The resolvent kernel on the discrete circle and twisted cosecant sums

Let $X_m$ denote the discrete circle with $m$ vertices. For $x,y\in X_{m}$ and complex $s$, let $G_{X_m,\chi_{\beta}}(x,y;s)$ be the resolvent kernel associated to the combinatorial Laplacian which acts on the space of functions on $X_{m}$ that are twisted by a character $\chi_{\beta}$. We will compute $G_{X_m,\chi_{\beta}}(x,y;s)$ in two different ways. First, using the spectral expansion of the Laplacian, we show that $G_{X_m,\chi_{\beta}}(x,y;s)$ is a generating function for certain trigonometric sums involving powers of the cosecant function; by choosing $\beta$ or $s$ appropriately, the sums in question involve powers of the secant function. Second, by viewing $X_{m}$ as a quotient space of $\mathbb{Z}$, we prove that $G_{X_m,\chi_{\beta}}(x,y;s)$ is a rational function which is given in terms of Chebyshev polynomials. From the existence and uniqueness of $G_{X_m,\chi_{\beta}}(x,y;s)$, these two evaluations are equal. From the resulting identity, we obtain a means by which one can obtain explicit evaluations of cosecant and secant sums. The identities we prove depend on a number of parameters, and when we specialize the values of these parameters we obtain several previously known formulas. Going further, we derive a recursion formula for special values of the $L$-functions associated to the cycle graph $X_{m}$, thus answering a question from arXiv:2212.13687v1.


