MathAction zeilberger

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(1) -> <spad>

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)abbrev package ZEIL Zeilberger
++ Author: $author
++ Date Created: $defaultdate
++ License: BSD
++ References:
++ Description:
++
Zeilberger() : Exports == Implementation where

  R   ==> Integer
  X   ==> Expression R
  XX  ==> (X,X)->X
  OF  ==> OutputForm

  SUP ==> SparseUnivariatePolynomial
  FPX ==> FRAC SUPX
  NNI ==> NonNegativeInteger
  SYM ==> Symbol

  SUPX ==> SUP(X)
  NIXX ==> (NNI,X,X) -> X
  EFSP ==> ElementaryFunctionStructurePackage(R,X)
  TSPR ==> TransSolvePackage R
  LEQX ==> List Equation X
  EQSR ==> Record(eqs:LEQX,unk:List X,cert:X)

  nz ==> normalize$EFSP

  Exports == with

    term : (XX,NNI,X) -> X
    termRatio : (XX,NNI,X) -> Record(p0:X,r:X,s:X)
    gosper : (SUPX,SUPX) -> Record(pa:SUPX,pb:SUPX,pc:SUPX)
    upperBound : (Record(pa:SUPX,pb:SUPX,pc:SUPX),SUPX) -> Union(Integer,"failed")
    equationsSetup : (XX,NNI,X) -> EQSR
    solveEquations: EQSR -> Union(LEQX,"failed")
    recEq: (XX,NNI,X) -> Union(Equation X,"failed")

  Implementation == add 

    term(F:XX,J:NNI,k:X):X ==
      n:X:='n::X
      t:List(X):=[subscript('a,[j::OF])::X*F(n+j::X,k) for j in 0..J]
      reduce(_+,t)

    termRatio(F:XX,J:NNI,k:X):Record(p0:X,r:X,s:X) ==
      n:X:='n::X
      --q:X:=nz( term(F,J,k+1::X)/term(F,J,k) )
      r12:=nz(F(n,k+1::X)/F(n,k))
      s12:=nz(F(n,k)/F(n-1::X,k))
      r1:=numerator r12
      r2:=denominator r12
      s1:=numerator s12
      s2:=denominator s12
      P1:=(j:NNI):X+->reduce("*",[eval(s1,n,n+j::X-i::X) for i in 0..j-1],1)
      P2:=(j:NNI):X+->reduce("*",[eval(s2,n,n+l::X) for l in j+1..J],1)
      p:List(X):=[subscript('a,[j::OF])::X * P1(j) * P2(j) for j in 0..J]
      p0:=nz(reduce(_+,p))
      u1:List(X):=[eval(s2,n,n+l::X) for l in 1..J]
      u2:List(X):=[eval(s2,[n=n+l::X,k=k+1::X]) for l in 1..J]
      r:= r1 * reduce("*",u1,1)
      s:= r2 * reduce("*",u2,1)
      qrs:=nz(r/s)
      [p0,numerator qrs,denominator qrs]

    gosper(r:SUPX,s:SUPX):Record(pa:SUPX,pb:SUPX,pc:SUPX) ==
      lcr:=leadingCoefficient r
      lcs:=leadingCoefficient s
      Z:=lcr/lcs
      r:=r/lcr
      s:=s/lcs
      h:X:='h::X
      single:=create()$SingletonAsOrderedSet
      Rh:X:=resultant(r,eval(s,single,single::SUPX+h::SUPX))
      Rh0:List Equation X:=solve(Rh=0,'h)$TransSolvePackage(R)
      nums:List X:=[rhs z for z in Rh0|integer?(rhs z)$IntegerRetractions(X)]
      ints:List Integer:=[retract(u) for u in nums]
      S:=[u for u in ints | u >=0 ]
      --output("** NNI zeroes of the resultant: ",S::OutputForm)$OutputPackage
      N:=#S
      p:=[subscript('p,[j::OF])::X::SUPX for j in 0..N]
      q:=[subscript('q,[j::OF])::X::SUPX for j in 0..N]
      w:=[subscript('w,[j::OF])::X::SUPX for j in 1..N]
      p.1:=r
      q.1:=s
      for j in 1..N repeat
        w.j:=gcd(p.j,eval(q.j,single,single::SUPX+(S.j)::SUPX))
        p.(j+1) := numer(p.j / w.j)
        q.(j+1) := numer(q.j / eval(w.j,single,single::SUPX-(S.j)::SUPX))
      a:=Z*p.(N+1)
      b:=q.(N+1)
      P1:=(i:NNI):SUPX+->reduce("*",[eval(w.i,single,single::SUPX-j::SUPX) for j in 1..S.i],1)      
      c:=reduce("*",[P1(i) for i in 1..N],1)
      [a,b,c]

