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volume 4, issue 5, article 100, 2003.

Received 16 June, 2003;

accepted 08 December, 2003.

Communicated by:F. Qi

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Journal of Inequalities in Pure and Applied Mathematics

ON INTEGRAL FORMS OF GENERALISED MATHIEU SERIES

P. CERONE AND C.T. LENARD

School of Computer Sciences and Mathematics Victoria University of Technology

PO Box 14428, MCMC 8001 VIC, Australia.

EMail:pc@csm.vu.edu.au URL:http://rgmia.vu.edu.au/cerone Department of Mathematics LaTrobe University PO Box 199, Bendigo, Victoria, 3552, Australia.

EMail:C.Lenard@bendigo.latrobe.edu.au

URL:http://www.bendigo.latrobe.edu.au/mte/maths/staff/lenard/

c

2000Victoria University ISSN (electronic): 1443-5756 080-03

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On Integral Forms of Generalised Mathieu Series

P. Cerone and C.T. Lenard

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J. Ineq. Pure and Appl. Math. 4(5) Art. 100, 2003

http://jipam.vu.edu.au

Abstract

Integral representations for generalised Mathieu series are obtained which re- capture the Mathieu series as a special case. Bounds are obtained through the use of the integral representations.

2000 Mathematics Subject Classification: Primary 26D15, 33E20; Secondary 26A42, 40A30.

Key words: Mathieu Series, Bounds, Identities.

1. Introduction

The series

(1.1) S(r) =

∞

X

n=1

2n

(n²+r²)², r >0

is well known in the literature as Mathieu’s series. It has been extensively stud- ied in the past since its introduction by Mathieu [12] in 1890, where it arose in connection with work on elasticity of solid bodies. The reader is directed to the references for further illustration.

One of the main questions addressed in relation (1.1) is to obtain sharp bounds. Alzer, Brenner and Ruehr [2] showed that the best constants a and bin

1

x²+a < S(x)< 1

x²+b, x6= 0

area = _2ζ(3)¹ andb = ¹₆ whereζ(·)denotes the Riemann zeta function defined by

(1.2) ζ(p) =

∞

X

n=1

1 n^p.

An integral representation forS(r)as given in (1.1) was presented in [6] and [7] as

(1.3) S(r) = 1

r Z ∞

0

x

e^x−1sin (rx)dx.

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Guo [10] utilised (1.3) and a lemma [3, pp. 89–90] to obtain bounds onS(r), namely,

(1.4) π r³

∞

X

k=0

(−1)^k k+¹₂ e(^k+¹₂)^π_r −1

< S(r)< 1

r² 1 + π r

∞

X

k=0

(−1)^k k+¹₂ e(^k+¹₂)^π_r −1

! . The following results were obtained by Qi and coworkers (see [4], [15] – [17])

4 (1 +r²) e⁻^π^r +e⁻^2r^π

−4r² −1 e⁻^π^r −1

(1 +r²) (1 + 4r²) (1.5)

≤S(r)

≤ (1 + 4r²) e⁻^π^r −e⁻^2r^π

−4 (1 +r²) e⁻^π^r −1

(1 +r²) (1 + 4r²) S(r)< 1

r Z ^π_r

0

x

e^x−1sin (rx)dx < 1 +e⁻^2r^π r²+ ¹₄ , and

S(r)≥ 1 8r(1 +r²)³

h

16r r² −3

+π³ r² + 13

sech² πr

2

tanh πr

2 i

. Guo in [10] poses the interesting problem as to whether there is an integral representation of the generalised Mathieu series

(1.6) Sµ(r) =

∞

X

n=1

2n

(n²+r²)^1+µ, r >0, µ >0.

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This is resolved in Section2.

Recently in [18] an integral representation was obtained forS_m(r), where m ∈N,namely

(1.7) S_m(r) = 2 (2r)^mm!

Z ∞

0

t^m

e^t−1cosmπ 2 −rt

dt

−2

m

X

k=1

"

(k−1) (2r)^k−2m−1 k! (m−k+ 1)

−(m+ 1) m−k

× Z ∞

0

t^kcos_π

2 (2m−k+ 1)−rt

e^t−1 dt

# . Bounds were obtained by Tomovski and Trenˇcevski [18] using (1.3).

It is the intention of the current paper to investigate further integral representations of the generalised Mathieu series (1.6).

