Appendix: Mode of Sums

This Appendix derives formulae for analytic continuation and limiting behaviour of mode sums used in Sections 2 and 3.

We consider first the one-dimensional massive mode sum (cf. (2.9))

We require an analytic continuation of valid for negative. Introducing the Jacobi function

one may use the integral representation of the gamma function to write

Applying Jacobi's transformation

becomes

This form provides an analytic continuation for all values of away from the pole at .

Performing the integral in (A.5) for each term in the sum (A.2) one obtains the large expansion

where is a modified Bessel function, and the prime indicates omission of the term in the sum. Some integral representations of the sum in (A.6) are given in (2.18).

An analytic continuation for multi-dimensional mode sums is obtained in direct analogy with (A.5) using the generalized Jacobi function defined by

The corresponding generalization of the large expansion (A.6) is thus

In the massless case , all mode sums may be expressed in terms of the Epstein zeta function (3.2)

An integral representation for this function in terms of the generalized Jacobi function is

where . This representation gives an analytic continuation for except for a pole at . The reflection formula (3.4)

follows.

Several further representations for appropriate in different limiting cases may be obtained from (A.10). In the limit one may sum first over , and then use an analogue of (A.8) with to obtain

The last term in both forms falls off exponentially with . When , the first term becomes , and thus exhibits a pole.

For , a convenient representation is

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