Slode package for Maple V Release 4

	This package proposes tools to construct formal power series 
solutions of linear ordinary differential equations with polynomial 
(over the rational number field) coefficients. The package lets one  
find power series solutions with nice coefficients.
	Slode package contains procedures to find series solutions with 
hypergeometric, polynomial and rational coefficients, m-sparse and 
m-sparse m-hypergeometric series solutions.

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                     The data representations.

	The differential equations are represented by usual Maple way:
in the diff notation or in the D operator notation.

#-----------------------------

	The formal power series is represented by list with two
elements:
 1. The expression in the form:

     v0+v1*(x-x0)+...+vM*(x-x0)^M + Sum(v(n)*(x-x0)^n,n=M+1..infinity)

 2. The recurrence for v(n):
	p[0]*v(n)+p[1]*v(n-1)+...+p[r]*v(n-r)

  Where 
	x is the independent variable;
	x0 is the expansion point;
	v(n) is the name of the sequence of series' coefficients;
	v0,v1,...,vM are initial elements of sequence v(n);
	p[0],p[1]...p[r] are polynomial on n over field Q(x0);
	r is the recurrence's order.
    The number M of initial elements is such that it is possible to 
calculate any series' coefficient for the initial coefficients and 
the recurrence (taking into account the order and the zeros of the 
leading coefficients of the recurrence). 

#-----------------------------

	The m-sparse power series is represented by list with two
elements:
 1. The expression in the form:

 v0+v1*(x-x0)+...+vM*(x-x0)^M + Sum(v(m*n+N)*(x-x0)^(m*n+N),n=s..infinity)

 2. The m-sprarse recurrence for v(n):
	p[0]*v(m*n+N)+p[1]*v(m*(n-1)+N)+...+p[r]*v(m*(n-r)+N)

  Where 
	x is the independent variable;
	x0 is the expansion point;
	v(n) is the name of the sequence of series' coefficients;
	v0,v1,...,vM are initial elements of sequence v(n);
	m is the sparse-order;
	N: 0 <= N < m;
	s: (M+1-N)/m;
	p[0],p[1]...p[r] are polynomial on n over field Q(x0).

	The number of initial elements is determined similarly the 
above cases and, additionally, taking into account that for all n > M
the sequence v(n) satisfies the condition:
                 v(n) <> 0 ==> (n = N) mod m.

#-----------------------------

	The power series with hypergeometric coefficients is 
represented by the follow expression:

 v0+v1*(x-x0)+...+vM*(x-x0)^M + Sum(F(n)*(x-x0)^n,n=M+1..infinity)

  Where 
	x is the independent variable;
	x0 is the expansion point;
	v0,v1,...,vM are initial series' coefficients;
	F(n) is a function such that F(n+1)/F(n) is rational on n.

    The number of initial elements is determined such that F(n) <> 0 
for all n > M. 

#-----------------------------

	The power series with rational coefficients is represented 
by the follow expression:

 v0+v1*(x-x0)+...+vM*(x-x0)^M + Sum(F(n)*(x-x0)^n,n=M+1..infinity)

  Where 
	x is the independent variable;
	x0 is the expansion point;
	v0,v1,...,vM are initial series' coefficients;
	F(n) is a rational function of n.

    The number of initial elements is determined such that the 
denominator of F(n) is not zero for all n > M. 

#-----------------------------

	The power series with polynomial coefficients is represented 
by the follow expression:

 v0+v1*(x-x0)+...+vM*(x-x0)^M + Sum(F(n)*(x-x0)^n,n=M+1..infinity)

  Where 
	x is the independent variable;
	x0 is the expansion point;
	v0,v1,...,vM are initial series' coefficients;
	F(n) is a polynomial in n.

#-----------------------------

	The m-sparse m- hypergeometric power series is represented 
by the follow expression:

 v0+v1*(x-x0)+...+vM*(x-x0)^M + Sum(F(n)*(x-x0)^(m*n+N),n=s..infinity)

  Where 
	x is the independent variable;
	x0 is the expansion point;
	v0,v1,...,vM are initial elements of sequence v(n);
	m is the sparse-order;
	N: 0 <= N < m;
	s: (M+1-N)/m;
	F(n) is a function such that F(n+1)/F(n) is rational on n.