Introduction
Finite trigonometric sums of the type have a long history and appear in various contexts.Two early points of reference are in Eisenstein's work and in the study of Dedekind sums [BY02].Modern appearances of these sums include the Hirzebruch signature defects and the Verlinde formulas in topology and mathematical physics [HZ74,Ve88,Do92,Za96], resistance in networks [Wu04,EW09,Ch12,Ch14b] as well as modeling angles in proteins and circular genomes [F-DG-D14].Many further instances are described in [BY02], such as the chiral Potts model in statistical physics [MO96], [Ch14a].Finite trigonometric sums are also related to Dedekind and Hardy sums and their generalizations.Several of those sums seem not to have known evaluations, but it is possible to establish reciprocal relations, see for example [BC13], [Ch18] or [MS20].
The sums C m (n) are also discrete analogs of the Riemann zeta function, as observed by Dowker in [Do92] and further developed in [FK17].This link is already implicitly present in [Ap73] where the asymptotics of the cotangent sums Recent contributions to the evaluation of trigonometric sums include [AH18], [AZ22], [GLY22] [CHJSV23] as well as [XZZ22].In [XZZ22], the authors found a precise formula for ζ(2n) as a finite linear combination, with universal constants, of the sums C m (2k) for 0 < k < n.These formulas do not involve asymptotic expansions.In the same paper, the authors obtain similar formulas for special values of Dirichlet L-functions and ask for a direct evaluation of the corresponding twisted trigonometric sums.One of the results in the present article is to provide an answer to this question posed by [XZZ22]; see section 6.We use the notation of the cosecant function csc(x) = 1/ sin(x) and the secant function sec(x) = 1/ cos(x) throughout this article.Let m and n be positive integers, and β be a positive real number, which we call a shift.Define where δ(β) = 1 if β ∈ Z and 0, otherwise.Similarly, we define the alternating sums (−1) j csc n (j + β) m π . (2) Chu and Marini proved in [CM99], among other formulas, that for any positive integer m one has that In words, the sequence of series (1) and (2) can be used to form a generating functions (3) and (4) which can be explicitly computed.As such, one can evaluate any given series (1) or (2) by computing the corresponding coefficient in the Taylor expansion on the right-hand-side of (3) or (4), respectively.Formulas for other generating functions have been derived in [CM99].For example, in [CM99] the authors evaluate the series defined by using the sequence of terms formed from the series in (1) and (2) when β = 1/2.More generally, it was proved in [WZ07] that A similar formula also is deduced for the generating function of the alternating cosecant sums (2).Other authors have studied twists of powers of cosecants by cosine function, see for example [Do92], Section 3 of [BY02] as well as Section 3 of [He20].Those authors also study secant sums and derive similar results.
In this paper we will study the cosecant sums with shift β ≥ 0 and twisted by an additive character.In doing so, we also derive results for analogously defined secant sums by suitably adjusting the shift β.More precisely, let m > 1 be an integer, let β be a positive nonintegral real number and take r ∈ {−(m − 1), . . ., 0, . . ., (m − 1)}.We define (the average value of) the twisted cosecant sums associated to those parameters and a positive integer n by The cosecant sums without the shift β are defined as Note that to get (7) from (6), one omits the term where j = 0 and then sets β = 0.The sums (7) appear in the formulas deduced in [Ta92] for the dimensions of a certain complex vector space at level k associated to a labeled Riemann surface of genus g ≥ 2. Specifically, in statement (12) of [Ta92] the aforementioned dimension is expressed in terms of (7) with r = 0, m = k + 2 and n = g − 1, while in statement (18) of [Ta92], the appropriate dimension of the "twisted" space is expressed in terms of (7) with even k, m = k + 2, r = m/2, and n = g − 1.Both expressions are special cases of Verlinde sums; see for example [Ve01,pp. 11,14].
Let m and r be as above, and let α be such that α − m 2 / ∈ Z.The (average value of) the twisted secant sums associated to those parameters and a positive integer n are defined as The (average) secant sums without the shift α are defined as where {j m } * is the empty set if m is odd and contains the single number j m such that j m ≡ m 2 (mod m) in the case when m is even.In this article we will also study powers (that are not necessarily even) of cosecant and secant functions evaluated at doubled arguments.We will derive an explicit evaluation of their generating functions as well as a finite recursion formula for computation.More precisely, for real number α such that α / ∈ Z when m ≡ 0 (mod 4), α / ∈ Z + 1 2 when m ≡ 2 (mod 4) and 2α / ∈ Z + 1 2 when m is odd, we will study the sum When m is not divisible by 4 and by taking α = 0 in (10), we immediately obtain the secant sums of double argument without the shift.If m ≡ 0 (mod 4), then one needs to exclude the value of j for which j ≡ m 4 (mod m) from the range 0, . . ., m − 1 of summation in (10).Such a sum equals zero when n + r is odd and equals S m/2,r/2 (n/2) when both r and n are even.We leave the study of the special case α = 0 of the sum Sm,r (α, n) when m ≡ 0 (mod 4) and both r, n are odd to the interested reader.For any real number β such that 2β / ∈ Z when m is odd and such that β / ∈ Z when m is even, we consider the sums The (average) cosecant sums of double argument, without the shift β are defined, with the definition of {j m } * as above by The above defined cosecant and secant sums, both with and without the shift β or twist by an additive character, have been extensively studied using various methods.For example, the authors in [CM99], [BY02], [WZ07], [CS12], [Do15] used contour integration, generating series and partial fraction decomposition to evaluate those sums as well as their generating functions.The approach in [dFGK17], [dFGK18] uses recurrence relations and generating series, while [He20] starts with Taylor series expansions of powers of tangent and cotangent.In [AH18] the starting point is to use varioius results in the theory of certain special functions.Also, a discrete form of sampling theorem was used in [Ha08], while [AZ22] describes an "automated approach" for proving some trigonometric identities.
In this article, we offer a different point of view and also study a more general situation, which includes series which may include a twist by an additive character.The approach is inspired by Dowker's computation of the heat kernel on a generalized cone [Do89] and the key observation is that the resolvent for the twisted heat kernel on a cycle graph can be viewed as a generating function for certain secant and cosecant sums.
Let us now describe our approach and state our main results.

