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Analytic continuation of the multiple Fibonacci zeta functions.

1. Introduction. The Riemann zeta function [zeta](s) is one of the most important objects in the study of number theory. It is a classical and well-known result that [zeta](s), originally defined on the half plane Re(s) > 1, can be analytically continued to a meromorphic function on the entire complex plane with the only pole at s = 1, which is a simple pole with residue 1 [3,6]. One way to generalize the Riemann zeta function is to define the "multiple (Euler-Riemann-Zagier) zeta function" of depth d as follows:

(1-1) [mathematical expression not reproducible]

Re([s.sub.d]) > 1, [d.summation over (j = 1)]Re([s.sub.j]) > d. Several authors have studied the analytic continuation of the multiple zeta function and proved that the multiple zeta function [zeta]([s.sub.1], ...,[s.sub.d]) of depth d can be analytically continued to a meromorphic function on all of [C.sup.d]. For example, Atkinson [5] first proved the analytic continuation of [zeta]([s.sub.1], [s.sub.2]), with applications to the study of the asymptotic behavior of the "mean values" of zeta-functions, using Poisson summation formula. In [4], Arakawa and Kaneko used analytic continuation of [zeta]([s.sub.1], ..., [s.sub.d]) as a function of one variable [s.sub.d] when [s.sub.1], ..., [s.sub.d-1] are positive integers and discussed the relation among generalized Bernoulli numbers. For a general d, Zhao [12] proved the analytic continuation of [zeta]([s.sub.1], ..., [s.sub.d]) as a function of d variables using the theory of generalized function and Akiyama, Egami and Tanigawa [1] proved the same result by applying the classical Euler-Maclaurin formula to the index of the summation [n.sub.d]. Recently, Mehta et al., in [9] obtained the meromorphic continuation of multiple zeta functions by means of an elementary and simple translation formula for this multiple zeta function.

The sequence of Fibonacci numbers is defined by the recurrence relation

[F.sub.n] = [F.sub.n-1] + [F.sub.n-2], n [greater than or equal to] 2

with initial values [F.sub.0] = 0, [F.sub.1] = 1. We denote the n-th term of the Fibonacci sequence by [F.sub.n] and the Binet form of [F.sub.n] is [[alpha].sup.n] - [[beta].sup.n]/[alpha] - [beta], where [alpha] = 1 + [square root of 5]/2 and [beta] = 1 - [square root of 5]/2. The Fibonacci zeta function is the series

[[zeta].sub.F](s) = [[infinity].summation over (n = 1)]1/[F.sup.s.sub.n]

and this series is absolutely convergent for Re(s) > 0. Also it can be considered as an analogue of the Riemann zeta function [zeta](s) = [[infinity].summation over (n = 1)] 1/[n.sup.s] for Re(s) > 1. Andre-Jeannin [2] proved that [[zeta].sub.F] (1) = [[infinity].summation over (n = 1)]1/[F.sub.n] is an irrational number. Duverney et al., in [7] proved that [[zeta].sub.F] (2m) for m = 1, 2, ... are all transcendental numbers and Elsner et al. [8] proved that [[zeta].sub.F](2), [[zeta].sub.F](4) and [[zeta].sub.F](6) are algebraically independent. In 2001, Navas [10] obtained analytic continuation of the Fibonacci Dirichlet series [[zeta].sub.F] (s). Also Ram Murty [11] obtained that [[zeta].sub.F] (2m) are transcendental for m [greater than or equal to] i using the theory of modular form and a result of Nesterenko, which is a slight modification of Duverney's proof [7].

There is a connection between the special values of zeta function at positive integers with theoretical physics. In eighteenth century, Euler investigated the double zeta values. Though the multiple zeta values are extremely important, we do not want to discuss in details here. Similarly, the arithmetic nature of the multiple Fibonacci zeta values are important.

The special values of the Fibonacci zeta function stimulate us to study the analytic continuation of the multiple Fibonacci zeta function. In particular, we investigate the multiple Fibonacci zeta function which is defined as

(1-2) [mathematical expression not reproducible]

where [F.sub.n] is the n-th Fibonacci number. In this situation, the sum [s.sub.1] + ... + [s.sub.d] is called the weight of [[zeta].sub.F] ([s.sub.1], ..., [s.sub.d]) and d is called its depth. In this paper, we study the analytic continuation of the defined series in (1.2) for d = 2 on all of [C.sup.2] with a complete list of poles and their corresponding residues. Moreover, we investigate the arithmetic nature of multiple Fibonacci zeta functions at negative integer arguments. To the best of our knowledge, this is the first work in this area.

