SetDirectory[NotebookDirectory[]];
data=Import["data.mx"];
Protect[n,i,j,k,s,t];
(*Calculates the fraction (x+s)_(y+t) / (x)_y*)
PFrac[x_,y_,s_,t_]:=Pochhammer[x+y,s+t]/Pochhammer[x,s]

Section 4.3

{teleSec43,certSec43}=data[[1]];
(*Calculates the fraction f2[n+dn,i+di,k+dk,s+ds,t+dt]/f2[n,i,k,s,t]*)
f2Frac[dn_,di_,dk_,ds_,dt_]:=(-1)^(ds)*2^(-dk-ds)PFrac[k-n,s,dk-dn,ds]PFrac[-1/2-i+2k-n/2+s,2i-k-s,-di+2dk-dn/2+ds,2di-dk-ds]/(PFrac[1,2i-k-s,0,2di-dk-ds]PFrac[1,s,0,ds]PFrac[1,k-1-2t,0,dk-2dt]PFrac[1,1-2t-k,0,-2dt-dk])

p1=Coefficient[teleSec43[[1]],(Subscript[S, i])]//Factor;
p2=Coefficient[teleSec43[[1]],(Subscript[S, t])]//Factor;
p3=teleSec43[[1]]/.{(Subscript[S, i])->0,(Subscript[S, t])->0}//Factor;
q4=certSec43[[1,1]];
q5=certSec43[[1,2]];

(*Tests the recursion in Section 4.3.*)
p1 f2Frac[0,1,0,0,0]+p2 f2Frac[0,0,0,0,1]+(p3-q4-q5) f2Frac[0,0,0,0,0]+(q4/.k->k+1)f2Frac[0,0,1,0,0]+(q5/.s->s+1)f2Frac[0,0,0,1,0]//Factor

p4=q4*(1+2 i-n) (2+2 i-k-s);
p5=q5*(1+2 i-n) (-k+n) (1+2 i-k-s) (2+2 i-k-s);

Factor[(-3/2-i+2k-n/2+s)*(-1/2+i+k-n/2)(s+1)(k-2t)*p1+
Pochhammer[1+2i-k-s,2]*(1+s)*Pochhammer[k-2-2t,3]*Pochhammer[-2t-k,2]*p2+
Pochhammer[1+2i-k-s,2]*(1+s)*(k-2t)*p3-(1+2i-k-s)*(1+s)*(k-2t)/(1+2i-n)*p4-(1+s)*(k-2t)/(1+2i-n)/(n-k)*p5+1/2*(s+1)(1-2t-k)*(-1/2+i+k-n/2)*(2i-k-s)*(2+2i-k-s)*(k-n+s)/(k-n)/Pochhammer[-1/2-i+2k-n/2+s,2]/(1+2i-n)*(p4 /. k->k+1)-1/2*(k-2t)*(2+2i-k-s)*(k-n+s)/(-1/2-i+2k-n/2+s)/(1+2i-n)/(n-k)*(p5 /. s->s+1)]

Section 5.1

{teleSec51,certSec51}=data[[2]];

f4Fracii[dn_,di_,dk_,ds_,dt_]:=(-1)^(dk+dt+ds)PFrac[1-k,s,-dk,ds]PFrac[-i+k-n+s,2i-1,-di+dk-dn+ds,2di]/(PFrac[1,s,0,ds]PFrac[1,i-t,0,di-dt]PFrac[1,t-k,0,dt-dk]PFrac[3-k+2n-2t,k,-dk+2dn-2dt,dk])

p1=Coefficient[teleSec51[[2]],(Subscript[S, i])(Subscript[S, n])]//Factor;
p2=Coefficient[teleSec51[[2]],(Subscript[S, n])]/.(Subscript[S, i])->0//Factor;
p3=Coefficient[teleSec51[[2]],(Subscript[S, i])]/.(Subscript[S, n])->0//Factor;
p4=teleSec51[[2]]/.{(Subscript[S, n])->0,(Subscript[S, i])->0}//Factor;
q5=certSec51[[2,1]];
q6=certSec51[[2,2]];
q7=certSec51[[2,3]];

(*Tests the recursion in Section 5.1.*)
p1 f4Fracii[1,1,0,0,0]+p2 f4Fracii[1,0,0,0,0]+p3 f4Fracii[0,1,0,0,0]+(p4-q5-q6-q7)f4Fracii[0,0,0,0,0]+(q5/.{k->k+1})f4Fracii[0,0,1,0,0]+(q6/.{s->s+1})f4Fracii[0,0,0,1,0]+(q7/.{t->t+1})f4Fracii[0,0,0,0,1]//Factor

