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Exponentially small expansions associated with a generalised Mathieu series R. B. Paris ∗ 6 Division of Computing and Mathematics, 1 0 University of Abertay Dundee, Dundee DD1 1HG, UK 2 n a J Abstract 8 2 We consider the generalised Mathieu series ] ∞ A nγ (µ>0) C (nλ+aλ)µ nX=1 . h whentheparametersλ(>0)andγ areevenintegersforlargecomplexainthesector t a arg a < π/λ. The asymptotics in this case consist of a finite algebraic expansion m | | together with an infinite sequence of increasingly subdominant exponentially small [ expansions. When µ is also a positive integer it is possible to give closed-form evaluations of this series. Numerical results are given to illustrate the accuracy of 1 v the expansion obtained. 1 5 Mathematics Subject Classification: 30E15,30E20, 34E05 7 Keywords: asymptotic expansions, exponentially small expansions, generalised 7 0 Mathieu series, Mellin transform method . 1 0 6 1. Introduction 1 : v This paper is a sequel to the asymptotic study of a generalised Mathieu series carried i X out by the author in [4]. The functional series r a ∞ n (1.1) (n2+a2)µ nX=1 in the case µ = 2 was introduced by Mathieu in his 1890 book [2] dealing with the elasticity of solid bodies. Considerable effort has been devoted to the determination of upper and low bounds for the series with µ = 2 when the parameter a > 0. Several integral representations for (1.1), and its alternating variant, have been obtained; see [8] and the references therein. ∗E-mail address: [email protected] 1 The asymptotic expansion of the more general functional series ∞ nγ S (a;λ) := (µ > 0, λ > 0, µλ γ > 1) (1.2) µ,γ (nλ+aλ)µ − nX=1 was considered by Zvastavnyi [11] for a + and by Paris [4] for a in the sector → ∞ | | → ∞ arg a < π/λ. In [4], the additional factor e := exp[ aλb/(nλ +aλ)] (with b > 0) was n | | − included in the summand which, although not affecting the rate of convergence of the series (since e 1 as n ), can modify the large-a growth, particularly with the n → → ∞ alternating variant of (1.2). Both these authors adopted a Mellin transform method and obtained the result1, when γ > 1, − S (a;λ) Γ(γ+λ1)Γ(µ− γ+λ1) 1 ∞ (−)kΓ(µ+k) ζ( γ λk) (1.3) µ,γ − λΓ(µ)aλµ γ 1 ∼ Γ(µ) k!aλ(k+µ) − − − − kX=0 as a in the sector arg a < π/λ, where ζ(s) denotes the Riemann zeta function. | | → ∞ | | In this paper we also employ the Mellin transform approach used in [4, 11], where our interest will be concerned with the parameter values µ > 0 and even integer values of λ (> 0) and γ. In this case, the asymptotic series on the right-hand side of (1.3) (the algebraic expansion) is either a finite series or vacuous on account of the trivial zeros of ζ(s). We shall find that the asymptotic expansion of S (a;λ) for large complex a in the µ,γ sector arg a < π/λ for these parameter values consists of a finite algebraic expansion | | together with an infinitesequence (whenµis notan integer) of increasingly subdominant exponentially small contributions. In the case of positive integer µ it is possible to give a closed-form evaluation of S (a;λ). µ,γ It is