CA for Combinatorics, 2022
cschneid@risc.jku.at

Section 4

Symbolic summation

<<Sigma.m

mySum=SigmaSum[1/(k+1)SigmaBinomial[2k,k]/(n-k)SigmaBinomial[2n-2k-2,n-k-1],{k,0,A}]

sol=SigmaReduce[mySum]

sol=sol/.A->n-1//CollectProdSum

SigmaReduce[sol,n,Tower->{SigmaBinomial[2n,n]}]

GeneratingFunctions

Exit[]

<<RISC`GeneratingFunctions`

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

tree=GuessRE[{1,1,2,5,14,42},a[n]]

de=RE2DE[tree[[1]],a[n],f[x]]

DSolve[de,f[x],x]

Section 5

Exit[]

<<RISC`GeneratingFunctions`

fSeries=Series[E^(-x),{x,0,10}]

fL=CoefficientList[fSeries,x]

rec=GuessRE[fL,f[n]]

RSolve[rec[[1]],f[n],n]

Section 7

Exit[]

<<RISC`GeneratingFunctions`

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

rec=(n+2)H[n+2]-(2n+3)H[n+1]+(n+1)H[n]==0

de=RE2DE[rec,H[n],h[x]]

DE2RE[de,h[x],H[n]]

rec=a[n+1]-(n+1)a[n]==0

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

DE2RE[de,f[x],a[n]]

Example 1

de=y'[x]-y[x]==0

de2=ACompose[de,y[x]^2==x^2/(1-4x),y[x]]

de2[[1]]/.{y[x]->E^(x/Sqrt[1-4x]), y'[x]:>D[E^(x/Sqrt[1-4x]),x],y''[x]:>D[D[E^(x/Sqrt[1-4x]),x],x]}//Simplify

Example 2

de1=y'[x]-y[x]==0

rec=F[n+1]-(n+1)F[n]==0

de2=RE2DE[rec,F[n],y[x]]

de3=DEPlus[de1,de2,y[x]]

Example 3

rec=(n+1)^3 F[n+1]-F[n]==0

rec1=RECauchy[rec,rec,F[n]]

recH= F[n+1]-(n+1)^3F[n]==0

rec2=REHadamard[rec1,recH,F[n]]

rec3=REHadamard[rec2,rec2,F[n]]

de=RE2DE[rec3,F[n],y[x]]

<<Sigma.m

mySum=SigmaSum[SigmaBinomial[n,k]^3,{k,0,n}]

recSigma=GenerateRecurrence[mySum]

Section 8: Symbolic summation

Exit[]

<<Sigma.m;

Indefinite summation

mySum=SigmaSum[SigmaFactorial[k]k,{k,a,b}]

SigmaReduce[mySum]

mySum=SigmaSum[SigmaPower[-1,k]SigmaBinomial[n,k]SigmaHNumber[k],{k,a,b}]

SigmaReduce[mySum]

mySum=SigmaSum[SigmaSum[SigmaBinomial[n,k],{k,0,r}]^2,{r,0,b}]

res=SigmaReduce[mySum]

mySum-res/.{b->10}//Together

Definite summation

mySum=SigmaSum[SigmaHNumber[k]^0SigmaBinomial[n,k],{k,0,n}]

SigmaReduce[mySum]

rec=GenerateRecurrence[mySum]

recSol=SolveRecurrence[rec[[1]],SUM[n]]

FindLinearCombination[recSol,mySum,n,3]