Internat. J. Math. & Math. Sci.
Vol. 23, No. 4 (2000) 289–295
S0161171200000466
© Hindawi Publishing Corp.
DOUBLE AND TRIPLE SUMMATION EXPRESSIONS
OBTAINED USING PERTURBATION THEORY
HARRY A. MAVROMATIS
(Received 4 May 1998)
Abstract. Using standard perturbation theory, simple new double and triple summation
expressions are obtained.
Keywords and phrases. Perturbation theory, hypergeometric functions, multiple summation expressions.
2000 Mathematics Subject Classification. Primary 40B05.
1. Introduction. Using standard nondegenerate perturbation theory, if the exact
energy of a system is known, we can compare this exact expression to the perturbation
expansion of the energy in powers of some small parameter. Equating equal powers of
this small parameter is shown to lead, in a conceptually direct way, to new summation
expressions with unexpectedly simple values.
2. Theory. In standard perturbation theory [7], it is shown that, given the 3-dimensional system
H = H0 + h(r ),
H0 = −
(2.1)
2
1 d
l(l + 1)
+
+ V (r ),
2 dr 2
2r 2
(2.2)
where V (r ) is a central potential and h(r ) is a small central perturbation, then
HΨnl (r ) = Enl Ψnl (r ),
(2.3)
(0)
Enl Rnl (r )
(2.4)
H0 Rnl (r ) =
=
(0)
Enl |nl,
and, if nl|h(r )|nl = 0, we can write
(0)
Enl = Enl +
∞
nl|h(r )|n l
(0)
n l|h(r )|nl
(0)
Enl − En l
∞
nl|h(r )|n l n l|h(r )|n l n l|h(r )|nl
+
(0)
(0)
(0)
(0)
Enl − En l Enl − En l
n ,n =n
n =n
290
HARRY A. MAVROMATIS
nl|h(r )|n l n l|h(r )|n l n l|h(r )|n l n l|h(r )|nl
(0)
(0)
(0)
(0)
(0)
(0)
Enl − En l Enl − En l Enl − En l
n ,n ,n =n
∞ ∞
nl|h(r )|n l n l|h(r )|nl nl|h(r )|n l n l|h(r )|nl
−
+ · · · (2.5)
(0)
(0)
(0)
(0) 2
Enl − En l
n =n
n =n
Enl − En l
+
∞
∞
which is a converging series for appropriate h(r )’s. Here, nl|h(r )|n l = 0 Rnl (r )×
h(r )Rn l (r )r 2 dr , etc.
The new summation results obtained are (2.6), (2.9), (2.10), (2.11), and (2.12). Of
these, the simplest is
∞
1
n> ! Γ (n< + β + 1)
1
=
,
β n =n n< ! Γ (n> + β + 1) (n − n)
β = 0 ,
(2.6)
where n> (n< ) is the biggest (smallest) of n, n . If β is an integer, equation (2.6) reduces to
∞
1
n> ! (n< + β)!
1
=
.
β n =n n< ! (n> + β)! (n − n)
(2.7)
If n = 0, this reduces to the standard 1-parameter expression [2, p. 11]
∞
(k − 1)!
1
=
.
ββ! k=1 (k + β)!
(2.8)
The other four expressions are:


∞
k−1
(k − 1)!  1 
1
=
,
β2 β! k=1 (k + β)! m=1 m
(2.9)

2
∞
∞
1 1
(i + β)!  (j − 1)! 
=
,
β3 β! i=1 ii! j>i (j + β)! 
(2.10)
∞
∞
∞
1 1
1 1 (k − 1)!
=
,
β3 β! i=1 i j<i j k>i (k + β)!


2
 ∞

∞
1 1
1 (i − 1)!  1 
1
=
.
−

β3 β!
2
(i + β)! 
j
βi 

i=1
(2.11)
(2.12)
j<i
If β is not an integer, all factorials involving β must be replaced by γ functions.
Thus, (2.9) becomes


∞
k−1
1
(k − 1)!  1 
1
=
,
β2 Γ (β + 1) k=0 Γ (k + β + 1) m=1 m
and so on.
β = 0 ,
(2.13)
DOUBLE AND TRIPLE SUMMATION EXPRESSIONS . . .
291
3. Derivation. Consider the case when in (2.2) V (r ) = (1/2)r 2 , i.e., we have a 3dimensional harmonic oscillator system. Then [4],
Rnl (r ) =
1/2 l −r 2 /2
2Γ n + l + 3/2
r e
3 2
.
1 F1 − n; l + ; r
n!
Γ (l + 3/2)
2
(3.1)
Here, n = 0, 1, 2, . . . , l = 0, 1, 2, . . . , and the confluent hypergeometric function
[3, p. 1045]
n
Γ (−n + m)Γ l + 3/2 r 2m
3 2
;
r
F
−
n;
l
+
=
1 1
2
Γ (−n)Γ l + 3/2 + m m!
m=0
(3.2)
is an n + 1-term polynomial in r 2 .
For this V (r ), the unperturbed energy in (2.4) is
3
(0)
Enl = 2n + l + .
2
(3.3)
The bit of ingenuity required here is to use, for the central perturbation in (2.1),
h(r ) = α
1
1
.
−
2r 2 2l + 1
(3.4)
Then,
H =−
l(l + 1) + α 1 2
α
1 d2
+
+ r −
2 dr 2
2r 2
2
2l + 1
(3.5)
l (l + 1) 1 2
α
1 d2
,
+
+ r −
2
2 dr
2r 2
2
2l + 1
(3.6)
can be written as
H =−
where
l2 + l − l2 − l − α = 0,
(3.7)
−1 + (2l + 1) 1 + (4α)/(2l + 1)2
l =
2
α2
2α3
5α4
α
−
+
−
+··· .
= l+
2l + 1 (2l + 1)3 (2l + 1)5 (2l + 1)7
(3.8)
i.e.,
Thus, for this h(r ), the exact energy in (2.3) and (2.5) is known, namely,
Enl = 2n + l +
α
3
−
.
2 2l + 1
(3.9)
292
HARRY A. MAVROMATIS
As is shown below, for this h(r ), nl|h(r )|nl = 0, for all values of n, and we have
(0)
Enl − Enl
α
3
3
α
−
− 2n + l +
= l − l −
2 2l + 1
2
2l + 1
2α3
5α4
α2
+
−
+···
=−
(2l + 1)3 (2l + 1)5 (2l + 1)7
∞ nl|1/2r 2 |n l n l|1/2r 2 |nl
= α2
2(n − n )
n =n
∞
nl|1/2r 2 |n l n l|1/2r 2 |n l n l|1/2r 2 |nl
+ α3
4(n − n )(n − n )
n ,n =n,
= 2n + l +
n =n
+α
4





∞



n ,n ,n =n,
n l|1/2r 2 |n l n l|1/2r 2 |n l n l|1/2r 2 |nl
2(n − n )2(n − n )2(n − n )
nl|1/2r 2 |n l
n =n ,n =n
−
∞
n =n




∞
nl|1/2r 2 |n l n l|1/2r 2 |nl nl|1/2r 2 |n l n l|1/2r 2 |nl 
+··· .

2(n − n )
4(n − n )2


n =n

(3.10)
Then we can equate equal powers of α to obtain
∞ nl|1/2r 2 |n l n l|1/2r 2 |nl
1
=
(2l + 1)3 n =n
2(n − n)
∞
nl|1/2r 2 |n l n l|1/2r 2 |n l n l|1/2r 2 |nl
1
=
(2l + 1)5 n ,n =n,
8(n − n )(n − n )
(3.11)
(3.12)
n =n
1
(2l + 1)7




nl|1/2r 2 |n l n l|1/2r 2 |n l n l|1/2r 2 |n l n l|1/2r 2 |nl
1
=
5
2(n − n )2(n − n )2(n − n )