perhaps rather surprising that such an innocent-looking series should possess such an intricate asymptotic structure. A similar phenomenon has been recently ob- served in the expansion of the generalised Euler-Jacobi series n wexp[ anp] as ∞n=1 − − the parameter a 0 when p and w are even integers; see [5]P. The leading terms in → the expansion of S (a;λ) when γ = 0 and λ = 2,4 have been given in [10] using the µ,γ Poisson-Jacobi formula. In the application of the Mellin transform method to the series in (1.2) and its alter- nating variant we shall require the following estimates for the gamma function and the Riemann zeta function. For real σ and t, we have the estimates 1 Γ(σ it) = O(tσ−2), ζ(σ it) = O(tΩ(σ)logαt) (t + ), (1.4) ± | ± | → ∞ where Ω(σ) = 0 (σ > 1), 1 1σ (0 σ 1), 1 σ (σ < 0) and α = 1 (0 σ 1), 2 − 2 ≤ ≤ 2 − ≤ ≤ α = 0 otherwise [9, p. 95]. The zeta function ζ(s) has a simple pole of unit residue at s = 1 and the evaluations for positive integer k (2π)2k ζ(0) = 1, ζ( 2k) = 0, ζ(2k) = B (k 1), −2 − 2(2k)!| 2k| ≥ 1The restriction γ >−1 was imposed in [4] to avoid the formation of a double pole for odd negative integer values of γ; in [11], theparameter γ was allowed to assume arbitrary real values. 2 B = 1, B = 1, B = 1 , B = 1 ,..., (1.5) 0 2 6 4 −30 6 42 where B are the Bernoulli numbers. Finally, we have the well-known functional relation k satisfied by ζ(s) given by [3, p. 603] ζ(s)= 2sπs 1ζ(1 s)Γ(1 s)sin 1πs. (1.6) − − − 2 2. An integral representation The generalised Mathieu series defined in (1.2) can be written as S (a;λ) = a δ ∞ h(n/a), h(x) := xγ , δ := λµ γ (2.1) µ,γ − (1+xλ)µ − nX=1 where the parameter δ > 1 for convergence. We employ a Mellin transform approach as discussed in [6, Section 4.1.1]. The Mellin transform of h(x) is H(s) = 0∞xs−1h(x)dx, where R xγ+s 1 1 τ(γ+s)/λ 1 (s) = ∞ − dx = ∞ − dτ H Z (1+xλ)µ λ Z (1+τ)µ 0 0 Γ(γ+s)Γ(µ γ+s) = λ − λ λΓ(µ) in the strip γ < (s) < δ. Using the Mellin inversion theorem (see, for example, [6, − ℜ p. 118]), we find ∞ h(n/a) = 1 ∞ c+∞i (s)(n/a) sds = 1 c+∞i (s)ζ(s)asds, − 2πi Z H 2πi Z H nX=1 nX=1 c−∞i c−∞i where 1< c < δ. The inversion of the order of summation and integration is justified by absolute convergence provided 1 < c< δ. Then, from (2.1), we have [4, 11] a δ 1 c+ i γ +s γ+s S (a;λ) = − ∞ Γ Γ µ ζ(s)asds, (2.2) µ,γ λΓ(µ) 2πi Z (cid:18) λ (cid:19) (cid:18) − λ (cid:19) c i −∞ where 1 <c < δ. From the estimates in (1.4), the integral in (2.2) then defines S (a;λ) µ,γ for complex a in the sector arg a < π/λ. The asymptotic expansion of S (a;λ) for µ,γ | | large a and real parameters λ, µ and γ, such that δ > 1 is given in (1.4); see [4, Theorem 3] for complex a and [11] for positive a and unrestricted γ. We now suppose in the remainder of this paper that µ > 0, with λ (> 0) and γ both chosen to be even integers. More specifically, we write λ = 2p (p =1,2,...), γ = 2m (m = 0, 1, 2,...). (2.3) ± ± 3 The integration path in (2.2) lies to the right of the simple pole of ζ(s) at s= 1 and the poles of Γ((γ+s)/λ) at s = γ λk (k = 0,1,2,...), but to the left of the poles of the − − second gamma function at s = δ+λk. When γ = 2,4,..., the poles at s = γ λk are − − cancelled by the trivial zeros of ζ(s) at s = 2, 4,.... When γ = 2m, m = 0,1,2,..., − − − however, there is a finite set of poles of this sequence on the left of the integration path situated in (s) 0 with 0 k k , where k is the index that satisfies ∗ ∗ ℜ ≥ ≤ ≤ m k p 0, m (k +1)p < 0. (2.4) ∗ ∗ − ≥ − The remaining poles of this sequence correspondingto k > k are cancelled by the trivial ∗ zeros of ζ(s) with the result that there are again no poles in (s)< 0. ℜ We consider theintegral in (2.2) taken roundthe rectangular contour with vertices at c iT and c iT, where c > 0 and T > 0. The contribution from the upper and lower ′ ′ ± − ± sidesoftherectangles = σ iT, c σ c,vanishesasT provided arg a < π/λ, ′ ± − ≤ ≤ → ∞ 1 |3 | sincefrom(1.4),themodulusoftheintegrandiscontrolledbyO(TΩ(σ)+2µ−4 log T e−∆T), where ∆ = π/λ arg a . Evaluation of the residues then yields −| | Γ(γ+1)Γ(µ γ+1) S (a;λ) = λ − λ +H (a;λ)+J(a), (2.5) µ,γ λΓ(µ)aδ 1 µ,γ − where the finite algebraic expansion H (a;λ) (with γ =2m) is given by µ,γ 0 (m = 1,2,...) H (a;λ) = a λµ k∗ ( )kΓ(µ+k) (2.6) µ,γ −  − ζ(2m 2kp) (m = 0, 1, 2,...), Γ(µ) k!aλk − − − kX=0  with the index k being defined in (2.4), and ∗ a δ 1 c′+ i γ+s γ+s J(a) = − − ∞ Γ Γ µ ζ(s)asds (c > 0). (2.7) ′ λΓ(µ) 2πi Z c′ i (cid:18) λ (cid:19) (cid:18) − λ (cid:19) − −∞ The values of ζ(s) at s = 0,2,4,... can be expressed in terms of the Bernoulli numbers, if so desired, by (1.5). The integrand in J(a) is holomorphic in (s) < 0, so that further displacement of ℜ the contour to the left can produce no additional terms in the algebraic expansion of S (a;λ). We shall see in the next section that J(a) possesses an infinite sequence of µ,γ increasingly exponentially small terms in the large-a limit. 3. The exponentially small expansion of J(a) In the integral (2.7), we make the change of variable s s γ to find → − − a λµ 1 d+ i s s J(a) = − ∞ Γ − Γ µ+ ζ( s γ)a−sds, d = c′ γ. λΓ(µ) 2πi Z (cid:18) λ (cid:19) (cid:18) λ(cid:19) − − − d i −∞ 4 We now employ (1.4) to convert the zeta function into one with real part greater than unity. With the parameters λ and γ in (2.3), the above integral can then be written in the form ( )ma λµ 1 d+ i sin(1πs) J(a) = (2−π)γλΓ−(µ) 2πi Z ∞ G(s)ζ(1+s+γ)(2πa)−s sin(2πs) ds, d i λ −∞ where Γ(s+γ+1)Γ(µ+s) G(s) := λ . (3.1) Γ(1+s) λ Making use of the expansion sin(1πs) sin(πps) p−1 π 2 λ = e iωrs, ω := (p 1 2r) , (3.2) sin(πs) ≡ sin(πs) − r − − λ λ λ rX=0 we then obtain ( )ma−λµ p−1 J(a) = − (a), (3.3) (2π)γλΓ(µ) Er rX=0 where 1 d+ i (a) = ∞ G(s)ζ(s+γ+1)(2πaeiωr) sds (3.4) r − E 2πiZ d i −∞ and d+γ = c >0. ′ The integrals (a) (0 r p 1) have no poles to the right of the integration path, r E ≤ ≤ − so that we can displace the path as far to the right as we please. On such a displaced path, which we denote by L, s is everywhere large. Let M denote an arbitrary positive | | integer. The quotient of gamma functions in G(s) may be expanded by appealing to the inverse-factorial expansion given in [6, p. 53] to obtain M 1 G(s) = λ1 µ − ( )jc Γ(s+ϑ j)+ρ (s)Γ(s+ϑ M) , ϑ := µ+γ, (3.5) − j M (cid:26) − − − (cid:27) jX=0 where c = 1 and ρ (s) = O(1) as s in arg s < π. An algorithm for the 0 M | | → ∞ | | evaluation of the coefficients c is discussed in Section 4. Substitution of the expansion j (3.5) into (3.4) then produces M 1 (a) = λ1 µ − ( )jc 1 Γ(s+ϑ j)ζ(s+γ+1)(2πaeiωr) sds+R , (3.6) r − j − M,r E (cid:26) − 2πi Z − (cid:27) jX=0 L where the remainders R are given by M,r 1 R = ρ (s)Γ(s+ϑ M)ζ(s+γ+1)(2πaeiωr) sds. (3.7) M,r M − 2πi Z − L The integrals appearing in (3.6) can be evaluated by making use of the well-known result 1 Γ(s+α)z sds = zαe z ( arg z < 1π), 2πi ZL′ − − | | 2 5 where L is a path parallel to the imaginary s-axis lying to the right of all the poles of ′ Γ(s + α); see, for example, [6, Section 3.3.1]. Upon expansion of the zeta function in (3.6) (since on L its argument satisfies (s)+γ +1> 1) we find ℜ 1 Γ(s+ϑ j)ζ(s+γ+1)(2πaeiωr) sds = ∞ (2πnaeiωr)ϑ−j exp[ 2πnaeiωr] − 2πi Z − n1+γ − L nX=1 = Xϑ je XrK (X ;µ), X := Xeiωr, X := 2πa, (3.8) r− − j r r where we have defined the exponential sum ∞ e−(n−1)Xr K (X ;µ) := . (3.9) j r n1 µ+j nX=1 − Thisevaluation isvalid providedthatthevariableX satisfies theconvergence conditions r arg a+ω < 1π (0 r p 1). | r| 2 ≤ ≤ − From the definition of ω in (3.2), it is easily verified that these conditions are met when r arg a < π/λ. It is then evident that K (X ;µ) 1 as a in arg a < π/λ. j r | | ∼ | | → ∞ | | Thus we find M 1 (a) = λ1 µe Xr − ( )jc Xϑ jK (X ;µ)+R . (3.10) Er − − − j r− j r M,r jX=0 Bounds for the remainders of the type R have been considered in [6, p. 71]; see also M,r [1, 10.1]. The integration path in (3.7) is such that (s)+γ +1 > 1, so that we may § ℜ employ the bound ζ(x+iy) ζ(x) for real x, y with x > 1. A slight modification of | | ≤ Lemma 2.7 in [6, p. 71] then shows that R = O(Xϑ Me Xr) (3.11) M,r r− − as a in the sector arg a < π/λ. | | → ∞ | | The expansion of S (a;λ) then follows from (2.5), (3.3), (3.10) and (3.11) and is µ,γ given in the following theorem. Theorem 1. Let µ > 0, γ = 2m, λ = 2p, where m = 0, 1, 2,... and p = 1,2,.... ± ± Further, let M denote a positive integer, ω = π(p 1 2r)/(2p) for 0 r p 1 and r − − ≤ ≤ − δ = λµ γ, ϑ = µ+γ. Then − Γ(γ+1)Γ(µ γ+1) ( )m π µ p−1 Sµ,γ(a;λ) = λλΓ(µ)aδ−1 λ +Hµ,γ(a;λ)+ Γ−(µ)(cid:18)p(cid:19) aµ−δ Er(a) (3.12) − rX=0 as a in the sector arg a < π/λ. The finite algebraic expansion H (a;λ) is µ,γ | | → ∞ | | defined in (2.6) and the exponentially small expansions E (a) are given by r M 1 E (a) = e Xr+iϑωr − ( )jc X jK (X ;µ)+O(X M) (0 r p 1), (3.13) r − (cid:26) − j r− j r r− (cid:27) ≤ ≤ − jX=0 where the leading coefficient c = 1 and X = 2πaeiωr. The infinite exponential sums 0 r K (X ;µ) are defined in (3.9). j r 6 When a is a real variable, the expansion in Theorem 1 can be expressed in a different form. We have the following theorem. Theorem 2. Let the parameters µ, γ, λ and δ, ϑ, ω be as in Theorem 1. Then with r N = 1p and X = 2πa, the expansion for S (a;λ) becomes ⌊2 ⌋ µ,γ Γ(γ+1)Γ(µ γ+1) S (a;λ) = λ − λ +H (a;λ) µ,γ λΓ(µ)aδ 1 µ,γ − +(Γ−(µ)m)(cid:18)πp(cid:19)µaµ−δ(cid:26)N−1Er∗(a)+(cid:18)1E0(a)(cid:19)(cid:27) (cid:26)pp eodvedn (3.14) rX=0 