Internat. J. Math. & Math. Sci.
VOL. 14 NO. 4 (1991) 737-740
737
SOME FORMULAS OF L. CARLITZ
ON
HERMITE POLYNOMIALS
S.K. CHATTERJEA and S.M. EAQUB ALl
Department of Mathematics
University of Calcutta
Calcutta- 700 019
India
(Received May 31, 1991)
We have used the idea of ’quasi inner product’ introduced by L. R.
Bragg in 1986 to consider generating series
ABSTRACT.
H2(X)n ll(y) t n
22n
n=o
(n!) 2
studied by L. earlitz in 1963. The pecularity of the
series is that there is
(n!) 2
usuaI generating
series
in the denominator, which has a striking deviation from the
containing n! in the denominator. Our generating function for the sid generating
series is quite different from that of Carlitz, but somewhat
analogous to generating
integrals derived by G.N. Watson (ltigher Transcendental function Vol.III, P 271-272
for the case of Legendre, Gegenbauer and Jacobi polynomiaiso
KEY WORDS AND PHRASES.
Hermite polynomials, Generating function, Quasi inner
product.
1980 AMS SUBJECT CLASSIFICATION CODE 33A65
I. INTRODUCTION.
In [I], L. Carlitz has obtained the following formula involving Hermite
polynomials
F.
n=o
H (x)
n
2
Hn(Y) t
22n(n!_)2
n
I
r=o
t
22r(r!) 2
r
(l+t) 2r+l
H2r(X
exp{
H2r(Y
2(x2+y2)t}
l+t
(1.1)
),
which is equivalent to
t
r.
n=o
n
22n(n! )2
Y.
r=o
"lt,
H2n( x ll_+t)l+t H2n( YI’iX’
")
22r( r! )2
(l+t) 2r+I
exp
He has also pointed out that the polynomial
n
F (x,y)= r.
n
r=o
(-1) r
H2r(X)H2r(Y) L2 r (2x2+2y2)
n-r
2
22r (r!)
has the generating function
(1.2)
738
S.K. CHATTERJEA AND S.M. EAQUB ALl
Fn(x,y)tn
n=o
r
(-1)
l
r=o
;I.r (x)
’!
(y) t
2r
(l-t) 2r+l
22r (r!)-
r
-2t(x2+y 2
exp
(1.4)
l-t
which suggests the problem of finding a closed expression for tile sum
(-l)r !o
2
r=o
2r
qor(y)
(x)
z
2
(r
r
The object of the present paper is first to prove the formula
Hn2(X q2n(Y)tn
22n (n! )2
n=o
e
2
x
+y
2
2
Y (I-2t cos 28+t2) -i
2
exp
O
(t-1(x2+y2)}{’+2t
cos +t
cos
[2/t(x2-y2)sing}
d@, (I .5)
I+2’t cos +t
which may be compared with the formula (I.I) due to Carlitz.
Indeed, on comparing (I.I) and (1.5) we obtain
- --
r.
(r!
r;o
where
$
o
I
H2r (x |-i- 2r (y
/t
(1-2t cos 2@+t 2)
exp
1-t
exp
,(t-1 (2/y2)
i+2/t cos /t
cos
2
-t
{’i’ (x2/y)}
2/t(x2-,2)sin
{i+2]t
cos O+t
I
(1 6)
0 }d.
(1.7)
The following particular cases f our result (1.5) see to be interesting,
2
dO
exp
(l-2t cos
cos
H(x)tn"
e2x
)22n(n!
.2
(2m!)_
(t/4)2m
(m!)4
n=o
y.
m=o
2I
20+t2)
o
2
Y
(1-2tcos20+t2)
2x2(t.-1)
{l+2/t
(1.8)
dO.
(1.9)
o
Similarly from our result (1.6) the following particular cases are noteworthy
t
l
22r(r!)2 (l+t)2r+l
r=o
e
r
2
H2r
(x)
2x 2 2
2
Z
r--o
Y (l-2t cos 28+t2) -=
exp
0
(.2r!)2
t
r
22r(r ’)4(l+t)2r+l
t-2(1,+t)x 2
I+2’t cos 8+t" dO.