    upperBound(g:Record(pa:SUPX,pb:SUPX,pc:SUPX),p0:SUPX):Union(Integer,"failed") ==
      single:=create()$SingletonAsOrderedSet
      a:SUPX:=g.pa
      b:SUPX:=g.pb
      c:SUPX:=g.pc * p0
      lc ==> leadingCoefficient
      BA:X:=lc(eval(b,single,single::SUPX-1::SUPX)-a)
      if degree(a) ~= degree(b) or lc a ~= lc b then
        D:List Integer:=[degree(c)-max(degree(a),degree(b))]
      else
        x:X:=BA/lc(a)
        if integer?(x)$IntegerRetractions(X) then
          D:=[degree(c)-degree(a)+1,retract x]
        else
          D:=[degree(c)-degree(a)+1]
      --output("** D:=",D::OutputForm)$OutputPackage
      --output("** BA:=",BA::OutputForm)$OutputPackage
      --output("** lca:=",lc(a)::OutputForm)$OutputPackage
      D:=[u for u in D | u>=0 ]
      empty? D => "failed"
      max(D)

    equationsSetup(F:XX,J:NNI,k:X):EQSR ==
      single:=create()$SingletonAsOrderedSet
      tr:=termRatio(F,J::NNI,k::X)
      ur:=univariate(tr.r,kernel 'k)
      us:=univariate(tr.s,kernel 'k)
      up0:=numer univariate(tr.p0,kernel 'k) --?
      pr:=retract(ur)@SUPX
      ps:=retract(us)@SUPX
      p0:=up0 --retract(up0)@SUPX
      g:=gosper(pr,ps)
      d:=upperBound(g,up0)
      d case "failed" => [[],[],0] 
      d:=d::Integer
      b:=[subscript('b,[j::OF])::X for j in 0..d]
      x0:=reduce(_+,[b.l*k^(l-1) for l in 1..d+1],0)
      x1:=reduce(_+,[b.l*(k+1)^(l-1) for l in 1..d+1],0)
      ux0:=numer univariate(x0,kernel 'k) 
      ux1:=numer univariate(x1,kernel 'k) 
      p:=g.pc * p0
      p2:=g.pa
      p3:=eval(g.pb,single,single::SUPX - 1::SUPX)
      eqx:SUPX:=p2*ux1-p3*ux0-p
      ceqx:List X:=coefficients(eqx)
      --output("** Equations:=",ceqx::OutputForm)$OutputPackage
      a:=[subscript('a,[j::OF])::X for j in 0..J]
      gp:=p3/p*ux0
      gpn:X:=retract eval(numer gp,single,k::SUPX)
      gpd:X:=retract eval(denom gp,single,k::SUPX)
      G:X:=gpn/gpd*term(F,J,k)
      [[u=0$X for u in ceqx],concat(a,b),G]

    solveEquations(e:EQSR):Union(LEQX,"failed") ==
      empty?(e.eqs) or empty?(e.unk) => "failed"
      s:=solve(e.eqs,[e.unk.i for i in 1..min(#e.unk-1,#e.eqs)])$TSPR
      empty? s => "failed"
      first s

    recEq(F:XX,J:NNI,k:X):Union(Equation X,"failed") ==
      es:=equationsSetup(F,J,k)
      sol:=solveEquations(es)
      if sol case "failed" then return "failed"
      else sol:=sol::LEQX
      S:=operator 'S
      n:='n::X
      rec:List X:=[rhs(sol.(i+1)) * S(n+i::X) for i in 0..J]
      rr:=reduce(_+,rec,0)   
      rr:=numerator nz(rr)
      u0:LEQX:=[es.unk.i=0 for i in #es.eqs+1..#es.unk-1]
      if not empty? u0 then rr:=nz(eval(rr,u0))
      rr:=eval(rr,[es.unk.(#es.unk)=1])
      rr=0$X</spad>

