Significance of Mathematical Analysis in Operational Methods [2014]

Ajay Shukla
Department of Mathematics,Department of Mathematics,
Sardar Vallabhbhai National Institute of
Technology, Surat-395 007, India.
e-mail: ajayshukla2@rediffmail.com
5th Lecture of Father Valles lecture series,
St. Xavier’s College, Ahmedabad
August 2nd , 2014

Laguerre polynomials occur in many fields of
research in science, Engineering and Numerical
Mathematics such as, in Quantum Mechanics,
Communication theory and Numerical Inverse
Laplace Transform. An attempt is made to discussLaplace Transform. An attempt is made to discuss
on Recent Development in Integral Transforms,
specially in Laguerre Transforms. We discuss about
Laguerre Transform in one variables as well as in
two variables and its properties.

Laguerre Polynomial in one VariablesLaguerre Polynomial in one Variables

Laguerre Polynomial:
The Laguerre Differential Equation can be written as
(1)
2
2
(1 ) 0
d y dy
x x ny
dx dx
+ − + =
( )
( )
th e n ,
0
n n 1n 2y ( x ) c 1 x x _ _
2 21 2 !
 
 
 
  
−
= − + −
(2)
(2) is the series solution of the differential equation (1).
4
( )
( )
( ) ( )
( )
( )
( ) ( )
0
0
2 !
n n n 1 ..... n r 1r rc 1 x
2r 0 r !
n r n ! rc 1 x
2r 0 n r ! r !
  
− − +
= −∑
=
= −∑
= −

If C0 = 1 then,
(3)
is known as simple Laguerre polynomial.
Now if the differential equation can be changed to
(4)
( ) ( ) ( )
( ) ( )
rx
n
0r 2r!!rn
2n!r1xnL ∑
= −
−=
2
2
(1 ) 0
d y dy
x x ny
dx dx
α+ + − + =
(4)
then,
(5)
is known as Generalized Laguerre polynomial.
5
2
(1 ) 0x x ny
dx dx
α+ + − + =
( ) ( ) ( )
( ) ( )
rx
n
0r r1!rnr!
n1r1
xnL ∑
= +−
+−
=
α
αα
( )






+
−+
=
;1
;
!
1
11
α
α xn
F
n

Orthogonality of Laguerre Polynomials:
(6)
Recurrence Relation of Laguerre Polynomials:
(7)
( )( ) ( )( ) ( )
( ) ( ) 1αRe,
!n
nα1Γ
1αRen,m0,dxxα
mLxα
nLxex
0
−>
++
=
−>≠=∫
−∞
( )
( ) ( )
( ) ( ) ( )
( )xLnxLnxLDx nnn
ααα
α 1−+−=
(8)
(9)
(10)
6
( )
( ) ( )
( ) ( )
( )xLxDLxDL nnn
ααα
11 −− −=
( )
( ) ( )
( )∑
−
=
−=
1
0
n
k
kn xLxDL
αα
( )
( ) ( ) ( )
( ) ( ) ( )
( )xLnxLxnxLn nnn
ααα
αα 21 112 −− +−−−+−=

Laguerre Transform in one VariablesLaguerre Transform in one Variables

Laguerre Transform in one Variables:
Laguerre transform can be used effectively to solve the heat
conduction problem (Debnath et al. [3]) in a semi-infinite
medium with variable thermal conductivity in the presence of a
heat source within the medium.
Definition:
Debnath (1960) introduced the Laguerre transform of aDebnath (1960) introduced the Laguerre transform of a
function defined in by means of the integral
(11)
where is the Laguerre polynomial of degree and
order discussed in the first Section (5), which
satisfies the ordinary differential equation given in (4). 8
)(xf ∞<≤ x0
∫
∞
−
==
0
)()()(
~
)}({ dxxfxLxenfxfL n
x αα
α
)(xLn
α
0≥n
1−>α

In view of the orthogonal property of the Laguerre polynomials
(12)
where is the Kronecker delta symbol and is given by
(13)
( ) mnnmnnm
x
n
n
dxxLxLxe ∂∂=∂+Γ




 +
=∫
∞
−
1)()(
0
α
αααα
mn∂ n∂
( )1+Γ




 +
=∂ α
α
n
n
n
The Inverse Laguerre transform is given by
(14)
9
 n
{ } )()(
~
)(
~
)(
0
11
xLnfnfLxf n
n
n
α
αα ∑
∞
=
−−
∂==

