[Abstract]
[PDF]
[HTML]
[Linked
References]
On
Hankel type transform of generalized Mathieu series
B.B.Waphare
MAEER’s MIT ACSC, Alandi, Pune412105, Maharashtra, India
Correspondence Addresses:
[email protected],
[email protected]
Abstract: In
this paper, by using integral representations for several Mathieu type
series, a number of integral transforms of Hankel type are derived here
for general families of Mathieu type series. These results generalize
the corresponding ones on the Fourier transforms of Mathieu type series,
obtained recently by Elezovic et.al. [4], Tomovski [19] and Tomovski and
Vu Kim Tuan [20].
Keywords:
Integral representations, Mathieu type series, Hankel transform, Fourier
transform, Bessel type function, FoxWright function,
SonineSchafheitlin formula, Fox Hfunction.
2000 Mathematics Subject
Classification:
44A10, 33A10.
Introduction:
The infinite series
(1.1)
is
named after Emile Leonard Mathieu (18351890), who investigated it in
his 1890 work [8] on elasticity of solid bodies. An integral
representation of (1.1) is given by (See [5])
(1.2)
Several interesting results dealing with integral representations and
bounds for a slight generalization of the Mathieu series with a
fractional power, defined as
(1.3)
can
be found in the works by Diananda [2]. Tomovski and Trencevski [16],
Cerone and Lenard [1] and Tomovski [18]. Srivastava and Tomovski [14]
defined a family of generalized Mathieu series
(1.4)
where it is tacitly
assumed that the positive sequence
is so chosen that the
infinite series in definition (1.4) converges, that is that the
following auxiliary series
is convergent. Comparing
the definitions (1.1), (1.3) and (1.4), we see that
.
Furthermore, the special cases
and have been
investigated by Qi [13], Diananda [2], Tomovski [17] and CeroneLenard
[1].
Definitions and formulas:
In
this section we give some definitions and formulas needed for
computation of the Hankel transforms of
.
In
order to evaluate the Hankel type transform of
, we first get
in SonineSchafheitlin formula {see example, [6,
p.692]) :
(2.1)
with
a corresponding expression for the case when
, which is obtained from (2.1) by interchanging a and
b, and also. In view of the relationship
(2.2)
we find from the
SonineSchafheitlin formula (2.1) that
,
(2.3)
together with the corresponding integral derived from the analogue of
(2.1) for the case when . Thus by further applying integral formula (2.3) and
its companion when, we obtain:
, (2.4)
where
.
If we apply the
SonineSchafheitlin formula (2.1), first with
,
and then with
we get
,
where .
(2.5)
.
(2.6)
. (2.7)
For
the evaluation of the Hankel type transform of the general Mathieu type
series , we need the definition of the Meijer Gfunction and
the Fox Hfunction (for more details, see e.g. in [3, Vol. 1, [12]).
Definition 1:
By a Fox’s Hfunction we mean a generalized hypergeometric function
defined by means of the MellinBarnestype contour integral.
(2.8)
where t is a suitable contour in
, the orders
are integers,
and the parameters, are such that
……..
Many
special functions are particular cases of the Hfunction.
For
example, if , it reduces to the Meijer Gfunction
(for definition and properties, See [3, Vol.1]) :
(2.9)
On
the other hand, the FoxWright
 function is the following special case of the
Hfunction, see for example in [15]:
(2.10)
Next
we use of the following Miller transform of a product of two
hypergeometric functions, proven by Miller and Srivastava [9] (see also
[12, Sect. 2.22, p. 333]).
(2.11)
.
Substituting the relation
(See [11, p. 727])
(2.12)
into the integral formula
(2.11) with
,
we get
(2.13)
.
In
order to evaluate the Hankel type transform, we apply the known integral formula from [11, p.355]
with:
(2.14)
.
Using relation (2.10) with
by (2.14) we get the following integral formula:
(2.15)
.
3. Evaluation of Hankel
type transforms:
The Hankel type transform
of order is defined by
(3.1)
where
is the Bessel function of the first kind and of order
with.
In view of the
relationship (2.1) and . (3.2)
The Hankel type transform
reduces to the SinFourier and CosFourier transforms:
.
For
a generalization of the Hankel type transform as well as the SinFourier
and CosFourier transforms, see Luchko and Kiryakova [7].
Using the integral formula (2.4), we first evaluate the Hankel type
transform of :
=
. (3.3)
Using the relation [6, p.1041]
,
we
get
,
.
Hence
, (3.4)
where the Cauchy Principal Value (PV) of the last integral is assumed to
exist. By a direct computation, the same formula (3.4) was recently
proved by Elezovic et. al [4].