I__. 2
$ (l-2t cos 28+t2) -1/2
2w
dg.
(.1o)
(1.11)
O
2. PROOF OF OUR RESULT (l. 5).
In proving the result (1.5) we recall the method of ’quasi inner product’,
introduced by L.R. Bragg [2], where the name ’quasi inner product’ is not appropriate,
-
on account of the fact that inner product has a different meaning, however one may
call it simply the product or multiplication like Hadamard’s multiplication.
We know that
Y.
n--o
H2(x)
t
n
(I-4t2)
exp
,,
(4x2t
f(x t)
(2 1)
SOME FORMULAS ON HERMITE POLYNOMIALS
739
Similar|y we write
9
’(y)t
Z
n
Brag’s
Then by
"
(1-4 t2)_
n!
n=o
f(y,t)
exPl+2t)
method we obtain
H(x)!(y)t 2n
E
f(x,t)Of(y,t)
(n!)2
n=o
2
(l-4t
2e2i0)_
(I-4 t2e_2i
0
)-
I__ f2 (l_qt2
2
cos
20+16t4 )-
exp{ 4x2tei0
i0 +
l+2te
2
o
4y2te-i0-}
l+2to
d9
-I
(x2+)cos 0 + 2t(x2+y)}
exp {4t
l+4t cos
o
9
9
e
+ 4t-
4t(x2-y2’) sine}
2
cos
de
l+4t cos 0 + 4t
In
other words
n
Z
(x)H
(y)tn
e
22n(n!)2
n=o
x2+y 2
f (1-2t cos
2
20+t2)
0
-
exp
2)
{(t-1)(x2+y
1/2t cos 0 + t
cos
1+2’t cos 0 +
t
dO
(1.5).
which is
3.
2/t(x2-y2)sin 8}
PARTICULAR CASES.
x in (1.5) we get
llsing y
!t4n(x)tn
y"
22n(n! )2
n=o
2x
e2
2 2w
of
(1-2t cos 20 +
t2)-1/2
2
2(t,71)x
exp{ l+2vt
cos
e +’ t
dO,
(3.1)
which is (1.8).
Again putting x
0 in (3.1) we get
2
(2m!)2 t 2m
(m’.)4 ()
Z
m--o
’ el
(1.9).
which is
(l-2tcos20+
t2)-
dO,
(3.2)
It may be noted that the integrnd is connected with the generating function
of the Legendre polynomials.
-
Indeed, using the well-known generating function of the Legendre polynomials
and term-by-term integration we obtain
2
Z tn(f
de=
2e)de).
I__ Ir(l-2t cos 2{9+
n=o
0
2w o
Now by the known result
t2)-1/2
f
Pn (cs 0)d0
0 when n is odd
(2m!)2 4_2m
4
(m!)
We can rediscover (3.2).
Pn(COS
when n
2m.
(3.3)
740
The derivation of the particular case (1.10) and (1.11) are similar in nature.
Furthermore, it may be of much interest to compare (1.9) and (1.11) as the left
members of two results are apparently different in nature. We now like to examine
the left members of (1.9) and (1.11). To do this we first observe that the left
member of (1.11) is
Z
m=o
’(2m!)2 tm
22m( m!)
4
(l+t)
2m+l
I
t
m
m
Z (-1
r=o r!
m=o
(2m-2r)! (2m-r)!
22(m-r)m-r)! 4
which, on comparison with the series in the left member of (1.9), yields the novel
identity
m
E
(-1
r=o r!
(2m-2r)’ (2m-r)
22(m-r)((m-r)!)
0
for odd n.
(m!) 2
l___ for even
((m/2)!) 4 4 m
m.
(3.4)
REFERENCES
I.
Bragg, L.R.
Quasi inner products of analytical functions with
application to special functions, SIAM J. MATH. ANAL.
Vol. 17 (1986), 220-230.
2.
Carlitz, L.
Some formulas of E. Feldheim, Acta .Math. Acad. Sci.
Hungar 14 (1963), P 21-29.