Overview of methods and illustration of results
Let X m denote the weighted Cayley graph with vertex set Z/mZ, generator set S = {−1, 1}, and weights given by the uniform probability distribution on S. Let β ∈ R be an arbitrary real parameter.Our starting point is the "twisted by an additive character" χ β (x) := exp(2πiβx) heat kernel on X m .We compute the heat kernel using two different means.First, we employ the method of averaging, by which we mean that we view Z/mZ as being covered by Z and then we sum the heat kernel on Z by the covering group mZ.Second, we use the discrete spectral expansion of the standard Laplacian on X m .Since the heat kernel under consideration is unique, the two different evaluations yield an identity.From this identity, we then compute the resolvent kernel G Xm,χ β twisted by the character χ β (or twisted Green's function, see [CY00]) for the Laplace operator on the graph X m .Essentially, the resolvent kernel is equal to the Laplace transform in the time variable of the heat kernel.
The above calculations yield an explicit identity for the resolvent kernel G Xm,χ β (x, y; s) for real β which is obtained by equating the two evaluations.The resulting formula admits a meromorphic continuation to all complex values of s.We then determine its analytic properties for different values of real parameter β at s = 0 and s = −1.The properties at s = 0 will yield results related to twisted even powers of secants and cosecants.The properties at s = −1 will yield results related to twisted, though not necessarily even, powers of shifted secants and cosecants at double arguments.Going further, we will apply the Gauss formula for primitive Dirichlet characters to get an explicit evaluation of the Dirichlet L-function associated to the cycle graph at positive integers.

Generating functions for twisted sums of even powers
To illustrate our results let us state the first main theorem.With the notation as above, let ℓ ∈ {0, . . ., m − 1} be such that ℓ ≡ r (mod m).For β / ∈ Z define the generating functions for the cosecant sums (6) and (7).The first main result is the following theorem.
Theorem 1.For all complex s with |s| sufficiently small, the series which defines f m,r (s, β) converges uniformly and absolutely.Furthermore, the function f m,r (s, β) admits a meromorphic continuation to all complex s, and we have that where T n and U n denote the Chebyshev polynomials of the first and the second kind, with the convention that U −1 (x) ≡ 0.
Similarly, for all complex s with |s| sufficiently small, the series which defines f m,r (s) converges uniformly and absolutely.Furthermore, the function f m,r (s) admits a meromorphic continuation to all complex s, and we have that For relevant information about Chebyshev polynomials see for example [GR07, Section 8.94].
For the convenience of the reader, we state the most relevant results regarding Chebyshev polynomials in the concluding section 7.3.With the contents of section 7.3 to the side, we can give a simple qualitative description of Theorem 1, which is the following: Both of the power series f m,r (s, β) and f m,r (s) are, in fact, rational functions is s with numerators and denominators given in terms of classical Chebyshev polynomials which are precisely defined in terms of the parameters m, r and β.
As the notation suggestions, (3) and ( 5) are special cases of Theorem 1 when r = 0, after one employs classical formulas for Chebyshev polynomials in terms of trigonometric and inverse trigonometric functions.Similarly, (4) follows from Theorem 1 by taking r = m/2, which is possible since it is assumed in this case that m is even.
From Theorem 1 one can derive a recurrence formula for the coefficients in the series expansion of f m,r (s, β).More or less, if P (s) is a convergent Taylor series at s = 0, and if we have that where Q 1 (s) and Q 2 (s) are polynomials, then one simply needs to equate the coefficents of s in the expression Q 2 (s)P (s) = Q 1 (s).As it turns out in this case, there are convenient formulas for the series expansions of the Chebyshev polynomials T m (z) and U m (z) at z = 1; see 7.3.From these computations, we arrive at the following corollary.
Then we have the recurrence relation that where ãm (0 Similarly, when β ∈ Z and n ≥ 0, define the numbers Then we have the recurrence relation that From Theorem 1 and Corollary 2 one can obtain an abundance of specific formulas, each one of which can be described as mathematically appealing.For example, we will show that for any k ≥ 1 one has that as well as that where ω is a primitive third root of unity.The recursive formulas in Corollary 2 allow one to readily evaluate series with higher powers.Again, these formulas are special evaluations of the above stated main Theorem.
Remark 3. In [Za96] Zagier proved a different recursion relation between certain cosecant sums.Our formula is simpler in the sense that it is linear and whereas the formula in [Za96] is quadratic.Our formulas are thus analogous to linear recursion relations between zeta values like those found from, for example, [F16], [FK17], [Me17] and references therein.
We shall now consider secant sums.Let the parameters m and r be defined as above.For any real number α such that α − m/2 / ∈ Z define the generating function S m,r (α, n + 1)s n associated to the sequence of series (8).Additionally, define the generating function associated to the sequence of secant sums (9).By taking β = α − m/2 in Theorem 1, we immediately deduce the following corollary.
Corollary 4. For all complex s with |s| sufficiently small, the series which defines h m,r (s, α) converges uniformly and absolutely.Furthermore, the function h m,r (s, α) admits a meromorphic continuation to all complex s, and we have that As for (18), there are two cases to consider.If n); hence, the evaluation for (18) in this case is given by (13).

