(Integer,"failed"):=retractIfCan(args.narg)
      p:Union(Integer,"failed"):=retractIfCan(args.(narg-1))
      p case "failed" => error "p?"
      q case "failed" => error "q?"
      ap:=args.(1..p)
      bq:=args.(p+1..q+1)
      fac:=args.(q+2)
      [ap,bq,fac]$HYPER

    -- :todo:check f(0)~=0
    sum(f:X,s:Symbol):X ==
      S:=summation(f,s)
      h:HYPER:=pFq(S)
      eval(f,s::X=0) * hypergeometricF(h.ap,h.bq,h.fac)

-- S1:=sum((-1)^k*(x/2)^(2*k+p)/factorial(k)/factorial(k+p),k)
-- S2:=sum(1/(2*n-1)/factorial(2*n+1),n)  --  display pFq(S2)</spad>

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Compiling FriCAS source code from file 
      /var/lib/zope2.10/instance/axiom-wiki/var/LatexWiki/5770435868577614772-25px001.spad
      using old system compiler.
   HYPGEOM abbreviates package HyperGeometric 
------------------------------------------------------------------------
   initializing NRLIB HYPGEOM for HyperGeometric 
   compiling into NRLIB HYPGEOM 
   compiling local rational0? : (Expression Fraction Integer,Symbol) -> Boolean
Time: 0.05 SEC.

   compiling exported rational? : (Expression Fraction Integer,Symbol) -> Boolean
Time: 0 SEC.

   compiling exported factoredForm : Expression Fraction Integer -> Expression Fraction Integer -> Fraction Factored SparseUnivariatePolynomial SparseMultivariatePolynomial(Fraction Integer,Kernel Expression Fraction Integer)
Time: 0.03 SEC.

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

   compiling exported admissible? : Fraction Factored SparseUnivariatePolynomial SparseMultivariatePolynomial(Fraction Integer,Kernel Expression Fraction Integer) -> Boolean
Time: 0 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 SEC.

   compiling exported convert : SparseUnivariatePolynomial SparseMultivariatePolynomial(Fraction Integer,Kernel Expression Fraction Integer) -> Expression Fraction Integer
Time: 0 SEC.

   compiling local buildConst : List SparseUnivariatePolynomial SparseMultivariatePolynomial(Fraction Integer,Kernel Expression Fraction Integer) -> Expression Fraction Integer
Time: 0 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: 0.02 SEC.

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

   compiling local checkGauss2F1 : Record(ap: List Expression Fraction Integer,bq: List Expression Fraction Integer,fac: Expression Fraction Integer) -> Boolean
Time: 0.01 SEC.

   compiling local checkKummer2F1 : Record(ap: List Expression Fraction Integer,bq: List Expression Fraction Integer,fac: Expression Fraction Integer) -> Boolean
Time: 0 SEC.

   compiling exported hyperLookup : Record(ap: List Expression Fraction Integer,bq: List Expression Fraction Integer,fac: Expression Fraction Integer) -> Expression Fraction Integer
Time: 0.18 SEC.

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

   compiling exported sum : (Expression Fraction Integer,Symbol) -> Expression Fraction Integer
Time: 0.05 SEC.

(time taken in buildFunctor:  0)
Time: 0 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:  l1 has no value
      [4] getParameters:  l2 has no value