2 N∗ as a + , where for arbitrary positive integer M → ∞ M 1 E (a) = 2e Xcosωr − ( )jc X jK (X;ω )+O(X M) (0 r N) (3.15) r∗ − (cid:26) − j − j∗ r − (cid:27) ≤ ≤ jX=0 and the infinite exponential sums K (X;ω ) are defined by j∗ r ∞ e−(n−1)Xcosωr K (X;ω )= cos[nXsinω +(j ϑ)ω ]. j∗ r n1 µ+j r − r nX=1 − When p is odd, the quantity ω = 0. N 4. The coefficients c j We describe an algorithm for the computation of the coefficients c that appear in the j exponentially small expansions E (a) and E (a) in (3.13) and (3.15). The expression for r r∗ the ratio of gamma functions in G(s) in (3.5) may be written in the form M 1 G(s) = λ1 µ − cj + ρM(s) , − Γ(s+ϑ) (cid:26) (1 s ϑ) (1 s ϑ) (cid:27) jX=0 − − j − − M where (α) = Γ(α + j)/Γ(α) is the Pochhammer symbol. If we introduce the scaled j 1 gamma function Γ∗(z) = Γ(z)/(√2πzz−2e−z), then we have 1 1 Γ(βs+γ) = Γ∗(βs+γ)(2π)2e−βs(βs)βs+γ−2 e(βs;γ), where γ e(βs;γ) := exp (βs+γ 1)log(1+ ) γ . (cid:20) − 2 βs − (cid:21) The above ratio of gamma functions may therefore be expressed as M 1 − cj ρM(s) R(s)Υ(s)= + (4.1) (1 s ϑ) (1 s ϑ) jX=0 − − j − − M 7 as s in arg s < π, where | |→ ∞ | | e(s;γ+1)e(s/λ;µ) Γ (s+γ+1)Γ (µ+s/λ) ∗ ∗ R(s):= , Υ(s) := . e(s/λ;1)e(s;ϑ) Γ (1+s/λ)Γ (s+ϑ) ∗ ∗ We now let ξ := s 1 and follow the procedure described in [6, p.47]. We expand R(s) − and Υ(s) for ξ 0 making use of the well-known expansion (see, for example, [6, p. 71]) → Γ (z) ∞ ( )kγ z k (z ; arg z < π), ∗ k − ∼ − | | → ∞ | | kX=0 where γ are the Stirling coefficients, with k γ = 1, γ = 1 , γ = 1 , γ = 139 , γ = 571 , ... . 0 1 −12 2 288 3 51840 4 −2488320 After some straightforward algebra we find that R(s)= 1+ 1(µ 1) (λ 1)µ 2γ ξ +O(ξ2), 2 − { − − } Υ(s) =1 1 (µ 1)(λ2 1)ξ2+O(ξ3), − 12 − − so that upon equating coefficients of ξ in (4.1) we can obtain c . The higher coefficients 1 can be obtained by matching coefficients recursively with the aid of Mathematica to find c = 1, c = 1(µ 1) 2γ (λ 1)µ , 0 1 2 − { − − } c = 1 (µ 1)(µ 2) 12γ(γ (λ 1)µ 1)+(λ 1)µ(5 3µ+λ(3µ 1)) , 2 24 − − { − − − − − − } c = 1 (µ 1)(µ 2)(µ 3) 2 2γ +(λ 1)µ 4γ(γ (λ 1)µ 2) 3 −48 − − − { − − }{ − − − +(λ 1)µ(3+λ(µ 1) µ) ,... . (4.2) − − − } The rapidly increasing complexity of the coefficients with j 4 prevents their presenta- ≥ tion. However, this procedure is found to work well in specific cases when the various parameters have numerical values, where many coefficients have been so calculated. In Table 1 we present some values2 of the coefficients c for 1 j 10, which are used in j ≤ ≤ the specific examples considered in Section 5. 4.1 The coefficients c when λ = 2, γ = 0 j When λ = 2 and γ = 0, it is possible to express the coefficients c in closed form for j arbitrary µ > 0. From (3.1), we have Γ(1+s)Γ(µ+ 1s) 2s G(s) = 2 = Γ(1 + 1s)Γ(µ+ 1s) Γ(1+ 1s) √π 2 2 2 2 2In the tables we write the valuesas x(y)instead of x×10y. 8 Table 1: The coefficients cj (1≤j ≤10) for different γ when µ=5/4 and λ=4. j γ = 0 γ = 2 γ = 2 − 1 4.6875000000( 1) +3.1250000000( 2) 2.1250000000(+0) − − − − 2 3.5888671875( 1) +1.5673828125( 1) 2.5546875000(+0) − − − − 3 4.0534973145( 1) +2.3551940918( 1) 6.8701171875(+0) − − − − 