Remark on the Hurwitz-Lerch zeta function

Various generalizations of the Hurwitz-Lerch zeta function have been actively investigated by many authors. Very recently, Srivastava presented a systematic investigation of numerous interesting properties of some families of generating functions and their partial sums which are associated with various classes of the extended Hurwitz-Lerch zeta functions. In this paper, firstly, we show that by using the Poisson summation formula, the analytic continuation of the Lerch zeta function can be explained and the functional relation for the Lerch zeta function can be obtained in a very elementary way. Secondly, we present another functional relation for the Lerch zeta function and derive the well-known functional relation for the Hurwitz zeta function from our formula by following the lines of Apostol’s argument. MSC: Primary 11M99; 33B15; 42A24; secondary 11M35; 11M36; 11M41; 42A16


Introduction and preliminaries
The Hurwitz-Lerch zeta function is defined as follows: (z, a, s) = ∞ n= z n (a + n) s a ∈ C \ Z - ; s ∈ C when |z| < ; (s) >  when |z| =  , (.) where C and Z - denote the set of complex numbers and the set of nonpositive integers, respectively. The function (z, a, s) in (.) has been studied in various ways (see, e.g., []). Recently, its generalizations have been investigated (see [-]). Very recently, Srivastava [], motivated essentially by recent works of several authors, presented a systematic investigation of numerous interesting properties of some families of generating functions and their partial sums which are associated with various classes of the extended Hurwitz-Lerch zeta functions (see also the references in []). Here we consider only the case |z| = , i.e., z = e π ix (x ∈ R), R being the set of real numbers: It is noted that, for convenience, (e π ix , a, s) in (.) is denoted simply by (x, a, s) throughout this paper.
which, by means of (.), is a meromorphic function with only a simple pole at s = . For nonintegral x, (x, a, s) becomes an entire function s. Lerch [] presented the functional equation showed that (.) could be derived from (.) by giving an elaborate argument.
It is pointed out that in order to obtain (.), a phrase in the fourth line of [, p.] 'replacing s by s, x by -a, a by x' should be changed to 'replacing s by s, x by a, a by -x' . In fact, if the phrase 'replacing s by s, x by -a, a by x' as it was in [, p.] is used, the following analogue of (.) is obtained: which, upon employing Apostol's method, yields another functional relation analogous to (.): Here we aim mainly at, first, showing that (x, a, s) in (.) becomes an entire function of s for nonintegral x ∈ R and the functional relation for (x, a, s) can be obtained by using Poisson's summation formula in a very elementary way; and, secondly, deriving the relations (.) and (.) and showing how the functional relation (.) for the Hurwitz zeta function ζ (s, a) can be obtained from (.) by just following Apostol's arguments [] and [], respectively.

An analytic continuation of (x, a, s)
where the prime denotes that n =  is omitted from the summation. The summation on the left refers to the integral values of n given by α ≤ n ≤ β; but, when either α or β is an integer, the corresponding term is halved. On the right, the summation means lim N→∞ N n=-N . It is supposed that http://www.fixedpointtheoryandapplications.com/content/2013/1/70 (p) f (t) and f (t) are continuous in α ≤ t ≤ β, the obvious one-sided continuity only being required at t = α or t = β; (q) f (t) and f (t) are such that the integrals β α f (t) dt and β α f (t) dt converge, and f is an integral of f . If either α or β is infinite, say α = -∞, further condition is required that f (t) →  and f (t) →  as t → α.
Applying Poisson's summation formula (.) to a function where, for convenience, n being an integer, Integrating by parts yields It is observed that the integral in (.) now converges for (s) > , and so I  (x, a, s) is analytic for (s) > . Integrating by parts again in (.), we get It is also observed that the integral in (.) now converges (s) > -, and so I  (x, a, s) is analytic for (s) > -. Continuing in this way, it is found that I  (x, a, s) can be continued to an entire function of s. Integrating by parts yields (.) http://www.fixedpointtheoryandapplications.com/content/2013/1/70 which converges for every fixed x with  < x < . Likewise, it is seen that It is noted that the last integral in (.) converges for (s) > , and the second summation in (.) converges for (s) > . Therefore (x, a, s) in (.) is analytic for (s) > . Integrating by parts in (.) and considering (.), we get (  .   ) http://www.fixedpointtheoryandapplications.com/content/2013/1/70 Adding (.) to (.) with the restriction of - < (s) < , we get Applying the following integral formulas to (.): we obtain, for - < (s) < , It is noted that (.) still holds for (s) <  since the two involved series converge uniformly in s for (s) ≤ δ <  (every δ < ). Finally, we get, for (s) < , which, upon replacing s by s, yields (.).

Proof of (1.11) and (1.12)
We Using (.), we have the functional equation Using the key formal identity to Riemann's second method where (x, a, s) is given in (.). In the second integral in (.), applying (.) and replacing z by /z, we have Here, if we replace s by s, x by a, a by -x in (.), and use θ (-a, x, z) = θ (a, -x, z) and the relation we are led to the relation (.). Instead, replacing s by s, x by -a, a by x in (.) as they were in [, p.], we obtain the desired identity (.). Now differentiating both sides of (.) with respect to a and using the following differential-difference equations satisfied by :