Bounds are obtained in Section3forS_µ(r).In Section4the open problem of obtaining an integral representation for

S(r;µ, γ) =

∞

X

n=1

2n^γ (n^2γ+r²)^µ+1 posed by Qi [15] is addressed.

We notice that

S(r; 1,1) =S₁(r) = S(r), the Mathieu series.

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2. Integral Representation of the Generalised Mathieu Series S

_µ

(r)

Before proceeding to obtain an integral representation for S_µ(r) as given by (1.6), it is instructive to present an alternative representation in terms of the zeta function ζ(p) presented in (1.2). Namely, a straightforward series expansion gives

(2.1) S_µ(r) = 2

∞

X

k=0

r^2k(−1)^k

µ+k k

ζ(2µ+ 2k+ 1) on using the result ^α_k

= (−1)^k ^k−α−1_k

withα=−(µ+ 1).

Theorem 2.1. The generalised Mathieu series S_µ(r) defined by (1.6) may be represented in the integral form

(2.2) S_µ(r) = C_µ(r) Z ∞

0

x^µ+¹² e^x−1J_µ−¹

2 (rx)dx, µ >0, where

(2.3) C_µ(r) =

√π

(2r)^µ−¹² Γ (µ+ 1) andJν(z)is theν^th order Bessel function of the first kind.

Proof (A). Consider

(2.4) T_µ(r) =

Z ∞

0

x^µ+¹² e^x−1J_µ−¹

2 (rx)dx.

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Then using the series definition forJ_ν(z)(Gradshtein and Ryzhik [9]), J_ν(z) =

∞

X

k=0

(−1)^k ^z₂ν+2k

k!Γ (ν+k+ 1)

in (2.4) produces after the permissible interchange of summation and integral,

(2.5) T_µ(r) =

∞

X

k=0

(−1)^k ^r₂µ+2k−¹₂

k!Γ µ+k+ ¹₂ Z ∞

0

x^2(µ+k) e^x−1dx.

Now, the well known representation [9]

(2.6)

Z ∞

0

x^p

e^x−1dx= Γ (p+ 1)ζ(p+ 1) gives from (2.5) withp= 2 (µ+k)

(2.7) T_µ(r) =

∞

X

k=0

(−1)^k ^r₂µ+2k−¹

2 Γ (2µ+ 2k+ 1)ζ(2µ+ 2k+ 1)

k!Γ µ+k+ ¹₂ .

An application of the duplication identity for the gamma function

√πΓ (2z) = 2^2z−1Γ (z) Γ

z+1 2

, withz =µ+k+ ¹₂ simplifies the expression in (2.7) to (2.8) T_µ(r) = (2r)^µ−¹² 2

√π

∞

X

k=0

(−1)^kr^2kΓ (µ+k+ 1)

k! ζ(2µ+ 2k+ 1).

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Repeated use of the identityΓ (z+ 1) =zΓ (z)gives Γ (µ+k+ 1)

k! =

µ+k k

Γ (µ+ 1)

and so from (2.8)

T_µ(r) = (2r)^µ−¹² Γ (µ+ 1)

√π 2

∞

X

k=0

(−1)^kr^2k

µ+k k

ζ(2µ+ 2k+ 1) produces the result (2.2) on reference to (2.1), (2.3) and (2.4).

Proof (A) is now complete.

Proof (B). From (2.4) we have

T_µ(r) = Z ∞

0

e^−x

1−e^−xx^µ+¹²J_µ−¹

2 (rx)dx (2.9)

=

∞

X

k=1

Z ∞

0

e^−nxx^µ+¹²J_µ−¹

2 (rx)dx.

Now Gradshtein and Ryzhik [9] on page 712 has the result Z ∞

0

e^−αxx^ν+1J_ν(βx)dx= 2α(2β)^νΓ ν+³₂

√π[α²+β²]^ν+³² , (2.10)

Re (ν)>−1, Re (α)>|Imβ|,

which is referred to in Watson [20] whom in turn attributes the result to an 1875 result of Gegenbauer.

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Takingα = n, ν = µ− ¹₂ andβ = r, all real, in (2.10) and substituting in (2.9) readily produces

T_µ(r) = (2r)^µ−¹² Γ (µ+ 1)

√π

∞

X

n=1

2n [n²+r²]^µ+1, giving from (1.6), (2.4) and (2.3) the result (2.2).

We note that the more restrictive condition ofµ > 0needs to be imposed for the convergence of the series although (2.10) requiresRe (ν) = µ− ¹₂ >

−1.