   Cumulative Statistics for Constructor HyperGeometric
      Time: 0.44 seconds

   finalizing NRLIB HYPGEOM 
   Processing HyperGeometric for Browser database:
--->-->HyperGeometric(constructor): Not documented!!!!
--------(rational? ((Boolean) (Expression (Fraction (Integer))) (Symbol)))---------
--------(factoredForm ((Fraction (Factored (SparseUnivariatePolynomial (SparseMultivariatePolynomial (Fraction (Integer)) (Kernel (Expression (Fraction (Integer)))))))) (Mapping (Expression (Fraction (Integer))) (Expression (Fraction (Integer))))))---------
--->-->HyperGeometric((factoredForm ((Fraction (Factored (SparseUnivariatePolynomial (SparseMultivariatePolynomial (Fraction (Integer)) (Kernel (Expression (Fraction (Integer)))))))) (Expression (Fraction (Integer)))))): Not documented!!!!
--------(admissible? ((Boolean) (Fraction (Factored (SparseUnivariatePolynomial (SparseMultivariatePolynomial (Fraction (Integer)) (Kernel (Expression (Fraction (Integer))))))))))---------
--------(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)))))))))---------
--------(convert ((Expression (Fraction (Integer))) (SparseUnivariatePolynomial (SparseMultivariatePolynomial (Fraction (Integer)) (Kernel (Expression (Fraction (Integer))))))))---------
--------(pFq ((Record (: ap (List (Expression (Fraction (Integer))))) (: bq (List (Expression (Fraction (Integer))))) (: fac (Expression (Fraction (Integer))))) (Mapping (Expression (Fraction (Integer))) (Expression (Fraction (Integer))))))---------
--->-->HyperGeometric((pFq ((Record (: ap (List (Expression (Fraction (Integer))))) (: bq (List (Expression (Fraction (Integer))))) (: fac (Expression (Fraction (Integer))))) (Expression (Fraction (Integer)))))): Not documented!!!!
--------(construct ((Expression (Fraction (Integer))) (Record (: ap (List (Expression (Fraction (Integer))))) (: bq (List (Expression (Fraction (Integer))))) (: fac (Expression (Fraction (Integer))))) (Symbol)))---------
--------(display ((OutputForm) (Record (: ap (List (Expression (Fraction (Integer))))) (: bq (List (Expression (Fraction (Integer))))) (: fac (Expression (Fraction (Integer)))))))---------
--->-->HyperGeometric((display ((OutputForm) (Record (: ap (List (Expression (Fraction (Integer))))) (: bq (List (Expression (Fraction (Integer))))) (: fac (Expression (Fraction (Integer)))))))): Improper first word in comments: Fancy
"Fancy display the output of \\spad{pFq}."
--------(hyperLookup ((Expression (Fraction (Integer))) (Record (: ap (List (Expression (Fraction (Integer))))) (: bq (List (Expression (Fraction (Integer))))) (: fac (Expression (Fraction (Integer)))))))---------
--->-->HyperGeometric((hyperLookup ((Expression (Fraction (Integer))) (Record (: ap (List (Expression (Fraction (Integer))))) (: bq (List (Expression (Fraction (Integer))))) (: fac (Expression (Fraction (Integer)))))))): Improper first word in comments: Lookup
"Lookup the output of \\spad{pFq} in a table."
--------(createHyperF ((Expression (Fraction (Integer))) (Record (: ap (List (Expression (Fraction (Integer))))) (: bq (List (Expression (Fraction (Integer))))) (: fac (Expression (Fraction (Integer)))))))---------
--------(hgFtoHYPER ((Record (: ap (List (Expression (Fraction (Integer))))) (: bq (List (Expression (Fraction (Integer))))) (: fac (Expression (Fraction (Integer))))) (Expression (Fraction (Integer)))))---------
--->-->HyperGeometric((sum ((Expression (Fraction (Integer))) (Expression (Fraction (Integer))) (Symbol)))): Not documented!!!!
--->-->HyperGeometric(): Missing Description
; compiling file "/var/aw/var/LatexWiki/HYPGEOM.NRLIB/HYPGEOM.lsp" (written 07 JUN 2026 05:13:20 PM):

; wrote /var/aw/var/LatexWiki/HYPGEOM.NRLIB/HYPGEOM.fasl
; compilation finished in 0:00:00.144
------------------------------------------------------------------------
   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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)version

"FriCAS 1.3.12 compiled at Sat  7 Jun 23:54:49 CEST 2025"

-- binomial(n,k)
-- expected: hypergeometricF([- n],[],- 1) = 2^n
sum(binomial(n,k),'k)

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

(1)

Type: Expression(Integer)

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sum(binomial(n,k),'k)$HYPGEOM

