read `W:\\mPOL.txt`:

HelpC:=proc(): print(`Celine1(F,n,k,N,K,degn,degN,degK)`):
print(`ApplyOper(F,n,k,P,N,K,n1,k1)`): 
end:


#Celine1(F,n,k,N,K,degn,degN,degK): Given an expression 
#in n and k, variables (symbols) n and k, symbols
#N and K denoting the fundamental shift operators
#in n and k resp. (NF(n,k):=F(n+1,k), KF(n,k):=F(n,k+1)
#outputs a conjectured linear recurrence operator
#of the form P(N,K,n) )of degrees degn,degN,degK
#resp.) in each of
#n,N,K
#(free of k) that annihilates
#F(n,k)

Celine1:=proc(F,n,k,N,K,degn,degN,degK) local P,a,var,k1,n1,eq,c,
var1,eq1,aek,i:


P:=GenPol([n,N,K],[degn,degN,degK],a,0):

var:=P[2]:

P:=P[1]:


eq:={}:

c:=0:

for n1 from 1 do
 for k1 from 0 to n1 do

 eq1:=ApplyOper(F,n,k,P,N,K,n1,k1):
 
  if eq1<>0 then
     c:=c+1:
     eq:=eq union {eq1}:
   fi:


if nops(eq)-nops(var)>8 then
 var1:=solve(eq,var):

   aek:={}:

     for i from 1 to nops(var1) do
        if op(1,var1[i])=op(2,var1[i]) then
          aek:=aek union {op(1,var1[i])}:
        fi:
   od:


if nops(aek)>1 then
  print(`Too much slack, reduce one of the degrees`):
fi:



 P:=subs(var1,P):
  if P=0 then
    RETURN(FAIL):
   else

    P:=subs({aek[1]=1,seq(aek[i]=0,i=2..nops(aek))},P):

    RETURN(P):
   fi:
fi:
 
 od:
od:




end:


#ApplyOper(F,n,k,P,N,K,n1,k1): P(n,N,K)F(n1,k1)
ApplyOper:=proc(F,n,k,P,N,K,n1,k1) local s,i,j:

s:=0:

for i from ldegree(P,N) to degree(P,N) do
 for j from ldegree(P,K) to degree(P,K) do
   s:=s+
  subs(n=n1,coeff(coeff(P,N,i),K,j))*eval(subs({n=n1+i,k=k1+j},F)):
 od:
od:

s:
   




end: