© Hindawi Publishing Corp. DOUBLE AND TRIPLE SUMMATION EXPRESSIONS OBTAINED USING PERTURBATION THEORY

Using standard perturbation theory, simple new double and triple summation expressions are obtained.


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.

Theory.
In standard perturbation theory [7], it is shown that, given the 3-dimensional system (2.1) where V (r ) is a central potential and h(r ) is a small central perturbation, then and, if nl|h(r )|nl = 0, we can write which is a converging series for appropriate h(r )'s.Here, nl|h(r The new summation results obtained are (2.6), (2.9), (2.10), (2.11), and (2.12).Of these, the simplest is where n > (n < ) is the biggest (smallest) of n, n .If β is an integer, equation (2.6) reduces to (2.7) If n = 0, this reduces to the standard 1-parameter expression [2, p. 11] The other four expressions are: If β is not an integer, all factorials involving β must be replaced by γ functions.Thus, (2.9) becomes and so on.

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], Here, n = 0, 1, 2,..., l = 0, 1, 2,..., and the confluent hypergeometric function [3, p. 1045] For this V (r ), the unperturbed energy in (2.4) is The bit of ingenuity required here is to use, for the central perturbation in (2.1), Then, can be written as where i.e., Thus, for this h(r ), the exact energy in (2.3) and (2.5) is known, namely, As is shown below, for this h(r ), nl|h(r )|nl = 0, for all values of n, and we have Then we can equate equal powers of α to obtain To evaluate (3.11), (3.12), and (3.13) we can use the general expression [5] I nn l (λ) = nl|r λ |n l where the generalized hypergeometric function 3 F 2 is an n + 1-term polynomial [3, p. 1045], namely, We note that if any of the terms −n, λ/2 where we have used Γ (β + 1) = βΓ (β), and 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 , (n > n). (3.18) Hence, 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.
By comparing the next power of alpha (α 3 ) in (3.12), we obtain a more complicated new double sum expression.For the special case n = 0, this reduces to Finally, comparing the fourth power of α, we obtain, for the simple case n = 0, β = integer, 3 triple sum results, namely, (3.23)

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/(β 4 β!), etc.
.20)This new result can be added to the known series of the form[6, p. 695] 2|nl is independent of the variable n.