2. Convergence of the multiple Fibonacci zeta functions. For the sake of completeness, we include here a small section on the convergence of the multiple Fibonacci zeta functions.

Proposition 1. The infinite sum

[mathematical expression not reproducible]

converges absolutely in the domain

[D.sub.2] := {([s.sub.1],[s.sub.2]) [member of] [C.sup.2] | Re([s.sub.1]) > 0, Re([s.sub.2]) > 0}.

Proof. One can observe that

(2.i) [mathematical expression not reproducible]

We know that Fibonacci numbers grow exponentially and also [F.sub.n] [greater than or equal to] [[alpha].sup.n]. Thus, for [[sigma].sub.1] := Re([s.sub.1]) > 0 and [[sigma].sub.2] := Re([s.sub.2]) > 0, we have

(2.2) [mathematical expression not reproducible]

and

(2.3) [mathematical expression not reproducible]

From (2.1), (2.2) and (2.3), we get

[mathematical expression not reproducible]

This finishes the proof of the proposition.

It is natural to ask whether the domain of convergence of [C.sub.F]([s.sub.1], [s.sub.2]) is extendable or not. In the following section, we give an affirmative answer to this question.

3. Analytic continuation of the multiple Fibonacci zeta functions.

Theorem 2. The multiple Fibonacci zeta function [C.sub.F]([s.sub.1], [s.sub.2]) of depth d = 2 can be analytically continued to a meromorphic function on [C.sup.2]. It has poles on the hyperplanes

[s.sub.2] = -2l + i[pi](l + 2n)/log [alpha](l,n [member of] Z,l [greater than or equal to] 0),

and

[s.sub.1] + [s.sub.2] = -2(k + l) + i[pi](k + l + 2m)/log [alpha] (k,lm [member of] Z,k,l [greater than or equal to] 0)

Moreover, all the poles are simple. Proof. For any z 2 C,

[mathematical expression not reproducible]

The above binomial series converges as [alpha] > i. Substituting this into the multiple Fibonacci zeta function in (1.2) for d = 2, we get

(3.1) [mathematical expression not reproducible]

Since [mathematical expression not reproducible] we have

[mathematical expression not reproducible]

Let us denote

[mathematical expression not reproducible]

By interchanging the order of summation in (3.1), we have

(3.2) [mathematical expression not reproducible]

For any [s.sub.1],[s.sub.2] [member of] C, we have

(3.3) [mathematical expression not reproducible]

for k [greater than or equal to] [k.sub.1], l [greater than or equal to] [l.sub.1] and [mathematical expression not reproducible] for l [greater than or equal to] [l.sub.2], where k1,'1 and '2 are constants given by [k.sub.1] = [k.sub.1] ([[sigma].sub.1], [[sigma].sub.2], [alpha]) [much greater than] 0, [l.sub.1] = [l.sub.1]([[sigma].sub.1], [[sigma].sub.2], [alpha]) [much greater than] 0 and [l.sub.2] = [l.sub.2]([[sigma].sub.2], [alpha]) [much greater than] 0. Define [l.sub.0] = max{[l.sub.1], [l.sub.2]}. Thus,

[mathematical expression not reproducible]

This bound is uniform, when ([s.sub.1],[s.sub.2]) varies over a compact subsets of [C.sup.2]. Thus, the series in (3.2) converges uniformly and absolutely on compact subsets of [C.sup.2] without containing any of the poles of the functions

[mathematical expression not reproducible]

Hence, (3.2) defines the analytic continuation of [[zeta].sub.F]([s.sub.1], [s.sub.2]) to a meromorphic function on [C.sup.2] with simple poles at [s.sub.2] = -2l + i[pi]/log [alpha] (l, n [member of] Z, l [greater than or equal to] 0) and [s.sub.1] + [s.sub.2] = -2(k + l) + i[pi](k + l + 2m)/log [alpha] (k, l, m [member of] Z, k, l [greater than or equal to] 0).

Remark. The poles [s.sub.2] = -2l + i[pi][l+2n)/log [alpha] lie on the lines Re([s.sub.2]) = -2l spaced at the intervals of length 2[pi]i/log [alpha]; [s.sub.2] = -2l is a pole, when l is even, and [s.sub.2] = -2l + [pi]i/log [alpha] is a pole, when l is odd. Similarly, [s.sub.1] + [s.sub.2] = -2k - 2l is a pole, when k,' are both even or both odd and [s.sub.1] + [s.sub.2] = -2k - 2l + [pi]i/log [alpha] is a pole when either k is odd and l is even or k is even and l is odd.