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Compiling FriCAS source code from file 
      /var/lib/zope2.10/instance/axiom-wiki/var/LatexWiki/3572774548778315158-25px001.spad
      using old system compiler.
   ZEIL abbreviates package Zeilberger 
------------------------------------------------------------------------
   initializing NRLIB ZEIL for Zeilberger 
   compiling into NRLIB ZEIL 
   compiling exported term : ((Expression Integer,Expression Integer) -> Expression Integer,NonNegativeInteger,Expression Integer) -> Expression Integer
Time: 0.03 SEC.

   compiling exported termRatio : ((Expression Integer,Expression Integer) -> Expression Integer,NonNegativeInteger,Expression Integer) -> Record(p0: Expression Integer,r: Expression Integer,s: Expression Integer)
Time: 0.06 SEC.

   compiling exported gosper : (SparseUnivariatePolynomial Expression Integer,SparseUnivariatePolynomial Expression Integer) -> Record(pa: SparseUnivariatePolynomial Expression Integer,pb: SparseUnivariatePolynomial Expression Integer,pc: SparseUnivariatePolynomial Expression Integer)
Time: 0.15 SEC.

   compiling exported upperBound : (Record(pa: SparseUnivariatePolynomial Expression Integer,pb: SparseUnivariatePolynomial Expression Integer,pc: SparseUnivariatePolynomial Expression Integer),SparseUnivariatePolynomial Expression Integer) -> Union(Integer,failed)
   processing macro definition lc ==> leadingCoefficient 
Time: 0.08 SEC.

   compiling exported equationsSetup : ((Expression Integer,Expression Integer) -> Expression Integer,NonNegativeInteger,Expression Integer) -> Record(eqs: List Equation Expression Integer,unk: List Expression Integer,cert: Expression Integer)
Time: 0.23 SEC.

   compiling exported solveEquations : Record(eqs: List Equation Expression Integer,unk: List Expression Integer,cert: Expression Integer) -> Union(List Equation Expression Integer,failed)
Time: 0.08 SEC.

   compiling exported recEq : ((Expression Integer,Expression Integer) -> Expression Integer,NonNegativeInteger,Expression Integer) -> Union(Equation Expression Integer,failed)
Time: 0.03 SEC.

(time taken in buildFunctor:  0)
Time: 0 SEC.

   Warnings: 
      [1] termRatio: 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] termRatio: 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))
      [3] termRatio:  k is BOTH a variable and a literal
      [4] equationsSetup:  k is BOTH a variable and a literal

