Weisner's methodic survey of modified laguerre polynomials

              Rev.Acad.Canar.Cienc., IX (Núm. 1), 23-32 (1997)
WEISNER'S METHODIC SURVEY
OF MODIFIED LAGUERRE POLYNOMIALS*
A.K.CHONGDAR - G.PITTALUGA - L.SACRIPANTE
ABSTRACT - A good number of generating functions involving modified Laguerre
polynomials La,b,m,n(x) have been derived by sorne researches ((4-7]) by suitable single
interpretation to (i) the index n, to (ii) the parameter m and by suitable double
interpretation to (iii) the index n and the parameter m simultaneously while applying group
theoretic method of obtaining generating functions introduced by L.Weisner in the study of
modified Laguerre polynomials. In this article the authors have made a modest attempt to
present a comprehensive Weisner's methodic su rvey on the polynotnials under consideration.
Moreover they have shown that the results obtained by double interpretation while studying
La,b,m, n(x) by the appl ication of We isner's method can be easi ly derived from the results
obtained by single interpreta.t.ion to the index n while investiga.ting La,b,m-n,n(x), a
modification of L a.b,m,n(x) by the sa111e method.
1980 AMS Sub]ect Classification: 33A75
Key words and phrases: Laguer re Polynomial, Generating function.
l. Introduction. In 1983, Goyal [l] defined the modified Laguerre polynomial as
follows:
(1.1) bn(m)n ( ax) La ,b,m,n(x) = --1-1F1 -n,m;-b
n.
satisfying the following ordinary differential equation
( 1.2)
rnc/0,- 1, -2, ... ,
d
dx ·
In 1955, Weisner [2] gave a rnethod of obtaining generating functions from the Lie
group view point, which is subsec¡uent ly known as "vVeisner's group theoretic method of
obtaining generating functions" while investigating HypergPornet ric polynomials.
Weisner's method of obtaining generating functions consists in constructing a partial
differential ec¡uation frorn an ordinary clifferential equation satisfied by a certain special
function by suitable interpretation to either the inclex or to the pararneter of the special
function under consicleration and then finding a non-trivial continuous transformations
*Work supported by the Consiglio Nazionale delle Ricerche of ltaly and by the Ministero dell'Universitá
e della Ricerca Scientifica e Tecnologica of lta.ly.
:n
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group admitted by the partial differential equation. The above method is lucidly
presented in the book "Obtaining generating functions" written by E.B.McBride [3].
Very recently this method has been extensively utilized for obtaining generating functions
of modified Laguerre polynomials as defined in (1.1) by Singh and Bala [4] with the
interpretation of the index n, Chongdar and Majumdar [5] by the interpretation of the
parameter m, Sen and Chongdar [6] and Chongdar, Pittaluga and Sacripante [7] with
the double interpretation of the index n and the parameter m simultaneously.
While investigating generating functions of La,b,m,n(x) by Weisner's group theoretic
method by the interpretation of the index n, Singh and Bala considered the set of
operators:
such that
a
A1 =ye,,
uy
-1 a a A2 = xy e, - e, ,
ux uy
éJ 2 éJ
A3 = bxyc:, + by e, + (bm - ax)y
ux uy
A1(La ,b,m ,n(x)y") =nL0 ,b,m,n(x)yn,
A2(Lo.b.m,n(x)y") = b(l - m - n)La,b,m ,n-1 (x)yn-l,
A3(Lo ,b,m ,n(x)y") = (n + l)La,b,m ,n+1(x)y"+ 1 .
The following commutator relations sa.tisfiecl by A1, A2, A3
[A1,A2] = - A2,
[A1, A3] =A3,
[A2, A3] = - 2bA1 - bm,
where
[A, B]u = (A.E - BA)u,
show that the set of operators { 1, A 1 , A2, A3 } generates a Líe algebra L1
For obtaining generating functions by suitable interpretation of the parameter m,
Chongdar ancl Majurnclar [5] consiclerecl the set of operators
a
B1 =y0- ,
y
b éJ
B2 = -y- - y,
a ax
-1º é) -1 B:3 = .ry - + - - y fü au
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such that
B1(La,b,m,n(x)ym) =mLo,b,m,n(x)ym,
B2(Lo,b,m,n(x)y"'·) = - La ,b,m+1,n(x)y"'+1,
B.3(Lo.b,m ,n(x)ym) = (n + m + l )L a,b,m-1,n(x)ym-l.