Remark 2.1. If we take µ = 1in (1.6) and (2.2) – (2.3) then S1(r) ≡ S(r), the Mathieu series given by (1.1) and its integral representation (1.3). This is easily seen to be the case sinceJ¹

2 (z) =q

2

πzsinz and takingµ= 1in (2.2) – (2.3) produces (1.3).

Remark 2.2. Gradshtein and Ryzhik [9] on page 712 also quote the result Z ∞

0

e^−αxx^νJ_ν(βx)dx= (2β)^νΓ ν+ ¹₂

√π(α²+β²)^ν+¹² (2.11)

Re (ν)>−1

2, Re (α)>|Im (β)|,

which Watson [20] again attributes to an 1875 result by Gegenbauer.

We note that formal differentiation of (2.11) with respect to αproduces the result (2.10).

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Following a similar process as in Proof (B) above, we may show that

(2.12)

Z ∞

0

x^µ−¹² e^x−1J_µ−¹

2 (rx)dx= (2r)^µ−¹² Γ (µ)

√π

∞

X

n=1

1 (n²+r²)^µ.

Gradshtein and Ryzhik [9] have an explicit expression which can be trans- formed by a simple change of variables to (2.12). Namely,

(2.13)

Z ∞

0

x^νJ_ν(bx)

e^πx−1 dx= (2b)^νΓ ν+¹₂

√π

∞

X

n=1

1 (n²π²+b²)^ν+¹²

,

Re (ν) > 0,|Im (b)| < π,which is attributed by Watson [20] to a 1906 result by Kapteyn.

An explicit integral expression forSµ(r)of the current form does not seem to have been available previously.

Finally, we note that (2.10) or (2.11) may be looked upon as an integral transform such as the Laplace or Hankel transform and the results may be found in tables of such.

Remark 2.3. S_µ(r) as given in (2.2) – (2.3) may be written in the alternate form

(2.14) S_µ(r) =

√π

2^µ−¹²r^2µ−1Γ (µ+ 1) Z ∞

0

x e^x−1

h

(rx)^µ−¹² J_µ−¹

2 (rx)i dx, which, forµ=m,a positive integer

(2.15) S_m(r) = 1

2^m−1r^2m−1m!

rπ 2

Z ∞

0

x

e^x−1R_m(rx)dx,

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where

(2.16)

rπ

2R_m(z) = rπ

2z^m−¹²J_m−¹

2 (z). Form= 1,2,3,4we have

rπ

2R_m(z) = sinz, sinz−zcosz, 3 sinz−3zcosz−z²sinz, and

15 sinz−15zcosz−6z²sinz+z³cosz, respectively.

Thus, for example,

S₁(r) = 1 r

Z ∞

0

x

e^x−1sin (rx)dx, S₂(r) = 1

4r³ Z ∞

0

x

e^x−1[sin (rx)−(rx) cos (rx)]dx, S₃(r) = 1

24r⁵ Z ∞

0

x e^x−1

3 sin (rx)−3 (rx) cos (rx)−(rx)²sin (rx) dx, and

S₄(r) = 1 192r⁷

Z ∞

0

x

e^x−1[15 sin (rx)−15 (rx) cos (rx)

−6 (rx)²sin (rx) + (rx) cos (rx) dx.

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The above results for integerm can also be obtained using the relationship from (1.1) and (1.3)

(2.17) S₁(r) = S(r) =

∞

X

n=1

2n

(n²+r²)² = 1 r

Z ∞

0

x

e^x−1sin (rx)dx.

Formal differentiation with respect torof (2.17) gives (−4r)S₂(r) =

Z ∞

0

x e^x−1

xcosrx

r − sinrx r²

dx

=−1 r²

Z ∞

0

x

e^x−1(sinrx−rxcosrx)dx

producing the result above. Continuing in this manner would produce further representations forSm(r).

The following theorem gives an explicit representation forS_m(r), m∈N. Theorem 2.2. Forma positive integer we have

(2.18) S_m(r) = 1

2^m−1 · 1 r^2m−1

× 1 m

m−1

X

k=0

(−1)b^3k2c

k! r^k[δ_k_evenA_k(r) +δ_k_oddB_k(r)], where

(2.19) A_k(r) = Z ∞

0

x^k+1

e^x−1sin (rx)dx, B_k(r) = Z ∞

0

x^k+1

e^x−1cos (rx)dx,

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withδ_condition = 1 if condition holds and zero otherwise andbxcis the greatest integer less than or equal tox.

