5 Application of the Uniformly Charged Sphere Stabilization
113
n
l
m
V st (r)|nlm =
V 0
b 3
Q
l +l+2
n l nl (bR) − bRQ
l +l+1
n l nl (bR)
N n l N nl δ l l δ m m ,
(5.20)
where Q m
n l nl (x) is the auxiliary integral
Q
m
n l nl (x) =
∞
x
e
−z z
m L
2l +2
n
(z)L
2l+2
n
(z)dz.
(5.21)
For the calculation of integrals (5.18)–(5.20) the method based on the recurrence
relation for product of Laguerre polynomials has been used [51]. The main relation of this method can be obtained by substituting the recursion (n + 1)L k
n+1 (z) −
(2n + k + 1 − z)L k
n (z) + (n + k)L k
n−1 (z) = 0 [40] for L 2l+2
n
or L
2l +2
n
polynomial in
(5.21) which gives two different three-term recursions for integrals over the indexes
n and n . Elimination of the integral Q
m+1
n l nl (x) from the both recursions leads to the
new relation
nQ
m
n l nl (x) =
n
+ 1
Q
m
n +1l n−1l (x) +
n
+ 2l
+ 2
Q
m
n −1l n−1l (x)
− (n + 2l + 1)Q
m
n l n−2l (x) + 2
n + l − n
− l
− 1
Q
m
n l n−1l (x).
(5.22)
This relation is applied for construction of the ascending recursion which starts from
the elements Q m
kl0l (x) with n − n + 1 ≤ k ≤ n + n − 1 and finishes at Q m
n lnl (x)
(assume that n ≤ n ). The integrals with m = 2l + 2 and x = 0 can be evaluated
directly using the orthogonality relation
Q
2l+2
n lnl (0) =
(n + 2l + 2)!
n!
δ n n .
(5.23)
For integrals with m < 2l + 2 and x = 0 the recursion (5.22) is initialized by the
elements
Q
m
0l0l (0) = m!,
(5.24)
Q
m
1l0l (0) = Q
m
0l1l (0) = m!(2l + 2 − m),
(5.25)
Q
m
1l1l (0) = m!
(m + 1)(m + 2) − 2(m + 1)(2l + 3) + (2l + 3)
2
. (5.26)
All elements Q m
kl0l (0) needed for the evaluation of integrals (5.18) and (5.19) are
calculated by means of the special three-term recursion (n + 1)Q m
n +1l0l (x) − (2n +
2l + 1)Q m
n l0l (x) + (2n + 2l + 1)Q
m+1
n l0l (x) + (n + 2l + 2)Q m
n −1l0l (x) = 0.
The case of the integrals (5.20) is more complicate. The elements Q m
n l0l (x) and
Q m
n l1l (x) (m = 2l + 2 or m = 2l + 1) needed for the initializing (5.21) can be taken
analytically
Q
m
n lnl (x) =
n +n
t=0
Γ (m + t + 1, x)W t
n
, l
, n, l
,
(5.27)
113
n
l
m
V st (r)|nlm =
V 0
b 3
Q
l +l+2
n l nl (bR) − bRQ
l +l+1
n l nl (bR)
N n l N nl δ l l δ m m ,
(5.20)
where Q m
n l nl (x) is the auxiliary integral
Q
m
n l nl (x) =
∞
x
e
−z z
m L
2l +2
n
(z)L
2l+2
n
(z)dz.
(5.21)
For the calculation of integrals (5.18)–(5.20) the method based on the recurrence
relation for product of Laguerre polynomials has been used [51]. The main relation of this method can be obtained by substituting the recursion (n + 1)L k
n+1 (z) −
(2n + k + 1 − z)L k
n (z) + (n + k)L k
n−1 (z) = 0 [40] for L 2l+2
n
or L
2l +2
n
polynomial in
(5.21) which gives two different three-term recursions for integrals over the indexes
n and n . Elimination of the integral Q
m+1
n l nl (x) from the both recursions leads to the
new relation
nQ
m
n l nl (x) =
n
+ 1
Q
m
n +1l n−1l (x) +
n
+ 2l
+ 2
Q
m
n −1l n−1l (x)
− (n + 2l + 1)Q
m
n l n−2l (x) + 2
n + l − n
− l
− 1
Q
m
n l n−1l (x).
(5.22)
This relation is applied for construction of the ascending recursion which starts from
the elements Q m
kl0l (x) with n − n + 1 ≤ k ≤ n + n − 1 and finishes at Q m
n lnl (x)
(assume that n ≤ n ). The integrals with m = 2l + 2 and x = 0 can be evaluated
directly using the orthogonality relation
Q
2l+2
n lnl (0) =
(n + 2l + 2)!
n!
δ n n .
(5.23)
For integrals with m < 2l + 2 and x = 0 the recursion (5.22) is initialized by the
elements
Q
m
0l0l (0) = m!,
(5.24)
Q
m
1l0l (0) = Q
m
0l1l (0) = m!(2l + 2 − m),
(5.25)
Q
m
1l1l (0) = m!
(m + 1)(m + 2) − 2(m + 1)(2l + 3) + (2l + 3)
2
. (5.26)
All elements Q m
kl0l (0) needed for the evaluation of integrals (5.18) and (5.19) are
calculated by means of the special three-term recursion (n + 1)Q m
n +1l0l (x) − (2n +
2l + 1)Q m
n l0l (x) + (2n + 2l + 1)Q
m+1
n l0l (x) + (n + 2l + 2)Q m
n −1l0l (x) = 0.
The case of the integrals (5.20) is more complicate. The elements Q m
n l0l (x) and
Q m
n l1l (x) (m = 2l + 2 or m = 2l + 1) needed for the initializing (5.21) can be taken
analytically
Q
m
n lnl (x) =
n +n
t=0
Γ (m + t + 1, x)W t
n
, l
, n, l
,
(5.27)