   Cumulative Statistics for Constructor Zeilberger
      Time: 0.66 seconds

   finalizing NRLIB ZEIL 
   Processing Zeilberger for Browser database:
--------constructor---------
--->-->Zeilberger((term ((Expression (Integer)) (Mapping (Expression (Integer)) (Expression (Integer)) (Expression (Integer))) (NonNegativeInteger) (Expression (Integer))))): Not documented!!!!
--->-->Zeilberger((termRatio ((Record (: p0 (Expression (Integer))) (: r (Expression (Integer))) (: s (Expression (Integer)))) (Mapping (Expression (Integer)) (Expression (Integer)) (Expression (Integer))) (NonNegativeInteger) (Expression (Integer))))): Not documented!!!!
--->-->Zeilberger((gosper ((Record (: pa (SparseUnivariatePolynomial (Expression (Integer)))) (: pb (SparseUnivariatePolynomial (Expression (Integer)))) (: pc (SparseUnivariatePolynomial (Expression (Integer))))) (SparseUnivariatePolynomial (Expression (Integer))) (SparseUnivariatePolynomial (Expression (Integer)))))): Not documented!!!!
--->-->Zeilberger((upperBound ((Union (Integer) failed) (Record (: pa (SparseUnivariatePolynomial (Expression (Integer)))) (: pb (SparseUnivariatePolynomial (Expression (Integer)))) (: pc (SparseUnivariatePolynomial (Expression (Integer))))) (SparseUnivariatePolynomial (Expression (Integer)))))): Not documented!!!!
--->-->Zeilberger((equationsSetup ((Record (: eqs (List (Equation (Expression (Integer))))) (: unk (List (Expression (Integer)))) (: cert (Expression (Integer)))) (Mapping (Expression (Integer)) (Expression (Integer)) (Expression (Integer))) (NonNegativeInteger) (Expression (Integer))))): Not documented!!!!
--->-->Zeilberger((solveEquations ((Union (List (Equation (Expression (Integer)))) failed) (Record (: eqs (List (Equation (Expression (Integer))))) (: unk (List (Expression (Integer)))) (: cert (Expression (Integer))))))): Not documented!!!!
--->-->Zeilberger((recEq ((Union (Equation (Expression (Integer))) failed) (Mapping (Expression (Integer)) (Expression (Integer)) (Expression (Integer))) (NonNegativeInteger) (Expression (Integer))))): Not documented!!!!
; compiling file "/var/aw/var/LatexWiki/ZEIL.NRLIB/ZEIL.lsp" (written 13 JUN 2026 01:20:09 PM):

; wrote /var/aw/var/LatexWiki/ZEIL.NRLIB/ZEIL.fasl
; compilation finished in 0:00:00.704
------------------------------------------------------------------------
   Zeilberger is now explicitly exposed in frame initial 
   Zeilberger will be automatically loaded when needed from 
      /var/aw/var/LatexWiki/ZEIL.NRLIB/ZEIL

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--)co zeilberger

X ==> EXPR INT

Type: Void

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S := operator 'S

$\label{eq1}S$

(1)

Type: BasicOperator?

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-- EJn ..... expected
-- SJn ..... calculated (recEq)

----
J:=1

$\label{eq2}1$

(2)

Type: PositiveInteger?

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----

F11:(X,X)->X:= (n,k) +-> binomial(n,k)

$\label{eq3}\mbox{theMap (...)}$

(3)

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

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S(n) = sum(F11(n,k),k)

$\label{eq4}{S \left({n}\right)}={\sum_{ \displaystyle k}{\hbox{\axiomType{BINOMIAL}\ } \left({n , \: k}\right)}}$

(4)

Type: Equation(Expression(Integer))

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E11:= - S(n + 1) + 2 * S(n) = 0;

Type: Equation(Expression(Integer))

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S11:=recEq(F11,1,k)

$\label{eq5}{-{S \left({n + 1}\right)}+{2 \ {S \left({n}\right)}}}= 0$

(5)

Type: Union(Equation(Expression(Integer)),...)

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F12:(X,X)->X:= (n,k) +-> binomial(n,k)^2

$\label{eq6}\mbox{theMap (...)}$

(6)

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

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S(n) = sum(F12(n,k),k)

$\label{eq7}{S \left({n}\right)}={\sum_{ \displaystyle k}{{\hbox{\axiomType{BINOMIAL}\ } \left({n , \: k}\right)}^{2}}}$

(7)

Type: Equation(Expression(Integer))

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E12:= (- n - 1)*S(n + 1) + (4*n + 2)*S(n) = 0;

Type: Equation(Expression(Integer))

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S12:=recEq(F12,1,k)

$\label{eq8}{{{\left(n + 1 \right)}\ {S \left({n + 1}\right)}}+{{\left(-{4 \ n}- 2 \right)}\ {S \left({n}\right)}}}= 0$

(8)

Type: Union(Equation(Expression(Integer)),...)

