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
1
Gamma[1+p]
*
Pochhammer[1,u]Pochhammer[-p,m]Pochhammer[-p,n]Pochhammer[1+p,u]
Pochhammer[1,m+u]Pochhammer[1,n+u]
*
m
(a
2
x
)
u
(cxy)
n
(b
2
y
)
m!n!u!
,{m,0,Infinity},{n,0,Infinity},{u,0,Infinity}//TraditionalForm
Out[]//TraditionalForm=
∞
∑
l=0
l
(cxy)
l
L
p
(a
2
x
)
l
L
p
(b
2
y
)
(l+p)!
=
∞
∑
m=0
∞
∑
n=0
∞
∑
u=0
(
(1)
u
(-p)
m
(-p)
n
(1+p)
u
)
m
(a
2
x
)
u
(cxy)
n
(b
2
y
)

Γ(1+p)(
(1)
m+u
(1)
n+u
)(m!n!u!)
In[]:=
f[a_,b_,c_,p_,x_,y_,M_]:=NSum
1
(l+p)!
*(cxy)^l*LaguerreL[p,l,ax^2]*LaguerreL[p,l,by^2],{l,0,M},30;
In[]:=
g[a_,b_,c_,p_,x_,y_,M_]:=NTotal@Total@Total@Table
1
Gamma[1+p]
*
Pochhammer[1,u]Pochhammer[-p,m]Pochhammer[-p,n]Pochhammer[1+p,u]
Pochhammer[1,m+u]Pochhammer[1,n+u]
*
m
(a
2
x
)
u
(cxy)
n
(b
2
y
)
m!n!u!
,{m,0,M},{n,0,M},{u,0,M},30;
In[]:=
Sum
1
(l+p)!
*(cxy)^l*LaguerreL[p,l,ax^2]*LaguerreL[p,l,by^2]/.a->2/.b->3/.c->4/.p->5/.x->1/.y->1,{l,0,Infinity}
Out[]=
-1143+19091
4

460800
In[]:=
N[%,30]
Out[]=
2.25952752231501013072960719753
In[]:=
A=f[2,3,4,5,1,1,1000]
Out[]=
2.25952752231501013072960719753
In[]:=
B=g[2,3,4,5,1,1,50]
Out[]=
2.25952752231501013072960719753
In[]:=
A-B
Out[]=
0.×
-30
10