<<RISC`GeneratingFunctions`
<<RISC`HolonomicFunctions`
<<RISC`Guess`

Package GeneratingFunctions version 0.8 written by Christian Mallinger
Copyright Research Institute for Symbolic Computation (RISC),
Johannes Kepler University, Linz, Austria

HolonomicFunctions Package version 1.7.3 (21-Mar-2017)
written by Christoph Koutschan
Copyright Research Institute for Symbolic Computation (RISC),
Johannes Kepler University, Linz, Austria

Guess Package version 0.52
written by Manuel Kauers
Copyright Research Institute for Symbolic Computation (RISC),
Johannes Kepler University, Linz, Austria

fibonacci = {a[n+2]-a[n+1]-a[n]==0,a[0]==0,a[1]==1};
lucas = {a[n+2]-a[n+1]-a[n]==0,a[0]==2,a[1]==1};

(* s[n] = Sum[LucasL[3k] Fibonacci[n-k],{k,0,n}] - HW 6/26 *)

?RESubsequence

lucas3 = RESubsequence[lucas,a[n],3n]

RECauchy[lucas3,fibonacci,a[n]]

data = Table[Sum[LucasL[3k] Fibonacci[n-k],{k,0,n}],{n,0,20}]

GuessRE[data,a[n]]

?GuessRE

GuessRE[data,a[n],{1,4},{0,5}]

?Guess*

GuessMinRE[data,a[n],Infolevel->3]

ann=Annihilator[Sum[LucasL[3k] Fibonacci[n-k],{k,0,n}],S[n]]

ApplyOreOperator[ann,a[n]]

ApplyOreOperator[Sum[LucasL[3k] Fibonacci[n-k],{k,0,n}]]

Sum[LucasL[3k] Fibonacci[n-k],{k,0,n}]

?Annihilator

Sum[k! HarmonicNumber[n-k],{k,0,n}]

factorial = {a[n+1]-(n+1) a[n]==0,a[0]==1}

harmonic = {ApplyOreOperator[First[Annihilator[HarmonicNumber[n],S[n]]],a[n]],a[0]==0,a[1]==1}

cp=RECauchy[factorial,harmonic,a[n]]

data = Table[Sum[k! HarmonicNumber[n-k],{k,0,n}],{n,0,40}];

GuessMinRE[data,a[n]]

guess=GuessRE[data,a[n],5,3][[1]]

guess2={-(((2+n))^2) (4+n) a[n]+(3+n) (35+21 n+3 (n^2)) a[1+n]-(4+n) (53+26 n+3 (n^2)) a[2+n]+(4+n) (43+14 n+(n^2)) a[3+n]-(6+n) (9+2 n) a[4+n]+(5+n) a[5+n]==0,a[0]==0,a[1]==-1,a[2]==-(5/2),a[3]==-(16/3),a[4]==-(155/12)}

REPlus[cp,guess2,a[n]]

cp

(* Catalan *)

AE2DE[x f[x]^2-f[x]+1==0,f[x]]

AE2DE[{x f[x]^2-f[x]+1==0,f[0]==1},f[x]]

DE2RE[AE2DE[{x f[x]^2-f[x]+1==0,f[0]==1},f[x]],f[x],a[n]]

Annihilator[LegendreP[n,x],S[n]]

Annihilator[LegendreP[n,x],{S[n],Der[x]}]

Annihilator[LegendreP[n,x],{Der[x],S[n]}]

Annihilator[Binomial[n,k],{S[n],S[k]}]

Annihilator[StirlingS2[n,k],{S[n],S[k]}]

legendre = {ApplyOreOperator[Annihilator[LegendreP[n,x],S[n]][[1]],p[n]]==0,p[0]==LegendreP[0,x],p[1]==LegendreP[1,x]}

ode=RE2DE[legendre,p[n],F[z]]

DSolve[ode,F[z],z]

<<RISC`fastZeil`

Fast Zeilberger Package version 3.61
written by Peter Paule, Markus Schorn, and Axel Riese
Copyright Research Institute for Symbolic Computation (RISC),
Johannes Kepler University, Linz, Austria

?Gosper

Gosper[k k!,k]

Gosper[k k!,{k,0,n}]

Gosper[k!,k]

Gosper[Binomial[2k,k]a^k,{k,0,n}]

Gosper[Binomial[2k,k](1/4)^k,{k,0,n}]

Gosper[Binomial[x+k,k],{k,0,n}]

Gosper[Binomial[n,k],{k,0,n}]

Zb[Binomial[n,k],{k,0,n},n]

<<Hyper.m

Hyper[(n-1)y[n+2]-(3n-2)y[n+1]+2n y[n],y[n],Solutions->All]