restart;

read "/Users/tsprenger/Universitaet/Softwareprojekte/multsum/multsum11/multsum10.mpl";

with(multsum);

infolevel[multsum]:=3;

Rectangular and P-maximal Structuresets

rect([1,2]);

opt(binomial(n,k)^2,k,n,[[3],4]);

plotopt(binomial(n,k)^2,k,n,[3,4],border=1);

plotopt(binomial(n,i)*binomial(i,j),[i,j],n,[[2,3],1],border=0,hyperplane=false);

The Procedure 'findrec'

'findrec' with specific structureset

findrec(binomial(2*n,k)^2,k,n,structureset={[0,0],[1,0],[1,1],[1,2]});

'findrec' without optional arguments is equivalent ...

findrec(binomial(2*n,k)^2,k,n);

... to the following input:

findrec(binomial(2*n,k)^2,k,n,structureset=optimal,certificate,strategy=2,direction=up,upperkbound=1);

Find certificate with rectangular structuresets and overhead strategy

findrec(binomial(2*n,k)^2,k,n,structureset=rectangular,certificate,strategy=50,direction=up,upperkbound=1);

Find certificate with rectangular structuresets and Wegschaider's strategy

findrec(binomial(2*n,k)^2,k,n,structureset=rectangular,certificate,strategy=0,direction=up,upperkbound=1);

Find k-free recurrence with rectangular structuresets and overhead strategy

findrec(binomial(2*n,k)^2,k,n,structureset=rectangular,certificate,strategy=50,direction=up,upperkbound=0);

Find k-free recurrence with rectangular structuresets and Wegschaider's strategy

findrec(binomial(2*n,k)^2,k,n,structureset=rectangular,certificate,strategy=0,direction=up,upperkbound=0);

Find k-free recurrence with optimal structuresets

rec:=findrec(binomial(2*n,k)^2,k,n,structureset=optimal,certificate,strategy=5,direction=down,upperkbound=0);

Miscellaneous Procedures

properterm((n+1)*binomial(n,k)*x^k,k,n);

orderrec(rec);

checkrec(rec,binomial(2*n,k)^2);

shiftrec(rec);

shiftrec(rec,direction=down);

sumrec(rec,S);

infolevel[multsum]:=0:

Holonomic Recurrence Equations for Multiple Sums and Closed Forms

rec:=multsumrecursion(binomial(n,j)*binomial(j,i)*x^i*y^(j-i)*z^(n-j),[i,j],S(n));

solverec(rec,S(0)=1);

'multsumrecursion' has the same optional arguments as 'findrec'. One optional argument is added: hypersol (of type boolean) decides

whether to search for a closed form or not. After trying to solve the recurrence directly (only for recurrences of hypergeometric type), Van Hoeij algorithm is used.

multsumrecursion(binomial(n,j)*binomial(j,i)*x^i*y^(j-i)*z^(n-j),[i,j],S(n),hypersol);

rec:=multsumrecursion(binomial(n,k)*binomial(n+k,k)*binomial(k,j)^3,[k,j],S(n),upperkbound=2);

rec:=multsumrecursion(binomial(n,k)^2*binomial(n+k,k)^2,k,S(n),upperkbound=2);

rec:=multsumrecursion(binomial(r,i)*binomial(s,j)*binomial(t,n-i-j),[i,j],S(n));

solverec(rec,S(0)=1);

rec:=multsumrecursion((k1-k2)*(k1-k3)*(k2-k3)*binomial(n,k1)*binomial(n,k2)*binomial(n,k3),[k1,k2,k3],S(n));

multsumrecursion(binomial(n,k),k,n,hypersol);

multsumrecursion(binomial(n,k)^2,k,n,hypersol);

multsumrecursion(k*binomial(n,k),k,n,hypersol);

multsumrecursion((-1)^k*binomial(n,k)^2,k,n,hypersol);

rec:=multsumrecursion((-1)^k*binomial(n,k)^3,k,S(n));

solverec(rec,{S(0)=1,S(1)=0});

rec:=multsumrecursion((-1)^k*binomial(n+b,n+k)*binomial(n+c,c+k)*binomial(b+c,b+k),k,S(n));

solverec(rec,S(0)=binomial(b+c,b));

multsumrecursion(binomial(j-i,n-i)*binomial(t-u-j+i,r-u-n+i)*binomial(u,i)*binomial(t-u,j-i)*binomial(v-t,s-j),[i,j],n,hypersol);

Applying Van Hoeij algorithm leads to ...

multsumrecursion((-1)^k*binomial(n,k)*binomial(5*k,n),k,n,hypersol);

... solving the following recurrence ...

rec:=multsumrecursion((-1)^k*binomial(n,k)*binomial(5*k,n),k,n);

... of order 4

orderrec(rec);

Holonomic Differential Equations for Multiple Sums

Generating Functions of Orthogonal Polynomials

Legendre Polynomials

diffeq:=multsumdiffeq(binomial(n,k)*binomial(-n-1,k)*((1-x)/2)^k*z^n,[k,n],F(z));

dsolve(diffeq,F(z));

Hermite Polynomials

diffeq:=multsumdiffeq(n!*(-1)^k/(n-2*k)!/k!*(2*x)^(n-2*k)*z^n,[k,n],F(z));

dsolve(diffeq,F(z));