$\label{eq2}hypergeometricF \left({{\left[ - n \right]}, \:{\left[ \right]}, \: - 1}\right)$

(2)

Type: Expression(Fraction(Integer))

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-- x^k/k!
-- expected: hypergeometricF([],[],x) = exp(x)
sum(x^k/factorial(k),'k)$HYPGEOM

$\label{eq3}hypergeometricF \left({{\left[ \right]}, \:{\left[ \right]}, \: x}\right)$

(3)

Type: Expression(Fraction(Integer))

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-- Example 3.3.1.
-- expected: hypergeometricF([],[1],2)
sum(2^k/factorial(k)^2,'k)$HYPGEOM

$\label{eq4}hypergeometricF \left({{\left[ \right]}, \:{\left[ 1 \right]}, \: 2}\right)$

(4)

Type: Expression(Fraction(Integer))

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-- Example 3.3.2.
--                            1   3  1  1
-- expected: - hypergeometricF([- -],[-, -],-)
--                            2   2  2  4
sum(1/(2*k-1)/factorial(2*k+1),'k)$HYPGEOM

$\label{eq5}-{hypergeometricF \left({{\left[ -{\frac{1}{2}}\right]}, \:{\left[{\frac{3}{2}}, \:{\frac{1}{2}}\right]}, \:{\frac{1}{4}}}\right)}$

(5)

Type: Expression(Fraction(Integer))

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-- Example 3.3.4. 
-- expected: hypergeometricF([- n],[1],1)
sum(binomial(n,k)*(-1)^k/factorial(k),'k)$HYPGEOM

$\label{eq6}hypergeometricF \left({{\left[ - n \right]}, \:{\left[ 1 \right]}, \: 1}\right)$

(6)

Type: Expression(Fraction(Integer))

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-- Example 3.3.5. (Bessel: J_p(x))
--                                    1  2  1   p
--       hypergeometricF([],[p + 1],- - x )(- x)
--                                    4     2
--       ----------------------------------------
--                          p!
sum((-1)^k*(x/2)^(2*k+p)/factorial(k)/factorial(k+p),'k)$HYPGEOM

$\label{eq7}\frac{{hypergeometricF \left({{\left[ \right]}, \:{\left[{p + 1}\right]}, \: -{{\frac{1}{4}}\ {{x}^{2}}}}\right)}\ {{\left({\frac{1}{2}}\ x \right)}^{p}}}{p !}$

(7)

Type: Expression(Fraction(Integer))

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-- Sec 3.4 / binomial(n,k)^2
-- expected: hypergeometricF([- n, - n],[1],1)
sum(binomial(n,k)^2,k)$HYPGEOM

$\label{eq8}hypergeometricF \left({{\left[ - n , \: - n \right]}, \:{\left[ 1 \right]}, \: 1}\right)$

(8)

Type: Expression(Fraction(Integer))

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-- sum(x^k*binomial(n,k)^2,k)$HYPGEOM --> hypergeometricF([- n, - n],[1],x)

-- Sec 3.4.1
-- expected: binomial(r,n)/(r-n+1)*hypergeometricF([-n,n-r-1,(s-r)/2,(s-r+1)/2],
--    [(1-r)/2,-r/2,s-r],1)
sum((-1)^k*binomial(r - s - k, k)*binomial(r - 2*k, n - k)/(r - n - k + 1),k)$HYPGEOM

$\label{eq9}\frac{{hypergeometricF \left({{\left[{{{\frac{1}{2}}\ s}-{{\frac{1}{2}}\ r}+{\frac{1}{2}}}, \:{{{\frac{1}{2}}\ s}-{{\frac{1}{2}}\ r}}, \: - n , \:{- r + n - 1}\right]}, \:{\left[{s - r}, \:{-{{\frac{1}{2}}\ r}+{\frac{1}{2}}}, \: -{{\frac{1}{2}}\ r}\right]}, \: 1}\right)}\ {\hbox{\axiomType{BINOMIAL}\ } \left({r , \: n}\right)}}{r - n + 1}$

(9)

Type: Expression(Fraction(Integer))