Generating functions for twisted sums at double arguments
As we will show, the resolvent kernel G Xm,χ β (x, y; s) at s = −1 yields the generating function the powers of secants and cosecants at double arguments.In particular, see Section 5, Theorem 11 for our second main result, which is the evaluation of the generating functions associated to the sequences of the sums (10) and of the sums (11).As an application of Theorem 11, we obtain the succinct formulas that where ω is a primitive third root of unity and k ≥ 1.As in the previous section, we state and prove recursive relations for the sequences of these sums.

Evaluation of the Dirichlet L-function of a cycle graph
Let m > 1 be an integer.The Dirichlet L-function of a cycle graph X m is the spectral L-function corresponding to the spectrum of a combinatorial Laplacian.Specifically, the function is defined for any even Dirichlet character χ of modulus m and any complex number s by see [F16,XZZ22].For odd Dirichlet characters the similar sum is identically 0. However, the authors in [XZZ22] propose a replacement.Specifically, it is suggested that one should consider the function The functions ( 21) and ( 22) can be used to evaluate the classical Dirichlet L-functions at even and odd integers, respectively; see [XZZ22].Hence, it is of interest to deduce an explicit evaluation of those functions.In Section 6 we will prove that for any even, primitive Dirichlet character χ one has where the coefficients c m,r (n − 1) are explicitly computable for all positive integers n when using the linear recurrence (15).In summary, from Theorem 1 and Corollary 2 one has a method by which (23) is explicitly computable in terms of coefficients of Chebyshev polynomials.The main theorem in [XZZ22] proves a relation involving the values of the Dirichlet L-functions at positive integers in terms of the values (23); see Theorem A of [XZZ22].In Section 5 of [XZZ22] the authors posed the question of determining a direct way by which one can evaluate (23), so then one can evaluate Dirichlet L-functions.Our results from Section 6 answer this question as stated in [XZZ22].
An explicit expression for values of ( 22) can be proved by differentiating the shifted L-function with respect to β.This computation is described in Section 7.2.

Organization of the article
In the next section we recall material from the literature regarding the continuous time heat kernel on a Cayley graph.As stated, for this paper the Cayley graph we consider is associated to Z/mZ, which is the group of integers modulo m with edges given by connecting an edge to its two nearest neighbors.In Section 3 we define and study the corresponding resolvent kernel, which amounts to the Laplace transform in the time variable of the heat kernel.In Section 4 we prove the main results as stated above, and in Section 5 we develop further general results associated to secant and cosecant sums with doubled arguments.In Section 6 we answer the aforementioned question posed in [XZZ22] which involves certain special values of spectral L-functions with a Dirichlet character.Finally, in Section 7, we present a few concluding remarks which suggest further studies which could be undertaken based on the results and methods presented in this article.
2 Heat kernel on Cayley graphs

Weighted Cayley graphs of abelian groups
Let G be a finite or countably infinite abelian group with composition law which is written additively.Let S ⊆ G be a finite symmetric subset of G.The symmetry condition means that if s ∈ S then −s ∈ S.
Let α : S → R >0 be a function such that α(s) = α(−s).The weighted and undirected Cayley graph X = C(G, S, α) of G with respect to S and α is constructed as follows.The vertices of X are the elements of G, and two vertices x and y are connected with an edge if and only if x − y ∈ S. The weight w(x, y) of the edge (x, y) is defined to be w(x, y) := α(x − y).One can show that X is a regular graph of degree If α is a probability distribution on S, then the degree of the graph X equals 1.In this case we will denote α by π S .
A function f : The set of L 2 -functions on G is a Hilbert space L 2 (G, C) with respect to the classical scalar product of functions We will denote by δ x the standard delta function, meaning δ x (x) = 1 and δ x (y) = 0 for x = y.The adjacency operator When X is finite, the adjacency operator when written with respect to the standard basis is called the adjacency matrix A X of the graph X.The (x, y)-entry of the adjacency matrix is A X (x, y) = α(x − y).Since α(x − y) = α(y − x), the matrix A X is symmetric.Moreover, when α = π S , A X has the property that the elements in any column, or any row, sum up to one.Given x in the finite abelian group G, let χ x denote the character of G corresponding to x in a chosen isomorphisms between G and its dual group; see, for example, [CR62].As proved in Corollary 3.2 of [Ba79], the character χ x is an eigenfunction of the adjacency operator A X of X with corresponding eigenvalue