Example 1:
If , then
This follows directly from the definitions, (11) and (12).
Example 2:
( ) ( )mf x L xα
= mnnm xLL ∂∂=)}({ α
Examples:
Example 2:
If and then
where result of Erdélyi et al. [4] (p 191) is used.
10
1a > −
ax
exf −
=)(
( )
10
)1(
)1(!
1
)(}{ ++
∞
+−−
+
++Γ
== ∫ α
αα α
n
n
n
axax
an
an
dxxLxeeL

Example 3:
Example 4:
If where s is a positive real number, then( ) 1s
f x x −
=
Example 5:
11

Example 6:
Example 7:
Example 8:
12

Basic Operational Properties of Laguerre
Transform in one VariablesTransform in one Variables

We obtain the Laguerre transform of derivatives of f(x) as
15
and so on for the Laguerre transforms of higher derivatives.

which is, due to Erdelyi (1954, vol. 2, p. 192)
16

Theorem 1
17
where L stands for the zero-order Laguerre transform.
Theorem 2
If exists, then( )( ) ( )L f x f nα= ɶ

where R[f(x)] is the differential operator given by
Theorem 3
(Convolution Theorem). If and
, then
( )( ) ( )L f x f nα= ɶ ( )( ) ( )L g x g nα= ɶ
18
where h(x) is given by the following repeated integral

Laguerre Polynomial of Two VariablesLaguerre Polynomial of Two Variables

In 1991, Ragab [12] defined Laguerre polynomials of two
variables as,
where is the well-known Laguerre polynomial of
one variable.
( )
( ),
,nL x y
α β
( )
( )
( ) ( ) ( ) ( )
( )
( ) ( )
,
0
1 1
,
! ! 1 1
r
n
n r
n
r
n n y L x
L x y
n r n r r
α
α β α β
α β
−
=
Γ + + Γ + + −
=
Γ + − + Γ + +
∑
( )
( )nL x
α
one variable.
The following explicit representation of is given
by Ragab [12] :
20
( )
( ),
,nL x y
α β
( )
( )
( ) ( )
( )
( )
( ) ( )
,
2
0 0
1 1
,
! ! 1 1!
r sn n r
n n r s
n
r s s r
n y x
L x y
r sn
α β α β
α β
−
+
= =
+ + −
=
+ +
∑ ∑

Generating Functions for :
where is a confluent hypergeometric function of two
variables and is defined as
( )
( ),
,nL x y
α β
( )
( )
( ) ( )
( )
[ ],
22
1 1
, ; 1, 1; ,
!
n n
nL x y n x y
n
α β α β
ψ α β
+ +
= − + +
2ψ
21
[ ]
( )
( ) ( )2
0 0
; , ; ,
! !
k n
n k
n k k n
a x y
a b b x y
k n b b
ψ
∞ ∞
+
= =
′ =
′∑ ∑

Further, in view of the following definition of Bessel’s
function , for n not a negative integer
the generating function can be written as
( )nJ z
( )
( )
( ) ( )2
0 1
2 ;1 ; ,
41
n
n
z
zJ z F n
n
= − + −
Γ +
the generating function can be written as
22
( ) ( )
( )
( )
( ) ( ) ( ) ( ) ( )
,
0
2 2
!
,
1 1
1 (1 ) 2 2
n
n
n n n
t
n t
L x y
xt yt e J x t J y t
α β
α β
α β
α β
α β
∞
=
− −
+ +
= Γ + Γ +
∑

If in the above generating relation , we replace
the generating relation can be written
as
( ) ( )
( )
( ) ( )
( )
,
2
0
! ,
1 ; 1, 1; ,
1 1 1 1
n
cnn
n n n
n c L x y t xt yt
t c
t t
α β
ψ α β
α β
∞
−
=
− − 
= − + + + + − − 
∑
( )
( ),
,nL x y
α β
by and let ,
t
t c
c
→ ∞
as
23
( )
( )
( )
( )
,
1
2
0
! ,
1 1 ;1 ,1 ; ,
1 1 1
n
n
n n
n L x y t xt yt
t
t t
α β
β
ψ β α β
α
∞
− −
=
 