Proof. From (2.17) we may differentiatem−1times with respect tor to produce

S₁^(m−1)(r) = (−1)^m−1m! (2r)^m−1S_m(r) (2.20)

= Z ∞

0

x

e^x−1 · d^m−1 dr^m−1

sinrx r

dx.

Now,

(2.21) d^m−1 dr^m−1

sinrx r

=

m−1

X

k=0

m−1 k

d^m−1−k

dr^m−1−k r⁻¹

· d^k

dr^k(sinrx) and

d^l

dr^l r⁻¹

= (−1)^ll!r^−(l+1), d^k

dr^k (sinrx) = (−1)b^k₂cx^k[δ_k_evensin (rx) +δ_k_oddcos (rx)]

whereδ_condition = 1if condition is true and zero otherwise.

Thus from (2.21) (2.22) d^m−1

dr^m−1

sin (rx) r

= 1 r^m

m−1

X

k=0

m−1 k

(−1)^m−1−k+b^k₂c

×(m−1−k)!r^kx^k[δ_k_evensin (rx) +δ_k_oddcos (rx)]

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= (−1)^m−1

r^m (m−1)!

m−1

X

k=0

(−1)b^3k₂c

k! r^kx^k[δ_k_evensin (rx) +δ_k_oddcos (rx)]. Substitution of (2.22) into (2.20) and simplifying produces the stated result (2.18).

Remark 2.4. The integral representation for S_m(r) given in Theorem 2.2 is simpler than that obtained in [18] as given by (1.7). Further, the derivation here is much more straight forward.

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3. Bounds for S

_µ

(r)

It was stated in the introduction that considerable effort has been expended in determining bounds for the generalised Mathieu series. More recently, bounds for the generalised Mathieu series (1.6) has been investigated in particular by Qi and coworkers and by Tomovski and Trenˇcevski [18].

In a recent article Landau [11] obtained the best possible uniform bounds for Bessel functions using monotonicity arguments. Of particular interest to us here is that he showed that

(3.1) |J_ν(x)|< bL

ν¹³

uniformly in the argumentxand is best possible in the exponent ¹₃ and constant (3.2) b_L = 2¹³ sup

x

Ai(x) = 0.674885· · · , whereAi(x)is the Airy function satisfying

w⁰⁰−xw= 0.

Landau also showed that

(3.3) |Jν(x)| ≤ c_L

x¹³

uniformly in the orderν >0and the exponent ¹₃ is best possible with c_L = sup

x

x¹³J₀(x) (3.4)

= 0.78574687. . . .

The following theorem is based on the Landau bounds (3.1) – (3.4).

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Theorem 3.1. The generalised Mathieu series S_µ(r) satisfies the bounds for µ > ¹₂ andr >0

(3.5) S_µ(r)≤b_L

√π (2r)^µ−¹²

· 1

µ− ¹₂¹₃ · Γ µ+ ³₂ Γ (µ+ 1)ζ

µ+3

2

, and

(3.6) S_µ(r)≤c_L·

√π 2^µ−¹²r^µ−¹⁶ ·Γ

µ+ 7

6

ζ

µ+ 7 6

, whereb_Landc_Lare given by (4.2) and (4.4) respectively.

Proof. From (2.2) and (2.3) we have

(3.7) S_µ(r)≤C_µ(r) Z ∞

0

x^µ+¹² e^x−1

J_µ−¹

2 (rx)

dx, r >0 and so from (3.1) we obtain, on utilising (2.6)

Sµ(r)≤Cµ(r)· b_L µ−¹₂¹₃Γ

µ+3

2

ζ

µ+3 2

, which simplifies down to (3.5).

Further, using (3.3) into (3.7) gives S_µ(r)≤C_µ(r)·c_L·

Z ∞

0

x^µ+¹² e^x−1 · 1

|rx|¹³dx

=C_µ(r)· c_L r¹³

Z ∞

0

x^µ+¹⁶ e^x−1dx

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which upon using (2.6) produces (3.8) S_µ(r)≤C_µ(r)· cL

r¹³ ·Γ

µ+ 7 6

ζ

µ+ 7

6

. Simplifying (3.8) and using (2.3) gives the stated result (3.6).

