Internat. J. Math. & Math. Sci.
Vol. i0, No.3 (1987) 531-534
531
SOME GENERATING FUNCTIONS OF LAGUERRE POLYNOMIALS
BIDYUT KUMAR GUHA THAKURTA
Department of Mathematics
R.K.M.V.C. College, Rahara,
24-Parganas, West Bengal
INDIA
(Received November 12, 1985)
In this note a class of interesting generating relation, which is stated in
the form of theorem, involving Laguerre polynomials is derived. Some applications of the
ABSTRACT.
theorem
are also given here.
KEY WORDS AND PHRASES. Lguerre polynomials, generating
I80 AMS SUBJECT CLASSIFICATION CODE. 33A65
i.
functions.
INDRODUCTION.
The Laguerre polynomials L
(a)
Ln
(x)
tl+Jn
(a)
n
IFI
n!
(x) are defined by,
(I I)
(-n; I+; x)
where n is a non-negative integer.
From [I] we have
_-_
l
(nil_)
L
n=o
(c)
n (x)
(l-w)
n
w
-l-m
(I 2)
exp( -xw
l-w
La)
m
x
)’
l-w
Obseing the existence of the above generating relation (1.2) the present author Is
interested to investigate the existence of more general generating relation by the
group-theoretic method.
In fact, the following theorem is obtained as the in result
of our investigation.
EO i.
If there exists a generating relation of the fo
G(x,w)
n=oZ an w Ln
then
(I--w)
where
--I -m exp
n
--
l-w
(1.3)
(x)
G(
n=o
X
WE
l-w
l-w
f (z)
n
)
() (x)
en
(1.4)
k
n
Z
z
a
k
k
k=o
The portance of the above theorem lies in the fact that one can get a good number
of generating relations from (i 4) by attributing different suitable values to an in the
relation (1.3).
f (z)
n
B.K. GUHA THAKURTA
532
2.
DERIVATION OF THE THEOREM.
THEOREM I.
Using the differential recurrence relation [23
(=)
d
x
(x))
(Ln+m
L,a,r
n+m+l
(n+m+l)
(x)
()
(x)
x) an+m
(n + m + a +
(2.1)
We find the following partial differential operator,
3
xy
R
+
2
y
3
+ (-x+m+l)y
y
such that
()
y+n Ln+m
(n+m+l) ya+n+l
(x))
L
()
(x)
n+m+l
(2.2)
The extended form of the group generated by R is given by,
e
w f(x y)
-m-I
(l-wy)
f(x
exp( -wxy
l-wy
l[wy
l-wy
Let us consider the generating relation of the form:
G(x w)
Z
a
n=o
n
(C) (x) wn
(2.3)
Ln+m
Replacing w by wyz and then multiplying both sides by y
n=o7" an
G(x,wyz)
y
a
n=o
n
(wyz)
(wz)n
n
we get
yS (=)m (x)
+n -() (x)
-Ln+
Y
Ln+m
w ),
Operating both sides of the above expression by (exp
(ya
(exp w]R.)
G(x,wyz))
we get
(no an(wz)n ya+n L(a)n+m
(exp wlR
(x))
(2.4)
The left member of (2.4) becomes
(l-wy)
-l-m
exp( -wxy
’1 _)rwy
l-wy
wyz
l-wy
x
l-wy
G(
The right member of (2.4) is equal to
k
(a) (x))
(wz)n w R k
n=o
n+k
n+a+k
n
w
z (n+m+l) y
a
Z
k
k!
n=o k o n
kZ=o
(yn+a Ln+m
an
Z=
t
y
=y
nZ=o kZ__o
n
nZ=o kZo
o
y
(2.5)
n=o
a
n
z
n
(wy)
(wy) n
(wy) n
f
a
(z)
z
L
(n+m)
n-k
k! (n-k+m)!
(a)
n+m
(x)
(x)
Ln+m+k
k (n+m)
(an_k
n
(n+m+k)
n+k
()
Ln+m+k
)
L
(a)
n+m
(x)
(2.6)
(x)
where
fn
n+m
n
(z)
kZ=o
ak
k+m
z
k
Equating (2.5) and (2.6) and then putting y=l, we get
(l-w)
--WX
exp
n
fn(Z)
k
(n+m.
.k+m
X
I-W
wn fn (z) L/r
n=o
where,
G
I-W
a
k
z
n
WZ
l-w
)
(x)
(2.7)
this completes the proof of the theorem.