Heat kernel on weighted Cayley graphs
Let X denote the weighted Cayley graph C(G, S, π S ).The standard, or random walk, Laplacian ∆ X is defined to be the operator on L 2 (G, C) given by when viewed as a function of x ∈ G for a fixed y ∈ G, and with initial condition It can be shown that ( 24) and (25) also holds if we interchange the roles of x and y.
When the graph X is countable with bounded vertex degree, it is shown in [Do06] and [DM06] that the continuous time heat kernel exists and is unique among all bounded functions.

Twisted heat kernel on Z/mZ
Let G = Z, and consider the Cayley graph X = C(G, S, π S ) when S = {−1, 1} and with π S (1) = π S (−1) = 1/2.Then an elementary computation involving properties of the I-Bessel function shows that the heat kernel on X is given by K X (x, y; t) = e −t I x−y (t); see section 3 of [KN06].In subsequent computations, we will use that I ν (t) = I −ν (t) for any ν ∈ N.For an explicit solution of a more general type of diffusion equation on X, we refer the interested reader to [SS14] and [SS15].
Let m > 1 be a positive integer, and let G m = Z/mZ be the cyclic group of order m with addition modulo m.Denote by X m the Cayley graph C(G m , S, π S ) where S = {−1, 1} and π S (1) = π S (−1) = 1/2; in case m = 2 then X 2 has two edges.For β ∈ [0, 1), χ β (x) := exp(2πiβx) is an additive character of Z.The χ β -twisted heat kernel on the Cayley graph X m is defined to be a function and it has the following properties.For a fixed y ∈ G m , and viewed as a function of x, (26) satisfies the transformation property Similarly, one has the analogue of (27) when the heat kernel is viewed as a function of y for a fixed x ∈ G m after replacing χ β by its complex conjugate.Additionally, when viewed as a function of t, (26) satisfies the heat equation (24) with the initial condition lim t↓0 K Xm,χ β (x, y; t) = δ x (y).
Using the method of images, as in [KN06], [Do12] and [CHJSV23], one has the following expression for the twisted heat kernel K Xm,χ β (x, y; t).
Lemma 5.With the notation as above, the twisted heat kernel K Xm,χ β (x, y; t) is given by Proof.First, we observe that the series on the right-hand side of (28) converges uniformly and absolutely for all t ≥ 0, due to the property that I ν (t) = I −ν (t) for ν ∈ N and the bound Finally, we have that e −t I x−y+km (t) satisfies the equation for all k ∈ Z.With all this, we conclude that (28) is indeed the heat kernel on X m twisted by χ β .
We can reformulate the lemma to give a slightly different expression for the twisted heat kernel K Xm,χ β that is more suitable for our purposes.
Lemma 6.With the notation as above, let ℓ ∈ {0, . . ., m−1} be such that ℓ ≡ (x−y) (mod m).Then The twisted heat kernel on X m has a spectral expansion in terms of eigenfunctions and eigenvalues of the Laplacian ∆ Xm .Namely, the eigenfunctions {ψ j } m−1 j=0 are given in terms of the normalized twisted characters, meaning that The normalization is chosen so that the L 2 -norm of ψ j (x) on G m equals one.The eigenvalues are described in section 2.1 for the adjacency operator, which gives that for j = 0, . . ., m − 1.With this notation, the spectral expansion of K Xm,χ β (x, y; t) is given by e −λ j t ψ j (x)ψ j (y) for x, y ∈ G m and t ≥ 0. (33) This identity can, of course, also be verified directly.

