Exponentially small expansions associated with a generalised Mathieu series

0.16 MB·

by  R B Paris

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