<< HolonomicFunctions.m

HolonomicFunctions package by Christoph Koutschan, RISC-Linz, Version 1.5.1 (09.08.2011)
⟶ Type  ?HolonomicFunctions  for help

ann=Annihilator[Fibonacci[n],{S[n]}]

ApplyOreOperator[ann,f[n]]

ann2 = Annihilator[n!,{S[n]}]

receqn =ApplyOreOperator[ann2[[1]],b[n]]==0

RSolve[receqn,b[n],n]

ApplyOreOperator[ann2,n!]

FullSimplify[%]

?DFinite*

annSum = DFinitePlus[ann,ann2]

Factor[annSum]

FullForm[annSum]

rec =ApplyOreOperator[annSum,Fibonacci[n]+n!]

Table[rec[[1]],{n,0,10}]

?DFinite*

Annihilator[n!+Fibonacci[n],S[n]]

Support[annSum]

Annihilator[Sqrt[n],{S[n]}]

Annihilator[Sqrt[n]-Sqrt[n],{S[n]}]

Annihilator[Sqrt[n^2],{S[n]},Assumptions->Element[n,Integers]&&n≥0]

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

Annihilator[Sum[Binomial[n,k] Fibonacci[2k+1],{k,0,n}],{S[n]}]

annF=Annihilator[x/(1-x-x^2),{Der[x]}]

annE = Annihilator[Exp[x],{Der[x]}]

annFE=DFiniteTimes[annF,annE]

ApplyOreOperator[annFE[[1]],Exp[x] x/(1-x-x^2)]

FullSimplify[%]

ode=ApplyOreOperator[annFE,f[x]]

DSolve[ode[[1]]==0,f[x],x]

?DFinite*

DFiniteDE2RE[annFE,x,n]

DFiniteRE2DE[%,n,x]

annFE

<< GeneratingFunctions.m

GeneratingFunctions Package by Christian Mallinger — © RISC Linz — V 0.69 (28-Sep-2009)

?RE2DE

rec=RECauchy[{a[n+2]-a[n+1]-a[n]==0,a[0]==0,a[1]==1},{(n+1)a[n+1]-a[n]==0,a[0]==1},a[n]]

ode = RE2DE[rec,a[n],f[x]]

rec1 = DE2RE[ode,f[x],a[n]]

ode1=RE2DE[rec1,a[n],f[x]]

Annihilator[Integrate[Sin[x]^2,{x,0,y}],{Der[y]}]

(* end of univariate *)

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

ApplyOreOperator[ann,a[n,k]]

UnderTheStaircase[ann]

annstir=Annihilator[StirlingS1[n,k],{S[k],S[n]}]

UnderTheStaircase[annstir]

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

ann=Annihilator[StirlingS1[n,k]+StirlingS2[n,k],{S[k],S[n]}]

ApplyOreOperator[ann,StirlingS1[n,k]+StirlingS2[n,k]]

Table[%,{k,0,5},{n,0,5}]

Annihilator[Sum[StirlingS1[n,k],{k,0,n}],{S[n]}]

Annihilator[Sum[StirlingS1[n,k] Binomial[m,k],{k,0,m}], {S[n],S[m]}]

{P,Q} =CreativeTelescoping[Binomial[n,k],S[k]-1,S[n]]

ApplyOreOperator[P[[1]],Binomial[n,k]]+ApplyOreOperator[(S[k]-1)**Q[[1]],Binomial[n,k]]

%//FullSimplify

Annihilator[Sum[Binomial[n,k],{k,0,n}],{S[n]}]

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

Annihilator[Sum[LegendreP[k,x],{k,0,n}],{S[n],Der[x]}]