= − + + + + − − 
∑

Laguerre Transform in Two VariablesLaguerre Transform in Two Variables

Significance of Mathematical Analysis in Operational Methods [2014]

BASIC OPERATIONAL PROPERTIESBASIC OPERATIONAL PROPERTIES

INVERSE OF LAGUERRE TRANSFORMINVERSE OF LAGUERRE TRANSFORM

Some Properties of Laguerre Transform in
two Variablestwo Variables

Applications:
The study of Laguerre Transform is gaining more importance
among the researchers due to its applications in the area of
Mathematical Physics. Debnath et al.[3](p 520) had applied
Laguerre Transform to solve Heat Conduction problem as well
as to solve Diffusion equation. In his paper “An Application of
Laguerre Transform on the problem of Oscillation of a very
long and heavy chain” Debnath et al have applied Laguerre
Transform to solve Mechanical Vibration problem.Transform to solve Mechanical Vibration problem.
G. Evangelista of Switzerland developed a new method for real
time frequency warping of sounds using short-time Laguerre
Transform.
(courtesy:infoscience.epfl.ch/record/34067/files/Evangelista00
a.pdf)
42

Mandyam et al. [8] of The University of New Mexico has
applied Discrete Laguerre Transform to the Speech Coding.
(courtesy:
http://www.computer.org/portal/web/csdl/doi/10.1109/ACSSC.1
995.540894)
Sumita et al.[17] of The University of Rochester applied
Bivariate Laguerre Transform for Numerical Exploration of
Bivariate Process.Bivariate Process.
Keilson et al. [5] used Laguerre Transform as a tool for the
numerical solution of integral equations of convolution type.
Application of Laguerre Transforms in Two Variables.
43

References :
1. Andrews, G.E., Askey R. and Roy R.: Special Functions, Cambridge Univ.
Press, 1999.
2. Bailey W.N., Generalized Hypergeometric Series, Stechert-Hafner Service
Agency, New York and London, 1964.
3. Debnath L. and Bhatta D., Integral Transform and Their Applications,
Chapman & Hall/CRC, 2007.
4. Erdélyi, A., Magnus, W., Oberhettinger, F. and Tricomi, F. G., Higher
Transcendental Functions, McGraw Hill, New York, Vol.1,1981.
5. Keilson J. and Nunn W. R., Laguerre transformation as a tool for the
numerical solution of integral equations of convolution type, Appliednumerical solution of integral equations of convolution type, Applied
Mathematics and Computation, Vol. 5(4), 313-359, 1979.
6. Lebedev N. N., Special functions and their applications, Prentice – Hall, Inc.
Englewood Cliffs, N. J., 1965.
7. McBride E. B., Obtaining generating functions, Springer-Verlag, New York,
1971.
8. Mandyam G. , Ahmed N. and Magotra N. , Application of the Discrete
Laguerre Transform to Speech Coding, Proc. IEEE Asilomar Conf. on
Signals, Systems and Computers, 1995. 45

9. Prajapati J.C., Ph. D. Thesis submitted to S.V.N.I.T.,SURAT On
Generalization of Mittag-Leffler Function and Its Applications, 2008.
10. Rahman M., Gasper G., Basic Hypergeometric Series, Cambridge Univ.
Press, 1990.
11. Rainville E. D., Special Functions, The Macmillan Company, New York 1960.
12. Saran N., Sharma S.D., Trivedi T.N., Special Functions, Pragati Prakashan,
Meerut, 2001.
13. Sharma J.M. and Gupta R.K., Special Functions, Krishna Prakashan,
Meerut, 1996.
14. Shukla A.K., Salehbhai I.A., and Prajapati J.C., On the Laguerre transform14. Shukla A.K., Salehbhai I.A., and Prajapati J.C., On the Laguerre transform
in two variables, Integral Transforms Spec. Funct., 20 (2009) 459-470.
15. Slater L. J., Generalized Hypergeometric functions, University press,
Cambridge, 1966.
16. Sneddon I. N., Special function of mathematical physics and chemistry, 3rd
edition, Longman, London, 1961.
17. Sumita U. and Kijima M., The Bivariate Laguerre Transform and Its
Applications: Numerical Exploration of Bivariate Processes, Adv. Appl. Prob.,
Vol. 17, 683-708, 1985.
46

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