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F13:(X,X)->X:= (n,k) +-> (-1)^k*binomial(2*n,k)^3   -- Dixon

$\label{eq9}\mbox{theMap (...)}$

(9)

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

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S(n) = sum(F13(n,k),k)

$\label{eq10}{S \left({n}\right)}={\sum_{ \displaystyle k}{{{\left(- 1 \right)}^{k}}\ {{\hbox{\axiomType{BINOMIAL}\ } \left({{2 \ n}, \: k}\right)}^{3}}}}$

(10)

Type: Equation(Expression(Integer))

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E13:=(- n^2  - 2*n - 1)*S(n + 1) + (- 27 * n^2  - 27 * n - 6) * S(n) = 0;

Type: Equation(Expression(Integer))

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S13:=recEq(F13,1,k)

$\label{eq11}{{{\left(-{2 \ {{n}^{2}}}-{4 \ n}- 2 \right)}\ {S \left({n + 1}\right)}}+{{\left(-{{54}\ {{n}^{2}}}-{{54}\ n}-{12}\right)}\ {S \left({n}\right)}}}= 0$

(11)

Type: Union(Equation(Expression(Integer)),...)

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F14:(X,X)->X:= (n,k) +-> 4^(-k)*binomial(n,2*k)*binomial(2*k,k)

$\label{eq12}\mbox{theMap (...)}$

(12)

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

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S(n) = sum(F14(n,k),k)

$\label{eq13}{S \left({n}\right)}={\sum_{ \displaystyle k}{{{4}^{- k}}\ {\hbox{\axiomType{BINOMIAL}\ } \left({{2 \ k}, \: k}\right)}\ {\hbox{\axiomType{BINOMIAL}\ } \left({n , \:{2 \ k}}\right)}}}$

(13)

Type: Equation(Expression(Integer))

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E14:= (- n - 1)*S(n + 1) + (2*n + 1)*S(n) = 0;

Type: Equation(Expression(Integer))

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S14:=recEq(F14,1,k)

$\label{eq14}{{{\left(- n - 1 \right)}\ {S \left({n + 1}\right)}}+{{\left({2 \ n}+ 1 \right)}\ {S \left({n}\right)}}}= 0$

(14)

Type: Union(Equation(Expression(Integer)),...)

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F15:(X,X)->X:= (n,k) +-> binomial(n+k,2*k)*binomial(2*k,k)*(-1)^k/(k+1) -- only 2 eq 4 unk

$\label{eq15}\mbox{theMap (...)}$

(15)

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

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S(n) = summation(F15(n,k),k) -- sum hangs!!!

$\label{eq16}\begin{array}{@{}l} \displaystyle {S \left({n}\right)}= \ \ \displaystyle {\sum_{ \displaystyle k}{\frac{{{\left(- 1 \right)}^{k}}\ {\hbox{\axiomType{BINOMIAL}\ } \left({{2 \ k}, \: k}\right)}\ {\hbox{\axiomType{BINOMIAL}\ } \left({{n + k}, \:{2 \ k}}\right)}}{k + 1}}}$

(16)

Type: Equation(Expression(Integer))

fricas

E15:= (- n^2  - 3 * n - 2)*S(n + 1) + (n^2  + n)*S(n) = 0 ;

Type: Equation(Expression(Integer))

fricas

S15:=recEq(F15,1,k)

$\label{eq17}{{{\left(-{{n}^{2}}-{3 \ n}- 2 \right)}\ {S \left({n + 1}\right)}}+{{\left({{n}^{2}}+ n \right)}\ {S \left({n}\right)}}}= 0$

(17)

Type: Union(Equation(Expression(Integer)),...)

fricas

F16:(X,X)->X:= (n,k) +-> (-1)^k*binomial(n,k)*binomial(2*n-2*k,n+a)

$\label{eq18}\mbox{theMap (...)}$

(18)

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

fricas

S(n) = sum(F16(n,k),k)

$\label{eq19}\begin{array}{@{}l} \displaystyle {S \left({n}\right)}= \ \ \displaystyle {\sum_{ \displaystyle k}{{{\left(- 1 \right)}^{k}}\ {\hbox{\axiomType{BINOMIAL}\ } \left({n , \: k}\right)}\ {\hbox{\axiomType{BINOMIAL}\ } \left({{{2 \ n}-{2 \ k}}, \:{n + a}}\right)}}}$

(19)

Type: Equation(Expression(Integer))

fricas

E16:= (n - a + 1)*S(n + 1) + (- 2*n - 2)*S(n) = 0   ;  --***verify

Type: Equation(Expression(Integer))

fricas

S16:=recEq(F16,1,k)