4 3.3581793308( 1) +1.6646325588( 1) 2.5683746338(+1) − − − − 5 +7.5268601999( 1) 9.0884858742( 1) 1.1944799423(+2) − − − − 6 +6.4821335676(+0) 6.6501405553(+0) 6.6193037868(+2) − − 7 +2.6358910987(+1) 2.7627888119(+1) 4.2794038211(+3) − − 8 +4.5855530043(+1) 5.8401959193(+1) 3.1831413077(+4) − − 9 3.7955573596(+2) +2.8858407940(+2) 2.6901844936(+5) − − 10 5.1286970180(+3) +4.6231064924(+3) 2.5504879368(+6) − − upon use of the duplication formula for the gamma function. The inverse factorial ex- pansion of a product of two gamma functions with equal coefficients of s is given in [6, pp. 51–52] in the form Γ(s+α)Γ(s+β) ∼ 223−α−β−2s√π ∞ djΓ(2s+α+β−12−j) jX=0 as s in arg s < π, where the coefficients satisfy d = 1 and 0 | |→ ∞ | | 2 j j d = − (α β)2 (r 1)2 (j 1). j j! { − − − 2 } ≥ rY=1 Putting α = 1 and β = µ, with s 1s, we therefore obtain the coefficients in the 2 → 2 inverse factorial expansion of G(s) when λ = 2, γ =0 and µ > 0 given by ( 2) j j − c = − (µ r)(µ+r 1) (j 1). (4.3) j j! − − ≥ rY=1 4.2 The coefficients c when µ is an integer j A study of the coefficients c with the aid of Mathematica enables us to conjecture that j they possess the general form c = (µ 1)(µ 2)...(µ j)P (µ,γ,λ) (j 1), j j − − − ≥ where P denotes a polynomial of degree j in the parameters µ, γ and λ. This implies j thatthesequenceofcoefficients isfiniteforinteger valuesofµ; thatis,forpositiveinteger q, we have c = 0 (j q; µ = q, q = 1,2,...). j ≥ 9 This can also be seen from (3.1) where, with µ = q, Γ(s+γ+1)Γ(q+s) s G(s) = λ = Γ(s+γ+1) 1+ . (4.4) Γ(1+s) (cid:18) λ(cid:19) λ q 1 − Whenq = 1,wehaveϑ = 1+γ andtheexpansion(3.5)issatisfiedtriviallybyterminating the series at the leading term with ρ (s) 0. When q 2, the right-hand side of (3.5) 1 ≡ ≥ must terminate at M = q with ρ (s) 0 to yield q ≡ q 1 − G(s) = λ1 qΓ(s+γ+1) ( )jc (s+γ +1) (4.5) − j g j 1 jX=0 − − − in order to have the polynomials in s in (4.4) and (4.5) of the same degree. Then the exponential expansions E (a) in (3.13) become the finite sums r q 1 − E (a) = e Xr+iϑωr ( )jc X jK (X ;q), ϑ = q+γ, r − − j r− j r jX=0 where the infinite sums K (X ;q) may be expressed exactly in terms of derivatives of an j r exponential by K (X ;q) = eXr ∞ nq j 1e nXr = ( 1)q j 1eXrDq j 1 ∞ e nXr j r − − − − − − − − − nX=1 nX=1 = ( )q j 1eXrDq j 1(eXr 1) 1 ( arg a < π/λ), (4.6) − − − − − − − | | with D d/dX . The coefficients c (0 j µ 1) are obtained by recursive solution r j ≡ ≤ ≤ − of (4.4) and (4.5) and are given below3 for µ = 1,2,...,5: µ = 1 : c = 1 0 µ = 2 : c = 1, c =γ λ 1 0 1 − − µ = 3 : c = 1, c =2γ 3λ+3, c = (1+γ λ)(1+γ 2λ) 0 1 2 − − − µ = 4 : c = 1, c =3γ 6λ+6, c = 3γ2+9γ +7 18λ 12λγ +11λ2, 0 1 2 − − − c = (1+γ λ)(1+γ 2λ)(1+γ 3λ) 3 − − − µ = 5 : c = 1, c =4γ 10λ+10, c = 6γ2+24γ +25 60λ 30λγ +35λ2, 0 1 2 − − − c = (2γ 5λ+3)(2γ2 +6γ +5 15λ 10λγ +10λ2), 3 − − − c = (1+γ λ)(1+γ 2λ)(1+γ 3λ)(1+γ 4λ). 4 − − − − (4.7) The generalised Mathieu series when µ is an integer can therefore be expressed by the following closed-form evaluation. 3The values of c1, c2 and c3 follow from (4.2). 10

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