In[]:=
F[x_,k_,m_]:=Gamma[x+k+m-1]/(Gamma[x]*Gamma[k])
In[]:=
Manipulate[​​Plot[F[x,k,m],{x,0.1,10},PlotRange->All],​​{{m,0},-5,5,0.25,Appearance->"Labeled"},​​{{k,2},-5,5,0.25,Appearance->"Labeled"},SaveDefinitions->True]​​
Out[]=
​
m
0
k
2
2
4
6
8
10
2
4
6
8
10
In[]:=
B[x_,k_]:=Gamma[x+k-1]/(Gamma[x]*Gamma[k])
In[]:=
FullSimplify[B[x,k]==​​B[x-1,k]+B[x,k-1]​​]
Out[]=
True
Okay, let’s see ...
In[]:=
FullSimplify[​​F[x,k,m]-F[x-1,k,m]==​​F[x,k-1,m]*(k+m-1)/(k-1)​​]
Out[]=
True
In[]:=
FullSimplify[​​F[x,k,m]==B[x,k]*​​Pochhammer[x+k-1,m]​​]
Out[]=
True
In[]:=
FullSimplify[​​F[x,k-m,m]==B[x,k]*​​Pochhammer[k-m,m]​​]
Out[]=
True
is The same function, Scaled by a Factor
In[]:=
FullSimplify[​​F[x,k-m,m]/F[x,k,m]==Pochhammer[k-m,m]/Pochhammer[x+k-1,m]]
Out[]=
True
That’s it
In[]:=
​​FullSimplify[​​F[x,k-m,m]==B[x,k]*​​Gamma[k]/Gamma[k-m]​​]
Out[]=
True
Bonus Track
In[]:=
FullSimplify[​​F[x,1-m,m]==1/Gamma[1-m]​​]
Out[]=
True