# FSums

A Mathematica package for determining inhomogeneous recurrences for nested sums over hypergeometric terms with nonstandard boundary conditions by Flavia Stan

## Usage instructions

• UseMultiSum[fIn, N, listSumVars] computes a recurrence by calling the FindRecurrence and FindStructureSet functions from the package MultiSum. The list of summation variables can be reordered later by the Arrange functions.

• ShiftToPositive[rec, N, varList] uses the ShiftRecurrence command from MultiSum to shift the recurrence rec such that it contains only positive shifts.

• Arrange[rec, range] arranges the notation for the F's appearing in rec such that we have F[N, listSumVars] where listSumVars is given by range. This is used if the certificate recurrence was computed using an order of variables different from the one given by the range.

• InhRecurrence[rec, N, range, CoeffBound -> const] delivers a inhomogeneous recurrence starting with the certificate recurrence rec in the variable N. If const>0 the function SplitLargeCoeff is applied for large polynomial coefficients inside the delta parts.

CoeffBound: Option for the InhRecurrence function. Default value is 0 to obtain the smallest number of FSums in the inhomogeneous part of the recurrence.

• FSum[summand, range] structure for the nested sums where the range is given as a list of the form {{sig, 0, Infinity}, {j,0,N-2}, {k,0,j+1},{l,0,N-k}}.

• SubstituteSummandInRec[rec, fIn, N, listSumVars] substitutes the summand fIn as F[N, listSumVars] in the recurrence rec.

## Software requirements

The package works best in combination with the RISC Algorithmic Combinatorics Software Package MultiSum

http://www.risc.jku.at/research/combinat/software/