determine the recurrence satisfied by a polynomial in holonomic sequences - Maple Help

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gfun[poltorec] - determine the recurrence satisfied by a polynomial in holonomic sequences

Calling Sequence

poltorec(P, listrec, list_unknowns, u(n))

Parameters

P

-

polynomial in n and u1(n), u2(n), ... and possibly their shifts (u1(n+1), u2(n+1), ...) and repeated shifts

listrec

-

list containing, for each of u1(n), u2(n), ..., either a linear recurrence equation it satisfies or a set containing the equation together with initial conditions

list_unknowns

-

list of sequences [u1n,u2n,...] 

u

-

name; holonomic sequence name

n

-

name; variable of the holonomic sequence u

Description

• 

The poltorec(P, listrec, list_unknowns, u(n)) command returns the recurrence satisfied by the polynomial P.

  

If u1n, u2n, ... are holonomic sequence solutions of listrec[1], listrec[2], ..., the poltorec function returns a linear recurrence equation satisfied by Pn,u1n,....

Examples

withgfun:

rec1:=u1n+1=n+1u1n,u10=1:

rec2:=u2n+2=2u2n+13nu2n,u21=1,u20=1:

poltorecu1n2+2u1nu2n,rec1,rec2,u1n,u2n,un

3n739n6192n5462n4579n3363n290nun+5n5+54n4+209n3+354n2+254n+60un+1+n412n346n262n15un+2+n2+4nun+3,u0=3,u1=3,u2=12,u3=48

(1)

Cassini's identity:

fib:=Fn+2=Fn+1+Fn,F0=1,F1=1:

poltorecFn+2FnFn+12,fib,Fn,fn

fn+1+fn,f0=1

(2)

See Also

gfun, gfun['rec+rec'], gfun[`rec*rec`], gfun[parameters], gfun[poltodiffeq]


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