Twisted resolvent kernel on X m
In this section we compute the twisted resolvent kernel, meaning the Green's function on X m ; see [CY00] for related results on the certain graphs which require that the eigenvalues are non-zero and the additive shift β = 0. Note that throughout this paper √ s denotes the principal branch of the square-root.Our starting point in computing the twisted resolvent kernel on X m is the spectral expansion (33).For a complex number s with Re(s) > 0, the resolvent kernel, or Green's function, is defined as Since the heat kernel is well defined and bounded for all t ≥ 0, the integral in (34) converges and defines a holomorphic function of s in the half-plane Re(s) > 0. With all this, we have the following evaluation of the resolvent kernel (34).
Proposition 7.With the notation as above, write x − y ≡ ℓ ∈ {0, . . ., m − 1}.Then for s ∈ C with Re(s) > 0 we have that Proof.We begin with (30).From the bound (29), it is evident that for s ∈ C with Re(s) > 0 that the series ∞ j=−∞ e −2πiβj e −(s+1)t I ℓ+jm (t) = e −st K Xm,χ β (x, y; s) can be integrated as in (34) term by term.When computing these integrals, we get the expression that where, as stated above, we have used that I ν (t) = I −ν (t) for any integer ν.
The integral (36) is the Laplace transform of the I-Bessel function.Hence, we can apply [GR07], formula 109 on p. 1116 with ν = |ℓ + jm| ≥ 0 and a = 1; note that the variable s in this formula from [GR07] is our s + 1.The assumption from [GR07] that Re(s + 1) > a = 1 is fulfilled for s ∈ C with Re(s) > 0. So then, we have that Since ℓ ∈ {0, . . ., m − 1}, it is immediate that |ℓ + jm| = ℓ + jm for all j ≥ 0. Also, we have that |ℓ + jm| = −ℓ − jm for j < 0.Moreover, for real s > 0, one has that Using that exp(cosh −1 (s + 1) When combining (38) with ( 36) and (37), the proof of equation ( 35) is completed for real and positive s.Since the function on the right-hand side of ( 35) is holomorphic for Re(s) > 0, the proof for such s follows from the principle of analytic continuation.
We now will show that for β / ∈ Z the function on the right-hand side of ( 35) is holomorphic at s = 0. Lemma 8.For any ℓ ∈ {0, . . ., m − 1} and real number β with β / ∈ Z, the function Proof.Since g m,ℓ (s, β) is holomorphic in the half-plane Re(s) > 0, it suffices to show that g m,ℓ (s, β) is bounded as s → 0. Indeed, for any positive integer j, it is elementary that With the spectral expansion (33) of the heat kernel, we get another expression for the resolvent kernel upon integrating as in (34).Specifically, we have that From the formulas (31) and (32) for ψ j and λ j , we arrive at the expression that It is immediate that the right-hand-side of ( 39) is a meromorphic function with simple poles whenever s is one of the finite points for which s = −2 sin 2 π j+β m .In effect, our main results follow from the identity obtained by equating (35) and (39).

Proof of Theorem 1
We start by proving the first part of Theorem 1. Assume β / ∈ Z.As stated, for x, y ∈ X m and s ∈ C with Re(s) > 0, we have two expressions (39) and (35) for the Green's function G Xm,χ β (x, y; s).Therefore, the right-hand sides of those formulas are equal.Set r = (x − y), and ℓ as before.With this, we get, upon cancelling a factor exp(2πiβr/m), the identity that From the definition of the Chebyshev polynomials of the first and the second kind, we have for Re(s + 1) > 1 that Then, for Re(s + 1) > 1 we have that and, by using (40), The equality (42), which holds for Re(s + 1) > 1, extends to an equality of meromorphic functions which holds for all values of the complex variable s.In particular, for any fixed β ∈ (0, 1), Lemma 8 yields that the function F m,r (s, β) is holomorphic at s = 0. Moreover by differentiating the right-hand side of (42) n times with respect to s evaluating at s = 0 we get for s sufficiently close to zero.This proves the first part of Theorem 1, after the cosmetic change of variable for s obtained by replacing s with −2s.
To prove the second part, we notice that from (40) that one has the identity For all s ∈ C with Re(s) ≥ 0 the function on the right-hand side ( 43) is continuous at β = 0 from the right.Hence, the function on the left-hand side of (43) must also be right-continuous, so then we have that Trivially, from (41) we obtain that (45) The function on the right-hand side of ( 44) is holomorphic at s = 0. Therefore, F m,r (s) is also holomorphic at s = 0, and then This proves the second claim of Theorem 1, again after replacing s with −2s.
Example 9. Consider any positive m, β = 1/2, r = 0 and n = 1.Then by taking s = 0 in Theorem 1, we get that This yields the well-known evaluation that see [BY02, Corollary 2.6] and references therein regarding the appearance of those sums elsewhere in the literature.
Example 10.For any positive integer k, let m = 3k.Take β = 1/2 and r = k.Let ω denote the third root of unity.Then, for all positive integers n one has the identity that where When n = 1 this yields the formula (17).For n ≥ 1, one can use the expansions of U 2k−1 (z), U k−1 (z) and T 3k (z) at z = 1, as provided in Section 7.3 below, to get further evaluations.For example, one gets that If one takes β = 0 and the same values of m and r, one gets the formula that where .

