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gfun

  

rectodiffeq

  

convert a linear recurrence into a differential equation

 

Calling Sequence

Parameters

Description

Examples

Calling Sequence

rectodiffeq(eqns, u(n), f(z))

Parameters

eqns

-

single equation or set of equations

u

-

name; recurrence name

n

-

name; index of the recurrence u

f

-

name; function name

z

-

name; variable of the function z

Description

• 

The rectodiffeq(eqns, u(n), f(z)) command converts a linear recurrence into a differential equation.

  

Let f be the generating function associated with the sequence u(n) where fz=n=0unzn. The rectodiffeq function returns a linear differential equation with polynomial coefficients satisfied by f.

• 

The input syntax is the same as for rsolve.  The first argument is a single recurrence relation or a set containing one recurrence relation and boundary conditions. The recurrence relation is linear in the variable u, with polynomial coefficients in n. The terms of the sequence appearing in the relation should be of the form un+k, with k as an integer.

• 

The function returns either a single differential equation, or a set containing a differential equation and initial conditions.

Examples

withgfun:

deqrectodiffeq5n+10un+aun+1un+2,u0=0,u1=0,un,ft

deq:=at+10t21ft+5t3ⅆⅆtft

(1)

diffeqtorecdeq,ft,un

5n+10un+aun+1un+2

(2)

deqrectodiffeqn10un+1un,un,yz

deq:=z11yz+zⅆⅆzyz

(3)

dsolvedeq,yz

yz=_C1ⅇzz11

(4)

See Also

dsolve

gfun

gfun/parameters

gfun[diffeqtorec]

rsolve

 


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