On the other hand, if we consider the continuous transformations group defined by
SOME GENERATING FUNCTIONS OF LAGUERRE POLYNOMIALS
.
the infinitesimal operator
i
e
i
x
2
t
533
z
x
then the equations of finite transformations of the group are
x"
(exp
y"
Wl)X,
w l)y, z"
(exp
(exp
w
(2.8)
l)Z
where w is the parameter of the group under consideration.
Also we know that
w I)
f((exp w l)X, (exp
f (x’, y’, z’)
From [3] we see that the effect of the operator (exp
(exp
f(x,y,z)
w l)y, (exp
w I)
Wl)Z)
(2.9)
on the variables are as
follows
x"
y"
z"
x/ (1-we
t
t
log (1-we
z
x we
t
(2.10)
t/ 2(1-wet).
and
I Fn+m
(x,t,z)
Fn+m+1
(n+m+l)
(2.11)
(x,t,z)
where
Fn+m
Now replacing w by wye
exp
+
mt
t
-
t
+
z
x
-]
x
(a+1) 12
(a) (x)
Ln+m
in (2.3) and then multiplying both members by
+i
2
+
t
x
z
we get
G(x, wye t)-
a+l
+
exp [(n+m)t
(x,t,z)
x
exp{mt + a+l
2
+
t
n
l a (wy)
n--o n
(s+l)/2
z
x
-}
X
(a+1)/2
(2.12)
F
n+m (x,t,z)
(exp w
Operating both members of the above expression by
and using (2.8), (2.9)
and (2.11), we get
G(x
wye
t
a+l
2
exp{mt" +
l
n=o
wn
x"
t" + z
Fn+m
f (y)
n
(x’) (a+l)/2
(2.13)
(x,t,z)
where
n
fn (y)
Putting the values of
n+m
k--Zo
ak
k+m
k
Y
x’, y’, z" from (2.10) and then substituting
t
z
o we finally
obtain
if
G(x,w)
7.
n--o
a
n
n
w
L
(e)
n+m (x)
then
(l-w)
-l--m
--XW
exp( l-w
n=o
X
G( l-w
wn fn
W
(2.14)
() (x)
(y) Ln+m
where
fn (y)
n
kZ--o
n+m
k+m
ak
y
k
which is same as (2.7).
From above we see that if
IR
is used.
be used the calculation becomes much harder than when
B.K. GUHA THAKURTA
534
COROLLARY I.
If we put m=o in the above theorem, we get the following well-known
theorem derived by W.A. Ai-Salam [4], and the second author [5].
"If there exists a generating relation of the form
n
() (x)
E
a w
L
G(x,w)
n=o
n
n
then
(I--w)
-e-I
X
--WX
exp( l-w
n=oZ wn
WZ
G( l-w
l-w
fn(Z)
L
(s)n
(x)
where
n
n
kE--o (k)
fn (z)
APPLICATION.
a
k
z
k
As a nice application of our theorem, we consider the generating
relation given in (1.2), i.e.,
l
n=o
L(a)
(n+m.
n
n+m
n
(x) w
(l-w)-l-=-m
an
If we put
n
-xw
exp( l-w
(a)
L
m
x
l-w
in our theorem, we get
(l-w-wz)-l-a-m
-wx(l+z) )
l-w-wz
exp
n--o
wn
L
()
m
x
l-w-wz
Cs)
f (z) L
n
n+m (x)
where
fn (z)
ACKNOWLEDGEMENT.
m
k_E_o .n+m)
k+m
(m-nn)
z
k
I am grateful to Dr. A.K. Chongdar for his help in the preparation of
this work.
REFERENCES
I.
2.
RAINVILLE, E.D. Special Functions, Chelsea Publishing Company, Bronz, New York,
(1960), 211.
ERDELYI, A. et al. Higher Transcendental Functions, McGraw-Hill Book Company, Inc.,
New York (1953), 189.
Sur les Polynomes de Laguerre, du Point de vue de L’Algebra de Lie,
C.R. Acad. Sc. Paris 270A (1970), 380-383.
3.
DAS, M.K.
4.
AL-SALAM, W.A. Operational Representation for the Laguerre and Other Polynomials,
Duke Math. Jour. 31 (1964), 127-142.
5.
CHONGDAR, A.K. On a Class of Trilateral Generating Relations with Tchebychev
Polynomials from the View Point of One Parameter Group of Continuous
Transformations, Bull Cal. Math. Soc. 73 (1981), 127-140.