# Hypergeometric solutions for linear functional systems

The procedure HypergeometricSolution in the package LFS (Linear Functional Systems) is an implementation in Maple 2018 of an algorithm to construct a partiсular hypergeometric solutions for inhomogeneous systems of linear differential, difference and q-difference equations with rational-function coefficients and hypergeometric right-hand sides.

Additionally, a procedure RationalSolution construct a particular rational solutions for systems with rational-function coefficients and rational-function right-hand sides.

## A system can be represented in several forms.

The matrix form of a full-rank system:

An(x) ξ^n(y(x)) + … + A1(x) ξ(y(x)) + A0(x) y(x) = b(x),

where

• Ai(x) are matrices whose entries are rational functions of x;
• b(x) is a column vector of finite sums of hypergeometric terms of x;
• y(x) is a column vector of unknown functions;
• ξ is a pseudo-linear map.

The normal form of a first order system:

ξ(y(x)) = A(x) y(x) + b(x),

where

• A(x) is a matrix whose entries are rational functions in x.

For the differential case: ξ(y(x)) = y'(x).

For the difference case: ξ(y(x)) = y(x+1).

For the q-difference case: ξ(y(x)) = y(q x), where q is a name or an integer large then 1.

## Source

lfs.mpl - the Maple code of the package (implemented by A.A.Ryabenko, D.E.Khmelnov).

lfs_examples.mw - the Maple session file with examples.

lfs_examples.pdf - the pdf copy of that Maple session.