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-- Sec 3.4.2
-- expected hypergeometricF([-n, -n, -n], [1, 1], -1),
sum(binomial(n, k)^3, k)$HYPGEOM

$\label{eq10}hypergeometricF \left({{\left[ - n , \: - n , \: - n \right]}, \:{\left[ 1, \: 1 \right]}, \: - 1}\right)$

(10)

Type: Expression(Fraction(Integer))

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-- Example 3.6.1
-- expected: hypergeometricF([- 2 n, - 2 n],[1],- 1) 
sum((-1)^k*binomial(2*n,k)^2,k)$HYPGEOM

$\label{eq11}hypergeometricF \left({{\left[ -{2 \ n}, \: -{2 \ n}\right]}, \:{\left[ 1 \right]}, \: - 1}\right)$

(11)

Type: Expression(Fraction(Integer))

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-- Example 3.6.2
-- expected: 
--                        1                            1     4 n
--       hypergeometricF([-, - 2 n, - 2 n],[1, - 2 n + -],1)(   )
--                        2                            2     2 n
-- tricky (in principle violates b>0, however, cancelling negative factorials)
sum((-1)^k*binomial(2*n,k)*binomial(2*k,k)*binomial(4*n-2*k,2*n-k),k)$HYPGEOM

$\label{eq12}{hypergeometricF \left({{\left[{\frac{1}{2}}, \: -{2 \ n}, \: -{2 \ n}\right]}, \:{\left[ 1, \:{-{2 \ n}+{\frac{1}{2}}}\right]}, \: 1}\right)}\ {\hbox{\axiomType{BINOMIAL}\ } \left({{4 \ n}, \:{2 \ n}}\right)}$

(12)

Type: Expression(Fraction(Integer))

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----------------
-- 3.8 Exercises
----------------
sum(binomial(n,k)*binomial(n+a,k)*binomial(n-a,n-k),k)$HYPGEOM

$\label{eq13}{hypergeometricF \left({{\left[ - n , \: - n , \:{- n - a}\right]}, \:{\left[ 1, \:{- a + 1}\right]}, \: - 1}\right)}\ {\hbox{\axiomType{BINOMIAL}\ } \left({{n - a}, \: n}\right)}$

(13)

Type: Expression(Fraction(Integer))

fricas

sum(binomial(n,k)^5,k)$HYPGEOM

$\label{eq14}hypergeometricF \left({{\left[ - n , \: - n , \: - n , \: - n , \: - n \right]}, \:{\left[ 1, \: 1, \: 1, \: 1 \right]}, \: - 1}\right)$

(14)

Type: Expression(Fraction(Integer))

fricas

sum((-1)^k*binomial(k,n-k),k)$HYPGEOM

$\label{eq15}{hypergeometricF \left({{\left[ 1, \: 1, \: - n \right]}, \:{\left[{-{{\frac{1}{2}}\ n}+ 1}, \:{-{{\frac{1}{2}}\ n}+{\frac{1}{2}}}\right]}, \:{\frac{1}{4}}}\right)}\ {\hbox{\axiomType{BINOMIAL}\ } \left({0, \: n}\right)}$

(15)

Type: Expression(Fraction(Integer))

fricas

---- 3.a-g
sum(binomial(n,k)/binomial(2*n-1,k),k)$HYPGEOM

$\label{eq16}hypergeometricF \left({{\left[ 1, \: - n \right]}, \:{\left[{-{2 \ n}+ 1}\right]}, \: 1}\right)$

(16)

Type: Expression(Fraction(Integer))

fricas

sum(binomial(n,k)^2*binomial(3*n+k,2*n),k)$HYPGEOM

$\label{eq17}{hypergeometricF \left({{\left[{{3 \ n}+ 1}, \: - n , \: - n \right]}, \:{\left[{n + 1}, \: 1 \right]}, \: 1}\right)}\ {\hbox{\axiomType{BINOMIAL}\ } \left({{3 \ n}, \:{2 \ n}}\right)}$

(17)

Type: Expression(Fraction(Integer))

fricas

sum((-1)^k*binomial(n,k)*factorial(k+3*a)/factorial(k+a),k)$HYPGEOM

$\label{eq18}\frac{{hypergeometricF \left({{\left[{{3 \ a}+ 1}, \: - n \right]}, \:{\left[{a + 1}\right]}, \: 1}\right)}\ {{\left(3 \ a \right)}!}}{a !}$

(18)

Type: Expression(Fraction(Integer))

fricas

sum(4^k*(-1)^k*binomial(k+a,a)*binomial(n+k,2*k+2*a),k)$HYPGEOM

$\label{eq19}{hypergeometricF \left({{\left[{n + 1}, \:{- n +{2 \ a}}\right]}, \:{\left[{a +{\frac{1}{2}}}\right]}, \: 1}\right)}\ {\hbox{\axiomType{BINOMIAL}\ } \left({n , \:{2 \ a}}\right)}$

(19)

Type: Expression(Fraction(Integer))

fricas

sum(binomial(2*n+2,2*k+1)*binomial(x+k,2*n+1),k)$HYPGEOM

$\label{eq20}\begin{array}{@{}l} \displaystyle {\left({2 \ n}+ 2 \right)}\ \cdot \ \ \displaystyle {hypergeometricF \left({{\left[{x + 1}, \: - n , \:{- n -{\frac{1}{2}}}\right]}, \:{\left[{x -{2 \ n}}, \:{\frac{3}{2}}\right]}, \: 1}\right)}\ \cdot \ \ \displaystyle {\hbox{\axiomType{BINOMIAL}\ } \left({x , \:{{2 \ n}+ 1}}\right)}$

(20)

Type: Expression(Fraction(Integer))

fricas

sum(binomial(2*n+1,2*p+2*k+1)*binomial(p+k,k),k)$HYPGEOM

$\label{eq21}{hypergeometricF \left({{\left[{p - n +{\frac{1}{2}}}, \:{p - n}\right]}, \:{\left[{p +{\frac{3}{2}}}\right]}, \: 1}\right)}\ {\hbox{\axiomType{BINOMIAL}\ } \left({{{2 \ n}+ 1}, \:{{2 \ p}+ 1}}\right)}$

(21)

Type: Expression(Fraction(Integer))

fricas

sum((-1)^k*binomial(2*n,k)*binomial(2*x,x+n-k)*binomial(2*z,z-n+k),k)$HYPGEOM

$\label{eq22}\begin{array}{@{}l} \displaystyle {hypergeometricF \left({{\left[ -{2 \ n}, \:{- x - n}, \:{- z - n}\right]}, \:{\left[{z - n + 1}, \:{x - n + 1}\right]}, \: 1}\right)}\ \cdot \ \ \displaystyle {\hbox{\axiomType{BINOMIAL}\ } \left({{2 \ x}, \:{x + n}}\right)}\ {\hbox{\axiomType{BINOMIAL}\ } \left({{2 \ z}, \:{z - n}}\right)}$

(22)

Type: Expression(Fraction(Integer))

fricas

----

f:=-sum(1/(2*k-1)/factorial(2*k+1),'k)$HYPGEOM

$\label{eq23}hypergeometricF \left({{\left[ -{\frac{1}{2}}\right]}, \:{\left[{\frac{3}{2}}, \:{\frac{1}{2}}\right]}, \:{\frac{1}{4}}}\right)$

(23)

Type: Expression(Fraction(Integer))

fricas

hgFtoHYPER f

$\label{eq24}\left[{ap ={\left[ -{\frac{1}{2}}\right]}}, \:{bq ={\left[{\frac{3}{2}}, \:{\frac{1}{2}}\right]}}, \:{fac ={\frac{1}{4}}}\right]$

(24)

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

fricas

sum((-1)^k*binomial(n,k),'k)$HYPGEOM --> 0?

$\label{eq25}hypergeometricF \left({{\left[ - n \right]}, \:{\left[ \right]}, \: 1}\right)$

(25)

Type: Expression(Fraction(Integer))

Subject: Be Bold !!

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