In[]:=
$Version
Out[]=
14.1.0 for Microsoft Windows (64-bit) (July 16, 2024)
In[]:=
HoldFormSum
1
(l+p)!
*(cxy)^l*LaguerreL[p,l,ax^2]*LaguerreL[p,l,by^2],{l,0,Infinity}==Sum
j+k
(-1)
k
(a
2
x
)
j
(b
2
y
)
Binomial[p,-j+p]Binomial[p,-k+p]HypergeometricPFQ[{1,1+p},{1+j,1+k},cxy]
j!k!p!
,{k,0,p},{j,0,p}//TraditionalForm
Out[]//TraditionalForm=
∞
∑
l=0
l
(cxy)
l
L
p
(a
2
x
)
l
L
p
(b
2
y
)
(l+p)!

p
∑
k=0
p
∑
j=0
j+k
(-1)
k
(a
2
x
)
j
(b
2
y
)
p
-j+p
p
-k+p
2
F
2
(1,1+p;1+j,1+k;cxy)
j!k!p!
In[]:=
f[p_]:=Sum
1
(l+p)!
*(cxy)^l*LaguerreL[p,l,ax^2]*LaguerreL[p,l,by^2],{l,0,Infinity}
In[]:=
g[p_]:=Sum
j+k
(-1)
k
(a
2
x
)
j
(b
2
y
)
Binomial[p,-j+p]Binomial[p,-k+p]HypergeometricPFQ[{1,1+p},{1+j,1+k},cxy]
j!k!p!
,{k,0,p},{j,0,p}
In[]:=
Table[f[n]-g[n]//Simplify,{n,1,5}]
Out[]=
{0,0,0,0,0}