The
integral representation of, obtained by Cerone and Lenard in [1] is given by
. (3.5)
Applying integral formula (2.5), we obtain
, (3.6)
.
In
order to evaluate the Hankel type transform of
we apply its integral representation from [14]:
(3.7)
,and integral formula (2.13). Thus we have
. (3.8)
The
integral representation of obtained by Srivastava and Tomovski [14] is given by
(3.9)
.
Using this integral
representation and (2.15), we get
,
(3.10)
References:
[1]
P. Cerone, C.T. Lenard, On integral forms of generalized Mathieu
series, J. In equal. Pure Appl. Math. 4, No. 5, Article 100, pp. 111
(electronic), 2008.
[2]
P.H. Diananda, Some inequalities related to an inequality of
Mathieu, Math. Ann. 250, pp. 9598, 1980.
[3]
A. Erdelyi, W. Magnus, F. Oberhettinger and F.G. Tricomi (EdS),
Higher Transcedental Functions, Vols. 123, McGraw Hill, N.
YorkTorontoLondon, 1953.
[4]
N. Elezovic, H.M. Srivastava, Z. Tomovski, Integral
representations and integral transforms of some families of Mathieu type
series. Integral Transforms and Special Functions, 19; No.7, pp.
481495, 2008.
[5]
O. Emersleben, Uber die Reihe. Math. Ann. 125, pp. 165171, 1952.
[6]
I.S.Gradshteyn, I.M. Ryzhik, Table of Integrals, Series and
Products, Academic Press, N. YorkLondon, 1965.
[7]
Yu.F.Luchko, V.S.Kiryakova, Generalized Hankel transforms for
hyperBessel differential operators C.R. Acad. Bulgare Sci. 53. No.8,
pp. 1720, 2000.
[8]
E.L.Mathieu, Traite’ de Physique Mathematique. VIVII : Theory de
l’ Elasticite’ des Corps Solides (Part2). Gauthier Villars, Paris, 1890.
[9]
A.R. Miller and H.M. Srivastava, On the Mellin transform of a
product of hypergeometric functions, J. Austral. Math. Soc. Ser. B40,
pp. 222237, 1998.
[10]
T.K. Pogany, H.M. Srivastava, Z. Tomovski, some families of
Mathieu aseries and alternating Mathieu aseries, Appl. Math.
Computation 173, pp. 69108, 2006.
[11]
A.P. Prudnikov, Yu. A. Brychkov O.I. Marichev, Integrals and
series, Vol.1: Elementary Functions; Vol.2, Special Functions Gordon and
Breach Sci. Publ., N.York (1992): Russian Original Nauka, Moscow, 1981.
[12]
A.P. Prudnikov, Yu. A. Brychkov, O.I. Marichev, Integrals and
series (More Special Functions), Gordon and Breach Sci. Publ. N. York
(1990): Russian Original: Nauka, Moscow, 1981.
[13]
F. Qi, An integral expression and some inequalities of Mathieu
type series, Rostock, Math. Kolloq. 58, pp. 3746, 2004.
[14]
H.M. Srivastava, Z. Tomovski, Some problems and solutions
involving Mathieu’s series and its generalizations J. Inequal Pure Appl.
Math. 5 No.2, Article 45, pp. 113 (electronic), 2004.
[15]
H.M. Srivastava and H.L. Manocha, A Treatise of Generating
Functions, Halsted Press (Ellis Horwood Ltd. Chichester), J. Wiley &
Sons, N. YorkChichesterBrisbaneToronto, 1984.
[16]
Z. Tomovski ; K. Trencevski, On an open problem of BaiNi Guo and
Feng Qi. J. Inequal. Pure Appl. Math. 4, No.2, Article 29, pp. 17
(electronic), 2003.
[17]
Z. Tomovski, New double inequality for Mathieu Series, Univ.
Beograd Publ. Elektrotehn. Fak. Ser. Mat. 15 pp. 7983, 2004.
[18]
Z. Tomovski, Integral representations of generalized Mathieu
Series Via MittagLeffler type functions. Fract. Calc. and Appl. Anal.
10, No. 2, pp. 127138, 2007.
[19]
Z. Tomovski, New integral and series representations of the
generalized Mathieu Series. Appl. Anal. Discrete Math. 2, No.2, pp.
205212, 2008.
[20]
Z. Tomovski, Vu Kim Taun, On Fourier transforms and summation
formulas of generalized Mathieu Series. Math. Science Res. J; To appear,
2009.
