In[]:=
B[k_][x_]:=Gamma[x+k-1]/(Gamma[x]*Gamma[k])
In[]:=
DI[f_][x_]:=Sum[f[n],{n,1,x}]
In[]:=
DD[f_][x_]:=f[x]-f[x-1]
Let’s do This
In[]:=
FullSimplify[DI[B[k]][x]==B[k+1][x]]
Out[]=
True
In[]:=
FullSimplify[DD[B[k+1]][x]==B[k][x]]
Out[]=
True
Okay
In[]:=
G[r_][n_]:=r^n
In[]:=
DD[G[r]][x]
Out[]=
-
-1+x
r
+
x
r
In[]:=
DD[DD[G[r]]][x]
Out[]=
-2+x
r
-2
-1+x
r
+
x
r
In[]:=
DD[DD[DD[G[r]]]][x]
Out[]=
-
-3+x
r
+3
-2+x
r
-3
-1+x
r
+
x
r
In[]:=
DI[G[r]][x]
Out[]=
r(-1+
x
r
)
-1+r
In[]:=
DI[DI[G[r]]][x]
Out[]=
r(
1+x
r
+x-r(1+x))
2
(-1+r)
Let’s see...
In[]:=
DS[f_][x_]:=Sum[f[n],{n,0,x}]
In[]:=
DS[G[r]][x]
Out[]=
-1+
1+x
r
-1+r
In[]:=
DS[DS[G[r]]][x]
Out[]=
1+
2+x
r
+x-r(2+x)
2
(-1+r)
In[]:=
Simplify[DS[G[r]][x]==DI[G[r]][x]+1]
Out[]=
True
In[]:=
DS[DS[DS[G[r]]]][x]
Out[]=
-2+6r-6
2
r
+2
3+x
r
-3x+8rx-5
2
r
x-
2
x
+2r
2
x
-
2
r
2
x
2
3
(-1+r)
Nothing interesting, until Now
In[]:=
F[m_][r_][x_]:=r^(x-m)(r-1)^m
In[]:=
FullSimplify[F[2][r][x]==DD[DD[G[r]]][x]]
Out[]=
True
In[]:=
S[m_][r_][x_]:=Sum[B[m][x-n+1]r^n,{n,0,x}]
In[]:=
FullSimplify[S[2][r][x]==DS[DS[G[r]]][x]]
Out[]=
True
Fully interesting, right Now
In[]:=
FullSimplify[S[m][r][x]/​​Sum[B[m][x-n+1],{n,0,x}]]
Out[]=
mHypergeometric2F1[1,-x,1-m-x,r]
m+x
In[]:=
FullSimplify[S[m][r][x]/​​(B[m+1][x+1])]
Out[]=
mHypergeometric2F1[1,-x,1-m-x,r]
m+x
and, seen from the other side ...
In[]:=
FullSimplify[S[m][r][x]/S[1][r][x]]
Out[]=
(-1+r)Gamma[m+x]Hypergeometric2F1[1,-x,1-m-x,r]
(-1+
1+x
r
)Gamma[m]Gamma[1+x]
In[]:=
FullSimplify[S[m][r][x]/​​S[1][r][x]/(B[m][x+1])]
Out[]=
(-1+r)Hypergeometric2F1[1,-x,1-m-x,r]
-1+
1+x
r
Which is this HyperGeometric function ?
In[]:=
HG[a_,b_,c_,z_]:=Sum[​​(Pochhammer[a,n]*Pochhammer[b,n])/​​(Pochhammer[c,n]*Gamma[1+n])*z^n,​​{n,0,Infinity}​​]
In[]:=
Simplify[HG[1,-x,1-m-x,r]]
Out[]=
Hypergeometric2F1[1,-x,1-m-x,r]
In[]:=
Simplify[DD[Function[x,​​HG[1,-x,1-m-x,r]]][n]]
Out[]=
-1+Hypergeometric2F1[1,1,m,r]
In[]:=
Simplify[DI[Function[x,​​HG[1,-x,1-m-x,r]]][n]]
Out[]=
nHypergeometric2F1[1,1,m,r]
In[]:=
FunctionExpand[HG[1,1,m,r]]
Out[]=
Hypergeometric2F1[1,1,m,r]
Ok, I’ve come to a Conclusion:
In[]:=
S[m_][r_][x_]:=Sum[B[m][x-n+1]r^n,{n,0,x}]
In[]:=
HG2F1[1,b_,c_,z_]:=S[1+b-c][z][-b]/​​S[1][z][-b]/B[1+b-c][1-b]*​​(z^(1-b)-1)/(z-1)
In[]:=
FullSimplify[​​Hypergeometric2F1[1,b,c,z]​​==HG2F1[1,b,c,z]]
Out[]=
True
This is Good
In[]:=
DS[G[z]][-b]==S[1][z][-b]
Out[]=
True
In[]:=
HG2F1[1,b_,c_,z_]:=S[1+b-c][z][-b]/​​DS[G[z]][-b]/B[1+b-c][1-b]*​​(z^(1-b)-1)/(z-1)
In[]:=
DS[G[z]][-b]
Out[]=
-
-b
z
(-z+
b
z
)
-1+z