Corollary 3.2. The Mathieu seriesS(r)satisfies the following bounds

(3.9) S(r)≤ 3π

2¹¹¹²bLζ 5

2

and

(3.10) S(r)≤ 7c_L 36 ·

rπ 2 ·Γ

1 6

ζ

13 6

·r⁻⁵⁶, whereb_Landc_Lare given by (3.2) and (3.4) respectively.

Proof. Taking µ = 1 in (3.5) and (3.6), noting that S(r) = S₁(r) gives the stated results after some simplification.

The following corollary gives coarser bounds than Theorem3.1without the presence of the zeta function.

Corollary 3.3. The generalised Mathieu series S_µ(r)satisfies the bounds for µ > ¹₂ andr >0

(3.11) Sµ(r)≤2√

π· b_L

µ− ¹₂¹₃ · 1

r^µ−¹² · Γ µ+¹₂ Γ (µ+ 1)

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and

(3.12) S_µ(r)≤2²³√

π· c_L

r^µ−¹⁶ ·Γ µ+ ¹₆ Γ (µ+ 1)

withb_Landc_Lgiven by (3.2) and (3.4).

Proof. We use the well known inequality

e^−x < x

e^x−1 < e⁻^x² to produce from (3.7)

(3.13) Sµ(r)≤Cµ(r) Z ∞

0

e⁻^x²x^µ−¹² J_µ−¹

2 (rx) dx.

We know from Laplace transforms or the definition of the gamma function that (3.14)

Z ∞

0

e^−αxx^sdx= Γ (s+ 1) α^s+1 .

Hence, placing (3.1) into (3.13) and utilising (3.14) we obtain (3.11) after simplification. A similar approach produces (3.12) starting from (3.3) rather than (3.1).

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4. Further Integral Expressions for Generalised Mathieu Series

In [18], Tomovski and Trenˇcevski gave the integral representation

(4.1) S_µ(r) = 2

Γ (µ+ 1) Z ∞

0

x^µe^−r²^xf(x)dx, where

(4.2) f(x) =

∞

X

n=1

ne⁻ⁿ²^x, convergent for finitex >0, by effectively utilising the result (3.14).

They leave the summation of the series in (4.2) as an open problem.

If we placeν =µ−¹₂ andβ =r,all real in (2.9) then we obtain the identity (4.3) C_µ(r)

Z ∞

0

e^−αxx^µ+¹²J_µ−¹

2 (rx)dx= 2α

[α²+r²]^µ+1, whereCµ(r)is as given by (2.3).

Proof B of Theorem2.1 takesα =nand sums to produce the identity (2.1) – (2.2).

If we takeα=n^γ then we have from (4.3) on summing S(r;µ, γ) =

∞

X

n=1

2n^γ (n^2γ+r²)^µ+1 (4.4)

=C_µ(r) Z ∞

0

∞

X

n=1

e⁻ⁿ^γ^x

!

x^µ+¹²J_µ−¹

2 (rx)dx,

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giving an integral representation that was left as an open problem by Qi [15].

As a matter of fact, if we takeα = a_nwhere a = (a₁, a₂, . . . , a_n, . . .)is a positive sequence, then

S(r;µ;a) =

∞

X

n=1

2a_n (a²_n+r²)^µ+1 (4.5)

=C_µ(r) Z ∞

0

∞

X

n=1

e^−aⁿ^x

!

x^µ+¹²J_µ−¹

2 (rx)dx.

We note that fora⁺ = (1^γ,2^γ, . . .)then S r;µ;a⁺

=S(r;µ, γ). The series

∞

X

n=1

2a_n (a²_n+r²)² has been investigated in [16].

A closed form expression for F (a) =

∞

X

n=1

e^−aⁿ^x, x >0 wherea_nis a positive sequence, remains an open problem.

Ifa^∗ = (1,2,3, . . . , n, . . .), then

F (a^∗) = 1 e^x−1.

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[4] Ch.-P. CHEN AND F. QI, On an evaluation of upper bound of Mathieu series, Studies in College Mathematics (G¯aod˘eng Shùxué Yánj¯ıu), 6(1) (2003), 48–49. (Chinese).

[5] P.H. DIANANDA, Some inequalities related to an inequality of Mathieu, Math. Ann., 250 (1980), 95–98.

[6] A. ELBERT, Asymptotic expansion and continued fraction for Mathieu’s series, Period. Math. Hungar., 13(1) (1982), 1–8.

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