4. Residues of the multiple Fibonacci zeta functions at poles. From Theorem 2, we know that [s.sub.2] = -2l + i[pi](l + 2n)/log [alpha] for l [greater than or equal to] 0,n [member of] Z are the simple poles of [[zeta].sub.F]([s.sub.1], [s.sub.2]). We define the residue along the hyperplane given by the equation [s.sub.2] = -2l + i[pi](l + 2n)/log [alpha] to be the restriction to the hyperplane of the meromorphic function ([s.sub.2] + 2l - i[pi](l + 2n)/log [alpha])[[zeta].sub.F]([s.sub.1], [s.sub.2]). Here we obtain the corresponding residues.

Theorem 3. Let l, n [member of] Z and l [greater than or equal to] 0. Then the residue of the multiple Fibonacci zeta function [[zeta].sub.F]([s.sub.1],[s.sub.2]) at [a.sub.l,n] := -2l + i[pi](l + 2n)/log [alpha] is

[mathematical expression not reproducible]

Proof. First note that [mathematical expression not reproducible] is an analytic function with simple zeros at [a.sub.l,n] = -2l + i[pi](l + 2n)/log [alpha]. Then, we have

(4.1) [mathematical expression not reproducible]

The residue of the multiple Fibonacci zeta function [[zeta].sub.F]([s.sub.1],[s.sub.2]) at [a.sub.l,n] := -2l + i[pi](l + 2n)/log [alpha] is equivalent to take the restriction to the hyperplane [s.sub.2] = [a.sub.l,n]. Hence, the residue of [[zeta].sub.F]([s.sub.1],[s.sub.2]) along the hyperplane [s.sub.2] = [a.sub.l,n] is

[mathematical expression not reproducible]

Further, we compute the residues at the poles which lie on the hyperplane [s.sub.1] + [s.sub.2] = -2(k + l) + i[pi](k + l +2m)/log [alpha]. Similarly, we define the residue along the hyperplane given by the equation

[s.sub.1] + [s.sub.2] = -2(k + l) + i[pi](k + l + 2m)/log [alpha]

to be the restriction to the hyperplane of the meromorphic function

([s.sub.1] + [s.sub.2] + 2(k + l) - i[pi](k + l + 2m)/log [alpha])[[zeta].sub.F]([s.sub.1],[s.sub.2])

Theorem 4. Let k', l', m' [member of] Z with k', l' [greater than or equal to] 0. Then the residue of the multiple Fibonacci zeta function [[zeta].sub.F]([s.sub.1], [s.sub.2]) at [b.sub.k',l',m'] := -2(k' + l') + i[pi](k' + l' + 2m')/log [alpha] is

[mathematical expression not reproducible]

Proof. By proceeding as in Theorem 3, we have

[mathematical expression not reproducible]

Hence the residue of [[zeta].sub.F]([s.sub.1], [s.sub.2]) along the hyperplane

[mathematical expression not reproducible]

5. Values at negative integers. Now we discuss the values of [[zeta].sub.F] ([s.sub.1], [s.sub.2]) at the negative integers. We already know that [s.sub.2] = 0, -4, -8, ... are simple poles and there are also other poles which lie on the hyperplane [s.sub.1] + [s.sub.2] = -2k - 2l when both k, l are even or odd simultaneously.

Theorem 5. Let m, n [member of] [Z.sub.[greater than or equal to]0] with n = 2 (mod 4), m = 0 (mod 4) or n is odd, m = 0 (mod 4). Then

[[zeta].sub.F](-m, -n) [member of] Q.

Proof. From (3.2), we have

(5.1) [mathematical expression not reproducible]

Note that [mathematical expression not reproducible] are 0 for k > m and l > n respectively. Therefore this is a finite sum belonging to Q([square root of 5]). Let

[mathematical expression not reproducible]

and

[mathematical expression not reproducible]

and

IV := [n.summation over (l = 0)][m.summation over (k = 0)] [[sigma].sub.n-l][[theta].sub.m-k,(n-l)]

Put [[alpha].sub.l,k] = I + II + III + IV. Let [phi] [not equal to] Id be an automorphism of Gal(Q([square root of 5])/Q) and hence [phi]([alpha]) = [beta]. Consider,

(5.3) [mathematical expression not reproducible]

Using [alpha][beta] = -1 and [([alpha][beta]).sup.n-2l] = [(-1).sup.n-2l] = [(-1).sup.n], we have [[alpha].sup.n-2l] = [(-1).sup.n][[beta].sup.-n+2l]. Similarly, we get [[alpha].sup.m+n-2(k+l)] = [(-1).sup.m+n][[beta].sup.-(m+n)+2(k+l)]. Substituting the above expression in (5.3), we obtain

(5.4) [mathematical expression not reproducible]

Similarly,

(5.5) [mathematical expression not reproducible]

Case I: (m and n are both even or both odd). Note that (m + n)/2 is an integer. Therefore from (5.4) and (5.5), we conclude that [phi](I) = IV and [phi](II) = III. Thus, from (5.2), [[alpha].sub.l,k] [member of] Q. Since [5.sup.(m+n)/2] [member of] Q, we have [[zeta].sub.F](-m, -n) [member of] Q.