    The number of initial elements is determined such that F(n) <> 0 
for all n >= s.

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                         The main procedures.

`Slode/series`(eq,fn,vn,C,x0,M1)

eq - a linear ordinary differential equation with polynomial
     coefficients and polynomial right hand side: 
fn - the unknown function in the form y(x)
vn - a new function in the form v(n)
C  - a name of arbitrary constants (optional), _C by default
x0 - a point (optional argument), 0 by default
M1 - the required number of initial elements of the series 
     solutions (optional argument)

	This procedure returns a formal power series solution at 
the given point x0. The solutions involve arbitrary constants in 
the form C[j] (where 'i' is some integer).

#-----------------------------

`Slode/msparse_at_point`(eq,fn,vn,m,x0,C)

eq - a homogeneous linear ordinary differential equation with 
     polynomial coefficients
fn - the unknown function in the form y(x)
vn - the new function of the recurrence in the form v(n)
m  - a sparse-order
x0 - an m-point
C  - the name of arbitrary constants (optional argument),
     _C by default 

	This procedure returns a set m-sparse power series 
solution at the given m-point x0. The solutions involve arbitrary 
constants in form C[j] (where 'i' is some integer).

#-----------------------------

`Slode/msparse`(eq,fn,vn,C)

eq - a homogeneous linear ordinary differential equation with 
     polynomial coefficients
fn - the unknown function in the form y(x)
vn - the new function of the recurrence in the form v(n)
C  - the name of arbitrary constants (optional argument),
     _C by default 

	This procedure returns a set of m-sparse power series solution 
for all possible number m. The m-points of these series are chosen 
automatically. The solutions involve arbitrary constants in form C[i]
(where 'i' is some integer).

#-----------------------------

`Slode/hyper_at_point`(eq,fn,C,x0)
`Slode/rat_at_point`(eq,fn,C,x0)
`Slode/poly_at_point`(eq,fn,C,x0)

eq - a homogeneous linear ordinary differential equation with 
     polynomial coefficients
fn - the unknown function in the form y(x)
C  - a name of arbitrary constants (optional argument) (not _C),
     _B by default 
x0 - a point (optional argument), 0 by default

	These procedures return a power series solution with 
hypergeometric (polynomial, rational, correspondingly) coefficients 
at the given point x0. The solutions may involve arbitrary constants 
in form _C[i], C[j] (where i,j are some integer).

#-----------------------------

`Slode/mhyper_at_point`(eq,fn,m,x0,C)

eq - a homogeneous linear ordinary differential equation with 
     polynomial coefficients: 
fn - the unknown function in the form y(x)
m  - a sparse-order
x0 - an m-point
C  - a name of arbitrary constants (optional argument) (not _C),
     _B by default 

	This procedure returns a set m-sparse power series 
solution with m-hypergeometric coefficients at the given m-point x0.
The solutions may involve arbitrary constants in form _C[i], C[j]
(where i,j are some integer).

#-----------------------------

`Slode/hypercoeffs`(eq ,fn,C)
`Slode/ratcoeffs`(eq,fn,C)
`Slode/polycoeffs`(eq,fn,C)

eq - a homogeneous linear ordinary differential equation with 
     polynomial coefficients
fn - the unknown function in the form y(x)
C  - a name of arbitrary constants (optional argument) (not _C),
     _B by default 

	These procedures return a set of power series solutions with 
hypergeometric (polynomial, rational, correspondingly) coefficients. 
The points of these series are chosen automatically. The solutions 
may involve arbitrary constants in form _C[i], C[j] (where i,j are 
some integer).

#-----------------------------

`Slode/mhypercoeffs`(eq,fn,C)

eq - a homogeneous linear ordinary differential equation with 
     polynomial coefficients
fn - the unknown function in the form y(x)
C  - the name of arbitrary constants (optional argument) (not _C),
     _B by default 

	This procedure returns a set of m-sparse power series 
solution with m-hypergeometric coefficients for all possible 
number m. The m-points of these series are chosen automatically.
The solutions may involve arbitrary constants in form _C[i], C[j]
(where i,j are some integer).

#------------------------------------------------------------------