The following commutator relations satisfied by B; , i = l , 2, 3,
[B1, B2] =B2 ,
[B1,B3]=-B3,
[B2, B3] =l
show that the set of operators { 1, B1 , B2 , B3} generates a Lie algebra .C2
For obtaining generating functions by suitable interpretations of the index n and the
parameter m of the polynomial, Sen ancl Chongdar [6] considered the following operators
such that
C1(La,b,m,n(x)ymz 11 ) ==rri La. ,b,m,11(x)ym zn ,
C2(La,b,m,n(x)y"'zn) =nLa,b,m,n(x)ymzn,
C3(La,b,m,n(x)y"'zn) = (n + l)La,b,m-l ,n+1(x)ym-l zn+ l ,
C4(La,b,m ,n(:r)y"' zn) = - La ,b,m+l,n-1(x)ym+l Zn-l.
The following cornrnutator relations satisfiecl by C;, i = 1, 2, 3, 4
i = 1
i = 1,2
i = 3
; J = 2
; J = 3,4
; J = 4
show that the set of operators { 1, C; , i = 1, 2, 3, 4} genera tes a Lie algebra .C3 .
Finally while invest igating generating functions of the polynornial uncler consideration
with the suitable interpretations of the inclex n ancl the parameter m of the polynomial
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Chongclar, Pittaluga ancl Sacripante consiclerecl the following operators
such that
a
Di =y-a ,
y
a
D2 =:: a:: ,
b a
D.3 = -y- -y,
a ax
-1 a a -] D4 =xy -ax + - -y ay
D5 = bx y -1 z-aa + zb - aa - y -1 z( ax+ b) ,
y::-1 a
D6=---
a a:.r
X y
Di (Lo,b,111,n( :1: )y 111 .: 11 ) = niLa.b, 111 ,n(J:)y·m Zn,
D2(La,b,111, 11( .1·)y 111
::'
1
) == nLc,.b.111,,1(.r)ym ::: 11 ,
D3(Lu.b,111,,,(.T)Y 111
::'1 ) == - L 0 .b.,11 +1 ,11 (x)ym+l :: 11 ,
D,i(La.b.,,,,,,(x)y"' ::") =(n + m + l )La ,b,m-1.n(x)y"'-1 z" ,
D5(La.b.,,,.,,(x)y"'::") =(n + l)La,b,m-l .n+1(.r)y"'- 1::"+ 1 ,
D,;(La,1,.,,,,,,(.r)y"' ::") = - L a,l,,rn+ l,"-1(.r)y"'+l ::n - l .
Tlw following commutator relations sa tisfiecl by D¡, i = l , 2, ... , 6
[D;,D;]= o when i = 1 ;J =2
i = 2 ;] = 3,4
i = 3,4 ;J = 5,6
(- l f+i Di i = 1 ; ./ = 3,4
(-l)i+i+1 Di l = 1,2 ; J = 5,6
1 i = 3 ;J =4
i =5 ;) =6
show tha t the set of operators { 1, D;, 1 = l. 2 ... , 6} genera.tes a Lie algebra L4 ancl each
of thc suh sd.s {l, D,. i = l. 3. 4} ancl {l, D;. i = 1, 2,5, 6} generates a sub algebra of [,4 .
Tlw object of t lw ¡H·esPnt article i~ to lllake a comprehensive study on La,b ,m- n,n( :i; ) -
a moclification of L0 .1,.,,, ·" ( .r) - fnr oht,iininp; generating functions by the application of
vVeisner's group thcord.ic method with t!w interpretation of the inclex n of the polynomial
uncler consideration ancl to 111,1kt' " r<'YÍ<'w on t he previous works.
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In fact, in the present discussion it has been pointed out that the results obtained by
Sen and Chongdar [6] with the help of double interpretation during the application
of Weisner's group theoretic method of obtaining generating functions on the modified
Laguerre polynomials La,b,m,n( x ), may be derived as the particular cases of the results
derived here.
We also like to point it out that our results (2.17) - (2.18) together with the results due
to Chongdar and Majurnder [5] give rise to the main generating relation of the present
authors [7] obtainecl by clouble interpretations to n ancl m., the inclex ancl the parameter
of La,b,m,n(x).
2. St udy of La,b,m-n,n ( x ). The clifferential ec¡uation satisfiecl by La,b,m- n,n( x) is given
by
(2.1) :rD~u + ( m - n - bax ) Dxu + -a-¡ ;nu = O
d
Dx = - .
dx
In this Section, sorne generationg functions of La ,b,m-n,n(x) have been clerivecl by using
\ 1\Teisner's group theoretic metl10d with the suitable interpretation of the inclex n .
i) G roup theoretic cliscussion.