Case II: (Either m is even and n is odd or m is odd and n is even).

In this case, [5.sup.(m+n)/2] [member of] [square root of 5]Q. From (5.4) and (5.5), we have [phi](I) = -IV and [phi](II) = -III. Also we know that, if [phi](x) = -x, then x - [phi](x) [member of] [square root of 5]Q. Thus, from (5.2), [[alpha].sub.l,k] is of the form [[beta].sub.l,k][square root of 5] for some [[beta].sub.l,k] [member of] Q and hence [[zeta].sub.F](-m, -n) [member of] Q.

6. Concluding remark. We know that Fibonacci zeta function has trivial zero at -2, -6, -10, ... . The determination of the zeros of the multiple Fibonacci zeta functions depth d = 2 is still unknown. It seems to be a delicate problem to find out the zeros of [[zeta].sub.F]([s.sub.1], [s.sub.2]).

doi: 10.3792/pjaa.94.64

Acknowledgements. The authors are grateful to the referee for her/his useful and helpful remarks which improved the presentation of this paper. We thank Prof. R. Thangadurai for his useful suggestions during the preparation of this manuscript. The second author was supported in part by the "INFOSYS" scholarship for senior students at Harish-Chandra Research Institute (HRI).

References

[1] S. Akiyama, S. Egami and Y. Tanigawa, Analytic continuation of multiple zeta-functions and their values at non-positive integers, Acta Arith. 98 (2001), no. 2, 107 116.

[2] R. Andre-Jeannin, Irrationalite de la somme des inverses de certaines suites recurrentes, C. R. Acad. Sci. Paris Ser. I Math. 308 (1989), no. 19, 539 541.

[3] T. M. Apostol, Introduction to analytic number theory, Springer-Verlag, New York, 1976.

[4] T. Arakawa and M. Kaneko, Multiple zeta values, poly-Bernoulli numbers, and related zeta functions, Nagoya Math. J. 153 (1999), 189 209.

[5] F. V. Atkinson, The mean-value of the Riemann zeta function, Acta Math. 81 (1949), 353 376.

[6] K. Chandrasekharan, Introduction to analytic number theory, Die Grundlehren der mathematischen Wissenschaften, Band 148, SpringerVerlag New York Inc., New York, 1968.

[7] D. Duverney, Ke. Nishioka, Ku. Nishioka and I. Shiokawa, Transcendence of RogersRamanujan continued fraction and reciprocal sums of Fibonacci numbers, Proc. Japan Acad. Ser. A Math. Sci. 73 (1997), no. 7, 140 142.

[8] C. Elsner, S. Shimomura and I. Shiokawa, Algebraic relations for reciprocal sums of Fibonacci numbers, Acta Arith. 130 (2007), no. 1, 37 60.

[9] J. Mehta, B. Saha and G. K. Viswanadham, Analytic properties of multiple zeta functions and certain weighted variants, an elementary approach, J. Number Theory 168 (2016), 487 508.

[10] L. Navas, Analytic continuation of the Fibonacci Dirichlet series, Fibonacci Quart. 39 (2001), no. 5, 409 418.

[11] M. Ram Murty, The Fibonacci zeta function, in Automorphic representations and L-functions (Mumbai, 2012), 409 425, Tata Inst. Fundam. Res. Stud. Math., 22, Hindustan Book Agency, Gurgaon, 2013.

[12] J. Zhao, Analytic continuation of multiple zeta functions, Proc. Amer. Math. Soc. 128 (2000), no. 5, 1275 1283.

Sudhansu Sekhar ROUT (*) Nabin Kumar MEHER (**)

(Communicated by Masaki Kashiwara, M.J.A., May 14, 2018)

2010 Mathematics Subject Classification. Primary 11M99; Secondary 11B39, 30D30.

(*) Institute of Mathematics and Applications, Andharua, Bhubaneswar 751 029, Odisha, India.

(**) Harish-Chandra Research Institute, HBNI, Chhatnag Road, Jhunsi, Allahabad 211 019, India.
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