$\label{eq20}{{{\left(n - a + 1 \right)}\ {S \left({n + 1}\right)}}+{{\left(-{2 \ n}- 2 \right)}\ {S \left({n}\right)}}}= 0$

(20)

Type: Union(Equation(Expression(Integer)),...)

fricas

F17:(X,X)->X:= (n,k) +-> binomial(x,k)*binomial(y,n-k)

$\label{eq21}\mbox{theMap (...)}$

(21)

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

fricas

S(n) = sum(F17(n,k),k)

$\label{eq22}{S \left({n}\right)}={\sum_{ \displaystyle k}{{\hbox{\axiomType{BINOMIAL}\ } \left({x , \: k}\right)}\ {\hbox{\axiomType{BINOMIAL}\ } \left({y , \:{n - k}}\right)}}}$

(22)

Type: Equation(Expression(Integer))

fricas

E17:= (- n - 1)*S(n + 1) + (y + x - n)*S(n) = 0 ;  --***verify

Type: Equation(Expression(Integer))

fricas

S17:=recEq(F17,1,k)

$\label{eq23}{{{\left(- n - 1 \right)}\ {S \left({n + 1}\right)}}+{{\left(y + x - n \right)}\ {S \left({n}\right)}}}= 0$

(23)

Type: Union(Equation(Expression(Integer)),...)

fricas

F18:(X,X)->X:= (n,k) +-> k*binomial(2*n+1,2*k+1)

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

(24)

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

fricas

S(n) = sum(F18(n,k),k)

$\label{eq25}{S \left({n}\right)}={\sum_{ \displaystyle k}{k \ {\hbox{\axiomType{BINOMIAL}\ } \left({{{2 \ n}+ 1}, \:{{2 \ k}+ 1}}\right)}}}$

(25)

Type: Equation(Expression(Integer))

fricas

E18:= (- 2 * n^2  + n)*S(n + 1) + (8*n^2  + 4*n)*S(n) = 0 ;  --***verify

Type: Equation(Expression(Integer))

fricas

S18:=recEq(F18,1,k)

$\label{eq26}{{{\left({2 \ {{n}^{2}}}- n \right)}\ {S \left({n + 1}\right)}}+{{\left(-{8 \ {{n}^{2}}}-{4 \ n}\right)}\ {S \left({n}\right)}}}= 0$

(26)

Type: Union(Equation(Expression(Integer)),...)

fricas

F19:(X,X)->X:= (n,k) +-> (-1)^k*binomial(n-k,k)*2^(n-2*k)

$\label{eq27}\mbox{theMap (...)}$

(27)

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

fricas

S(n) = sum(F19(n,k),k)

$\label{eq28}{S \left({n}\right)}={\sum_{ \displaystyle k}{{{\left(- 1 \right)}^{k}}\ {{2}^{n -{2 \ k}}}\ {\hbox{\axiomType{BINOMIAL}\ } \left({{n - k}, \: k}\right)}}}$

(28)

Type: Equation(Expression(Integer))

fricas

E19:= (n + 1)*S(n + 1) + (- n - 2)*S(n) = 0 ; --***verify

Type: Equation(Expression(Integer))

fricas

S19:=recEq(F19,1,k)

$\label{eq29}{{{\left(n + 1 \right)}\ {S \left({n + 1}\right)}}+{{\left(- n - 2 \right)}\ {S \left({n}\right)}}}= 0$

(29)

Type: Union(Equation(Expression(Integer)),...)

fricas

J:=2

$\label{eq30}2$

(30)

Type: PositiveInteger?

fricas

F21:(X,X)->X:= (n,k) +-> binomial(2*k,k)*binomial(n,k)*(-1/2)^k -- Reed-Dawson

$\label{eq31}\mbox{theMap (...)}$

(31)

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

fricas

S(n) = sum(F21(n,k),k)

$\label{eq32}\begin{array}{@{}l} \displaystyle {S \left({n}\right)}= \ \ \displaystyle {\sum_{ \displaystyle k}{{{\left(-{\frac{1}{2}}\right)}^{k}}\ {\hbox{\axiomType{BINOMIAL}\ } \left({{2 \ k}, \: k}\right)}\ {\hbox{\axiomType{BINOMIAL}\ } \left({n , \: k}\right)}}}$

