restart; with(SumTools[Hypergeometric]):

po:=proc(i,j) local f;
f:=i->i-floor(i): if type(i,numeric) then
if i<=0 and f(i)=0 then return pochhammer(1,j)/product(j-h,h=0..-i):
else if i>0 and f(i)=0 then return
simplify(pochhammer(i,j)/pochhammer(f(i)+1,j))*pochhammer(f(i)+1,j)
else: if f(i) > 0 then return
simplify(pochhammer(i,j)/pochhammer(f(i),j))*pochhammer(f(i),j)
fi: fi: fi: else return pochhammer(i,j): fi: end:

UU:=proc(r1,r2,r3,r4,r5) global U,po; local en,qq,ne,A1,A2,A;
alias(p=pochhammer): en:=[r1,r2,r3,r4,r5]:
if r5=oo then ne:=1 else ne:=0 fi:
A1:=(a,b,c,d,e,k)->(-1)^(ne*k)*product(po(en[i],k),i=2..nops(en)-ne):
A2:=(a,b,c,d,e,k)->product(po(1+en[1]-en[i],k),i=2..nops(en)-ne):
A:=(a,b,c,d,e,k)->A1(a,b,c,d,e,k)/A2(a,b,c,d,e,k)*(a+2*k):
qq:=r1,r2,r3,r4,r5: U:=(n,k)->A(qq,k): end:

FF:=proc(r1,r2,r3,r4,r5) global z0,F; local o,p1,p2,t1,t2,T,pz,z1;
UU(r1,r2,r3,r4,r5): o:=Zeilberger(U(n,k),n,k,N)[1]:
p1:=-solve(factor(coeff(o,N,1)),n): p2:=-solve(factor(coeff(o,N,0)),n):
pz:=subs(n=1/n,-factor(coeff(o,N,1))/factor(coeff(o,N,0))):
z0:=subs(n=0,(simplify(pz))):
t1:=map(po,[p1],n): t2:=map(po,[p2],n):
T:=(n,k)->product(t1[i],i=1..nops(t1))/product(t2[i],i=1..nops(t2)):
F:=(n,k)->U(n,k)*T(n,k)*z0^n: z1:=Zeilberger(F(n,k),n,k,N)[1]:
if z1=N-1 or z1=1-N then return("F(n,k)"=F(n,k)) fi end:

CC:=proc(r1,r2,r3,r4,r5) local sF; global R,G,ss;
FF(r1,r2,r3,r4,r5):
sF:=simplify(Zeilberger(F(n,k),n,k,N)[2]/F(n,k)):
R:=(nn,kk)->simplify(subs(n=nn,k=kk,sF)):
G:=(n,k)->F(n,k)*R(n,k): return("R(n,k)"=R(n,k)): end:

SER:=proc(r1,r2,r3,r4,r5) local w,ss,res;
CC(r1,r2,r3,r4,r5): w:=factor(G(n,0)): Digits:=30:
ss:=evalf(Sum(w,n=0..infinity)):
if type(ss,numeric) then res:=identify(ss,extension=[Catalan]):
else res:="there is no sum from n=0": fi:
if abs(z0)>1 then res:="it is not convergent" fi:
return(Sum(w,n=0..infinity)=res): end:


entry:=1+4*n,1/2+2*n,1/2+1*n,1/2+1*n,oo; SER(entry); FF(entry); CC(entry);