. d f)
Replacmg -d by ~,
f)
n by y~, u by 1/(x,y) m (2.1), we get the following partial
X U X uy
clifferential eq ua tion:
(2.2) 82 v 82 v ( ax) ov ay o v
x-- - y-- + m- - -+-- = 0.
ox2 oxoy b ox b oy
Thus v1(x, y)= La ,b,m- n,n(x)y" is a solution of (2.2) since L a, l,,m-11,n(x) is a solution of
(2.1 ).
We now define the infinitesimal operators A, B, C as follows
(2.3)
such tha t
(2.4)
f)
A=y~,
uy
B = bxyf)- - by? -f)- - [ax+ (1 - m)bJ y , ax au
Y- 1 f) e-· - -;- 01' '
~4(Lo ,b,m - n, 11 (.1.:)y" ) == nL(/,{1 ,m-n,n(J·)y" ~
B (L u, /1 ,m-n ,11 (.1:)y'') =(11 + l)Lo ,/1 ,m- n- J.11+1(.l'),1/ 11+ 1 ,
C'(La,b,m-n,11(.T)y") = - Lrt ,b.m-n+I ,11-J (.r)y" - l.
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The commutator relations satisfied by .4., B , C' are
(2.5) [A,B] = B [A, C'] = -C [B,C'] = l.
The a bove commu tator relations show that the set of operators { 1, .4, B, C'} genera tes a
Lie algebra [, 5 _
It can be easily shown that t he partía! differential operator
which can be expressed as
(2.6) ~L = BC + A + l ,
a
commutes with each .-l, B , C i.e.
(2.7)
The extended form of the groups gcnerated by A, fl , C' are as follows
{
e" 1Af (.1:,y) = f (x,c" 1 y) ,
(2.8) e"'/J f (x, y) =( l + a2by)m- 1exp( - aa2xy)f (x(l + a2by). y b ) ,
1 + a2 y
e a3C. f(..l , )J ) -- J (X. + -a;3; )J - 1 , y ) .
Thus we get
(2.9)
ii ) Genera ti ng fo nct ions.
From (2.2) Wf' sef' that 11(.i-,y) = La,b_,,, _ ,,_,,(.r)y" is a solution of the system
(2.10) {
L 1/ = O
(.-l - n)I/ = O.
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It can be ea.sily verified tha.t
S · ~L(La,b,m-n,n(x)y") = ~L · S(La ,b,m-n,n(x)y" ) =O,
where
Thus the tra.nsforma.tion S(La,l•,m-n,n ( x )y") is annihila.ted by ! L.
Putting a¡= O a.ne! writing f(x , y ) = La ,b,m-n,n(x)yn in (2.9) , we get
(2.11)
( a3 _ 1 ) · La ,b,m-11,n (x + -¡;Y )(1 + a2by) .
But,
(2.12) eª :tC,,a1B\ La,b.11, -n,11(X)y11 ) =
n+J.: / . oo k
n ~ (-a3 y)1' ~ (a2y)
= Y L pi L - 1.-! - (n + l)kLa,b,m-n-k+p ,n+k-p(x ).
p=O k=O
Equa.ting (2. 11 ) a.nd (2. 12) we get
(2.13)
( a3 -1 ) · La,b,m-n,n( :r) (.i- + -¡;Y )(1 + a2by)
n+k / oo k ~ (-a3 y)P ~ (a2y)
L p i L --¡;:¡--(n + l)kLa,b,m-n-k+p ,n+k-p(x) .
p=O k=O
Vi/e now discuss the following particular ca.ses of the a.bove genera.ting rela.tion(2.13).
Ca.se 1 : putting a3 = O and then repla.cing a2y by t in (2.13) we get
(2.14) (1 + bt)"'-"- 1exp(-a.rt)La,b.m-n,n(x)(x(l + bt))
.:f!-..(n+l)k k
L J.:I La ,b,m-n-k,n+k(J·)t .
k=O
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Case 2: putti::.:; a2 = O a.nd then replacing -a3/y by t in (2.13), we get
(2 .15) La,b,m-11,n(x) (.1' - ~) =
n 1
= L P' La,b,m+p-11,11-p(x)tP.
p=O
Ca.se 3 : taking a2a3 f= O without a.ny loss of genera.lity we can choose a2y
-a3/y = t2 in (2.13) ancl then we get
(2. 16) ( 1 + bti )m-n-l exp[-( a.T - t2 )t 1 ]La,b,m-n,n ( ( X - ~) (1 + bt))
t 1 a.nd
Now if we replacr m by m + n on both sicles of (2.14)-(2.16), we get the results founcl
derivecl in [6].