(32)

Type: Equation(Expression(Integer))

fricas

E21:= (- n - 2)*S(n + 2) + (n + 1)*S(n) = 0 ;

Type: Equation(Expression(Integer))

fricas

S21:=recEq(F21,2,k)

$\label{eq33}{{{\left(- n - 2 \right)}\ {S \left({n + 2}\right)}}+{{\left(n + 1 \right)}\ {S \left({n}\right)}}}= 0$

(33)

Type: Union(Equation(Expression(Integer)),...)

fricas

F22:(X,X)->X:= (n,k) +-> 2^(-n)*(-1)^k*binomial(2*n-2*k,n-k)*binomial(n-k,k)*x^(n-2*k)

$\label{eq34}\mbox{theMap (...)}$

(34)

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

fricas

S(n) = sum(F22(n,k),k)

$\label{eq35}\begin{array}{@{}l} \displaystyle {S \left({n}\right)}= \ \ \displaystyle {\sum_{ \displaystyle k}{{{\left(- 1 \right)}^{k}}\ {{2}^{- n}}\ {{x}^{n -{2 \ k}}}\ {\hbox{\axiomType{BINOMIAL}\ } \left({{n - k}, \: k}\right)}\ {\hbox{\axiomType{BINOMIAL}\ } \left({{{2 \ n}-{2 \ k}}, \:{n - k}}\right)}}}$

(35)

Type: Equation(Expression(Integer))

fricas

-- Legendre polynomials
E22:= (n + 2)*S(n + 2) + (- 2*n - 3) * x * S(n + 1) + (n + 1)*S(n) = 0 ;

Type: Equation(Expression(Integer))

fricas

S22:=recEq(F22,2,k)

$\label{eq36}{{{\left(n + 2 \right)}\ {S \left({n + 2}\right)}}+{{\left(-{2 \ n}- 3 \right)}\ x \ {S \left({n + 1}\right)}}+{{\left(n + 1 \right)}\ {S \left({n}\right)}}}= 0$

(36)

Type: Union(Equation(Expression(Integer)),...)

fricas

F23:(X,X)->X:= (n,k) +-> 2^(-k)*factorial(n)/factorial(k)/factorial(n-2*k)

$\label{eq37}\mbox{theMap (...)}$

(37)

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

fricas

S(n) = sum(F23(n,k),k)

$\label{eq38}{S \left({n}\right)}={\sum_{ \displaystyle k}{\frac{{n !}\ {{2}^{- k}}}{{k !}\ {{\left(n -{2 \ k}\right)}!}}}}$

(38)

Type: Equation(Expression(Integer))

fricas

E23:= - S(n + 2) + S(n + 1) + (n + 1)*S(n) = 0 ; --***verify

Type: Equation(Expression(Integer))

fricas

S23:=recEq(F23,2,k)

$\label{eq39}{-{S \left({n + 2}\right)}+{S \left({n + 1}\right)}+{{\left(n + 1 \right)}\ {S \left({n}\right)}}}= 0$

(39)

Type: Union(Equation(Expression(Integer)),...)

fricas

SJn:= [S11,S12,S13,S14,S15,S16,S17,S18,S19,S22,S23];

Type: List(Union(Equation(Expression(Integer)),"failed"))

fricas

EJn:= [E11,E12,E13,E14,E15,E16,E17,E18,E19,E22,E23];

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

fricas

check:=[integer?(lhs SJn.i / lhs EJn.i) for i in 1..#SJn]

$\label{eq40}\begin{array}{@{}l} \displaystyle \left[ \mbox{\rm true} , \: \mbox{\rm true} , \: \mbox{\rm true} , \: \mbox{\rm true} , \: \mbox{\rm true} , \: \mbox{\rm true} , \: \mbox{\rm true} , \: \mbox{\rm true} , \: \mbox{\rm true} , \: \mbox{\rm true} , \: \mbox{\rm true} \right]$

(40)

Type: List(Boolean)

fricas

reduce(_and,check)

$\label{eq41} \mbox{\rm true}$

(41)

Type: Boolean