Secant and cosecant sums of a double argument
In this section we will study the resolvent kernel G Xm,χα (x, y; s), which equals F m,r (s, α) for r = x − y, in the neighbourhood of s = −1.In doing so, we will prove the following theorem.
Proof.Our starting point is the equation Equation (49) stems from ( 40) and (41) with β = α, which comes from two different ways to write F m,r (s, α).For real values of α such that 2α / ∈ m 2 Z when m is even and for 2α / ∈ Z + 1 2 when m is odd, it is obvious that the left-hand side of ( 49) is analytic at s = −1.Therefore, F m,r (s, α) is analytic at s = −1, for the given values of m and α.By differentiating the left-hand side of (49) n times with respect to s, we get, after applying the trigonometric identity −1 + 2 sin 2 x = − cos 2x, that Therefore, equation (47) holds for fm,r (z, α) = F m,r (z − 1, α).As in previous discussion, the recursion formula (48) follows from the uniqueness of the Taylor series expansion.
By letting α = β − m 4 in the above theorem, and using that cos x = sin(π/2 + x), we arrive at the following corollary where both sides of (50) are holomorphic for all complex z with 0 < Re(z) < δ when The left-hand sides of the above two displayed equations are holomorphic functions at z = 0, hence so are the right-hand sides.Moreover, for even m we have for the sum (12) is given for all complex z in a neighbourhood of z = 0 by where δ(m) = 1 if m is odd and δ(m) = 2 if m is even.Furthermore, hm,r (z) admits a meromorphic continuation to all complex z.
Given that we have different generating functions, those relations yield further identities satisfied by functions f and h and their derivatives.We studied both types of sums because there are instances when one sum cannot be reduced to another one, such as when taking odd powers in (10) and (11) or odd m in (8) and (6).

Sums twisted by a multiplicative character
In this section we will relate the results in this article to that from [F16, FK17] and [XZZ22].
In particular, we will prove formula (23) for evaluation of the special values of the spectral L-function associated to the cycle graphs X m at positive integers, thus providing an answer to the question raised at the end of [XZZ22].
More precisely, we will consider the generating function for the L-function defined on X m for any even Dirichlet character χ of modulus m and any complex number s.This L-function is given in (21).When the character is trivial, L Xm (s, χ) becomes the spectral zeta function ζ Xm (s) on X m which was studied in [FK17].Note that the special values of ζ Xm (s) at positive integers n is the non-twisted cosecant sum C m (0, 2n) as defined in (1).
The following corollary evaluates the generating function for the special values of L Xm (n+1, χ) associated to a primitive Dirichlet character modulo m and n ≥ 0.
Corollary 16.Let m > 1 be an integer and assume χ is a primitive Dirichlet character modulo m.The generating function coverges for |s| sufficiently small.Furthermore, we have that where τ (χ) denotes the Gauss sum associated to the character χ, and the value of F m,χ (s) at s = 0 is obtained by taking the limit as s → 0. Furthermore, the series F m,χ (s) admits a meromorphic continuation to all complex s and, by (51), is equal to a rational function in s.
Therefore, for s in a neighborhood of s = 0, we deduce from (46) that By observing that F m,r (s) is defined by (45), the proof is them complete.
The proof of (23) now readily follows by conjugating (52) and recalling that, according to (45) and ( 46), one has that in a neighbourhood of s = 0, where c m,r (n) is defined by (14).Note that the terms c m,r (n) satisfy the recurrence relation (15).
This yields an interesting evaluation of L Xm (1, χ), namely that When n = 1, further calculations easily produce that L Xm (2, χ) is given by Furthrmore, Fm,χ (s) extends to a meromorphic function in s which, indeed, is equal to a rational function.