Now we proceecl to deriH· the main result of the present authors [( 3.3) in [7]] by making
use of our results (2 .14H2.15) togetll('r with t he results of Congelar and Ma.jumda.r.
In fact, we see the right hancl si ele of ( 3.3) in [7], with the help of the rela.tions ( Case 2,
of[5]), (Case 1 of [5]), (2. 15) ancl (2.14). is
· ( 11 + 1 ),.( -'17 - P - m + 1 h L a ,b, 111+p-k-r+s,n+r-s( X)
11.1 1 o 5::: 1 a5 y a4 . .
( )
,n- 1 = ( ) r11+r ( )s l+ - - - --(1+-J
u ; ·, ! u( 1 + 7i) ; s! z y
00 1 ( a4 )
1
· L 1 - 0:3.1¡(1 + -) ' (11 + l),.La. l,,m+p - r+s,11+r - s ( ,1'(1 + -a4) )
p=O p . y y
Cl.4 1 [ · (1.4 = ( 1 + -)'"- exp -a:iu(l + - l]
U y
( a.1 h )
·La,l,,m-r+.,.11+1·-, (1 + ---;¡H-r + ~1-a3y) =
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00
( ) r CL¡ 111 ¡ a4 1 a5 Z = ( 1 + - ) - exp [-a,y( 1 + - )] ¿ 1 ( a
Y Y r=O r. Y l + 7)
·(n+l),.Labm-rn+r ( (l+-a4) (x+-ba 3y+ª6-Y.) ) =
' ' ' y a za
(l+a-4 + -ba-s::) m -e1 xp [ -a3y( l+a-4 +b-as-z )- -aa-sz (x+ -a6)y ]
y y y y y az
( b a6y a4 basz )
· La,b,m,, (x + -a3y + -)(1 + - + --) ,
' a za y y
which is the left han el si ele of ( 3.3 )of the result founcl clerivecl in [7].
Other variants of ( 3.3) in [7] can be clerivecl by using the four generating relations, two
of Chongclar a.ne! Maju11clar a.ne! two of this Section.
It may be notecl that the importance of the result (3 .3) in [7] lies in the fact tha.t whenever
one knows the sum, in the closecl form, of a. c¡ua.clruple generating series like the right
member of (3.3) in [7], one can verify the sa.me by cla.ssica.l methocl, but prior to the
existence of a result like (3 .3) in [7] nobocly coulcl even guess such a. generating relation
without the help of group theoretic method.
iii) Rela.tion nf Lo. L¡. L5.
It may be notecl that the commutatm rela.t.ions sat isfi ecl by the operators generating L2
can be comparecl with the following commut.ator relations of L4 :
[D1,D3J = D3
It, therefore, follows tha.t the Lie algebra L2 is isomorpliic \\·ith the sub-Lie algebra
generatecl by Di. D3, D.1 .
Again the commutator relations satisfiecl by the operators of the Lie algebra í:,5 can be
comparecl with the folliwing commutator relations of L4:
[D2,Ds] = Ds
It,therefore, follows tha.t the Lie algebra L.s is 11lso ismnorphic with the sub-Lie algebra
generatecl by D2 , D.s . D" .
Therefore we can st.at.e th11t L., = L2 c-i~ L.s .
References.
[l] G.K.Goyal, Vijn;,n,1 l'.,ris.,d :'ums.rndhm1 Pat.rika . 2G(l983), 263-266.
[2] L.\Veisner, Group t.l,rn.,., ·/.1.c origrn. of cnla.in genern.iing fnnciions , Pacific J. Math.
5(1955), 1033-1039.
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[3] E.B.McBride, Obtaining generating f1mctions, Springer Verlag, Berlin, Heidelberg,
New York (1971).
[4] S.N.Singh and R.N.Bala, Gronp Theoretic origins of certain generating functions of
the modified Laguerre polynomials , Inclian J. Pure and Appl. Math. 17( 4)( 1986),512-521.
[5] A.I<.Chongdar ancl N.K.Majumclar, Group theoretic stiidy of certain generating
functions involving modified Lag·nerre polynomials, Mathematica Balkanica, NS7(1993) ,
142-147.
[6] B.K.Sen and A.K.Chongclar, Some generating functions of modified Laguerre
polynomials by group theoretic method, Communicated.
[7] A.K.Chongdar,G.Pittaluga ancl L.Sacripante, On generating fnnctions of modi.fied
Laguerre polynomials by Lie algebraic method, Quacl.Dip.Mat.Univ.Torino,n.32( 1996 ),
to appear In Rev. Acacl. Canar. Cienc ..
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