 n ,n ,n =n,
n =n ,n =n


∞ 

∞
2
2
2
2
nl|1/2r |n l n l|1/2r |nl nl|1/2r |n l n l|1/2r |nl 
−

2(n − n )
4(n − n )2


n =n
n =n

(3.13)
··· = ···
To evaluate (3.11), (3.12), and (3.13) we can use the general expression [5]
DOUBLE AND TRIPLE SUMMATION EXPRESSIONS . . .
Inn l (λ) = nl|r λ |n l
1/2 Γ λ/2 + l + 3/2 Γ n − λ/2
Γ n + l + 3/2
=
n! n ! Γ n + l + 3/2
Γ l + 3/2 Γ (−λ/2)
3 λ
3
λ
λ
× 3 F2 −n, + l + , + 1; l + , −n + + 1; 1 , (n ≥ n),
2
2 2
2
2
293
(3.14)
where the generalized hypergeometric function 3 F2 is an n + 1-term polynomial [3,
p. 1045], namely,
=
n
Γ (−n + p)Γ λ/2 + l + 3/2 + p Γ λ/2 + l + p
Γ (−n)Γ λ/2 + l + 3/2 Γ λ/2 + 1
p=0
Γ l/2 + 3/2 Γ − n + λ/2 + 1
.
× Γ l/2 + 3/2 + p Γ − n + λ/2 + 1 + p p!
(3.15)
We note that if any of the terms −n, λ/2 + l + 3/2, λ/2 + 1 is zero, the 3 F2 in (3.14) is
equal to 1.
The integral expression equation (3.14) (with n = n , λ = −2) confirms that nl|h(r )|
nl = 0 since
1 1 Γ l + 1/2 Γ (n + 1)
3
1
nl1/2r 2 nl =
3 F2 − n, l + , 0; l + , −n; 1
2 n! Γ l + 3/2 Γ (1)
2
2
(3.16)
1
1
=
,
2 (l + 1/2)
where we have used Γ (β + 1) = βΓ (β), and Γ (m + 1) = m! if m is an integer. Hence,
!
1
1
(3.17)
nl
2r 2 − 2l + 1 nl = 0.
This result was recently discussed in an interesting paper [1], where it is obtained
using the Hellman-Feynman theorem. Here, this result is shown to also follow from
(3.14) (with n = n , and λ = −2). We note a very interesting property of this result,
namely, that nl|1/2r 2 |nl is independent of the variable n.
Equation (3.11) gives the simplest new summation expression. Using (3.14) again
(this time for n = n ), we obtain
1/2
1
Γ (n + l + 3/2)
Γ (l + 1/2)Γ (n + 1)
nl1/2r 2 n l =
2 n! n ! Γ (n + l + 3/2)
Γ (l + 3/2)Γ (1)
3
1
× 3 F2 − n, l + , 0; l + , −n ; 1
2
2
1/2
1
1 n ! Γ (n + l + 3/2)
, (n > n).
=
2 n! Γ (n + l + 3/2)
(l + 1/2)
(3.18)
Hence,
1
1 n> ! Γ (n< + l + 3/2)
1
,
=
3
(2l + 1)
8 n =n n< ! Γ (n> + l + 3/2) (l + 1/2)2 (n − n)
(3.19)
where n> , (n< ) is the biggest (smallest) of n, n . This is just a result of (2.6) if we
substitute β = l + 1/2.
294
HARRY A. MAVROMATIS
By comparing the next power of alpha (α3 ) in (3.12), we obtain a more complicated
new double sum expression. For the special case n = 0, this reduces to
∞
1
(k − 1)!
1
=
β2 Γ (β + 1) k=1 Γ (k + β + 1)
" k−1
#
1
,
m
m=1
(β = 0).
(3.20)
This new result can be added to the known series of the form [6, p. 695]
ak
" N(k)
#
bm .
(3.21)
If β is an integer, equation (3.20) reduces to
∞
1 1
(k − 1)!
=
β2 β! k=1 (k + β)!
" k−1
#
1
.
m
m=1
(3.22)
Finally, comparing the fourth power of α, we obtain, for the simple case n = 0, β =
integer, 3 triple sum results, namely,

2
∞
∞
1 1
(i + β)!  (j − 1)! 
=
,
β3 β!
ii! 
(j + β)! 
i=1
j>i
∞
∞
∞
1 1 (k − 1)!
1 1
=
,
3
β β! i=1 i j<i j k>i (k + β)!
(3.23)


2


∞
∞

1
1
(i − 1)! 
1
1 1
=
.
−
β3 β!
2 i=1 (i + β)! 
βi 
 j<i j

4. Conclusions. We have used a term-by-term comparison between the exact expression for the energy of a particular system and the perturbation expansion of this
energy in powers of a parameter to obtain new, multiple summation expressions. The
results of the summations are remarkably simple given the multiple summations involved. All these expressions were verified for a few specific values of β using the
Mathematica program [8, p. 69]. One expects that continuing, as above, to higher orders, one can obtain quadruple sums related to 1/(β4 β!), etc.
Acknowledgement. The author would like to acknowledge KFUPM support.
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[4]
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Mavromatis: Physics Department, King Fahd University of Petroleum and Minerals,
Dhahran 31261, Saudi Arabia