Cotangent and tangent sums
In view of the standard identity csc 2 x = 1 + cot 2 x, we can also deduce results complementary to [He20, Theorem 2.2], where evaluations of cotangent sums twisted by multiplicative character were obtained.Specifically, it is clear that computing even powers of cotangent sums reduces to computing even powers of cosecant sums of the same argument and with the same twist by an additive character.In other words, an application of the recurrence relation in Corollary where we define C m,r (β, 0) to be equal to 1 for all values of m, r, β.We assume that m and r are chosen as above and that β is such that β / ∈ Z. Similar reasoning applies to the computation of cotangent sums without the shift β and to the computation of even powers of tangents, which reduces to an application of the binomial theorem to and secant sums.These results can be compared to those of [EL21] where the authors compute, by using a different method, sums of any powers of cotangent and tangent functions at arguments of the form j+β m π.Their result is more general in the sense that they treat both even and odd powers.On the other hand, we look only at even powers, but employ a character twist.As shown above, the use of the character twist is necessary in other situations, such as when one wants to apply the Gauss formula and pass to multiplicative character twists; see Section 6 above.

Differentiating or integrating with respect to β
A further possibility that presents itself is to differentiate or integrate the formulas above with respect to β.Let us illustrate an approach.Let χ be a primitive, odd Dirichlet character, from which we seek to study the function LXm (s, χ) defined by (22).To do so, let us start with the shifted L-function which we define for By proceeding analogously as in the proof of Corollary 16, it is immediate that the generating function kπ/(2m + 1)) as m → ∞ are used to evaluate the Riemann zeta values ζ(n), ultimately recovering Euler's formula in case n is even.These computations indicate the delicate nature of these trigonometric sums, including C m (n), because the values of ζ(2n) are known while the values of ζ(2n + 1) are far from understood.

Proof.
It suffices to relate L Xm (n + 1, χ) to the sum of twists of C m,r (n + 1) and apply the second part of Theorem 1. Recall the identity m−1 r=0 χ(r)e 2πir m j = χ(j)τ (χ), which holds for primitive Dirichlet characters.From this, we immediately deduce that m−1 r=0 )(r − 2m)(r − m)r(r + m).These two formulas suggest a general pattern.From the recurrence relation (15), it is immediate that mc m,r (n − 1) is a polynomial of degree 2n in two variables m and r.Hence, L Xm (n, χ) can be expressed asL Xm (n, χ) = m−1 r=0 χ(r)P 2n (r, m)for a certain explicitly computable polynomial P 2n (r, m) of degree 2n.Using results of Section 5, it is possible to deduce further evaluations of secant and cosecant sums of double arguments twisted by multiplicative characters, thus complementing results of[BZ04] and[BBCZ05].For example, consider a positive integer m which is not divisible by 4 and a primitive Dirichlet character χ modulo m.By reasoning as in the proof of Corollary 16, with the starting point being Theorem 11 with α = 0, one will deduce the evaluation of the L-function given by LXm (w, χ) = m−1 j=1 χ(j) sec w 2jπ m whenever w = n and n is a positive integer.From this, we have the following corollary.Corollary 18.Let m > 1 be an integer not divisible by 4, and assume that χ is a primitive Dirichlet character modulo m.The generating function Fm,χ (s) = ∞ n=0 LXm (n, χ)s n is given for all complex s with |s| sufficiently small by Fm,χ (s) := − m τ (χ) m−1 r=0 χ(r) U m−r−1 (s) + U r−1 (s) T m (s) − 1 .