Section 5.2

{teleSec52,certSec52}=data[[3]];

f4Frac[dn_,di_,dj_,dk_,ds_,dt_]:=(-1)^(dk+dt+ds)PFrac[1-k,s,-dk,ds]PFrac[-i+k-n+s,2i-1,-di+dk-dn+ds,2di]/(PFrac[1,s,0,ds]PFrac[1,j-t,0,dj-dt]PFrac[1,t-k,0,dt-dk]PFrac[3-k+2n-2t,k,-dk+2dn-2dt,dk])

p1=Coefficient[teleSec52[[3]], (Subscript[S, j])(Subscript[S, n])]//Factor;
p2=Coefficient[teleSec52[[3]], (Subscript[S, n])]/.(Subscript[S, j])->0//Factor;
p3=Coefficient[teleSec52[[3]], (Subscript[S, j])]/.(Subscript[S, n])->0//Factor;
p4=teleSec52[[3]]/.{(Subscript[S, n])->0,(Subscript[S, j])->0}//Factor;

q5=certSec52[[3,1]];
q6=certSec52[[3,2]];
q7=certSec52[[3,3]];

p1 f4Frac[1,0,1,0,0,0]+p2 f4Frac[1,0,0,0,0,0]+p3 f4Frac[0,0,1,0,0,0]+(p4-q5-q6-q7) f4Frac[0,0,0,0,0,0]+(q5/.{k->k+1})f4Frac[0,0,0,1,0,0]+(q6/.{s->s+1})f4Frac[0,0,0,0,1,0]+(q7/.{t->t+1})f4Frac[0,0,0,0,0,1]//Factor

d[i_]:=1/2*(-1)^i*3^(i-1)*(i-1)!

p2

Factor[p2+p4]

help=(Numerator[q5]/(3+2 n-2 t) /.{k->1,s->0})//Factor;
c3=Coefficient[help,t^2]//Factor;
c2=Coefficient[help-c3(t-2)(t-1),t]//Factor;
c1=help-c3(t-2)(t-1)-c2(t-1)//Factor;

Factor[c3 (-1)^4 (2+j-3)/PFrac[-n-1,j+2,2,-2]+
c2 (-1)^3(2+j-2)/PFrac[-n-1,j+2,1,-1]+
c1 (-1)^2 (2+j-1)/PFrac[-n-1,j+2,0,0]]

Factor[p2+p4]

(*linux*)
Get["/users/aigner/ucloud/Shared/FloundIlse/FloDiFrancesco/hyp/riscergosum-1.2.0/RISC/HolonomicFunctions.m"];

HolonomicFunctions Package version 1.7.3 (21-Mar-2017)
written by Christoph Koutschan
Copyright Research Institute for Symbolic Computation (RISC),
Johannes Kepler University, Linz, Austria

Section 4.3 using the Package

f2[n_,i_,k_,s_,t_]:=(-1)^(1+s)*2^(-k-s)Pochhammer[k-n,s]Pochhammer[-1/2-i+2k-n/2+s,2i-k-s]/(Pochhammer[1,2i-k-s]Pochhammer[1,s]Pochhammer[1,k-1-2t]Pochhammer[1,1-2t-k]);
{teleSec43,certSec43}=FindCreativeTelescoping[f2[n,i,k,s,t],{S[k]-1,S[s]-1},{S[n],S[i],S[t]}];

Section 5.2 using the Package

f4[n_,i_,j_,k_,s_,t_]:=(-1)^(k+s+t)*Pochhammer[1-k,s]*Pochhammer[-i+k-n+s,2i-1]/Pochhammer[1,s]/Pochhammer[1,j-t]/Pochhammer[1,t-k]/Pochhammer[3-k+2n-2t,k];

(i=j)

{teleSec51,certSec51}=FindCreativeTelescoping[f4[n,i,i,k,s,t],{S[k]-1,S[s]-1,S[t]-1},{S[i],S[n]}]

(i≠j)

{teleSec52,certSec52}=FindCreativeTelescoping[f4[n,i,j,k,s,t],{S[k]-1,S[s]-1,S[t]-1},{S[i],S[j],S[n]}]

(i+1=j)

AbsoluteTiming[{teleSec53,certSec53}=FindCreativeTelescoping[f4[n,i,i+1,k,s,t],{S[k]-1,S[s]-1,S[t]-1},{S[n],S[i]}];]