General Mathematics Vol. 9, No. 1–2 (2001), 49–52
Note on a class of delta operators
Emil C. Popa
Abstract
Let Q be a delta operator with basic set (pn ), and
An (x) = x(x + nb)n−1 , n = 1, 2, ...
the Abel polynomials.
For each natural number n, n ≥ 1 let us consider the delta operator defined on
the algebra of polynomials, by
(1)
αn,Q = Q(Q + nbI)n−1 , b 6= 0.
The purpose of this paper is to give a representation theorem for basic set (qm )
relative to αn,D , and we define a linear operator tn,D whence we obtain the Rn,a
and αn,D operators (see [1], [2]).
2000 Mathematical Subject Classification: 05A40
1
Theorem 1 If m = min(k, n), we have
m
(2)
1 X (−n)j (−k)j
(nb)n−1 pk−j (x)
(αn,Q pk )(x) =
n j=1 (j − 1)!
where (a)s = a(a + 1) · ... · (a + s − 1).
49
50
Popa C. Emil
Proof.
An (x) =
n−1
X
Ã
j=0
=
n
X
Ã
j=1
Hence
αn,Q =
n
X
n−1
j
n−1
j−1
Ã
j=1
!
(nb)n−j−1 xj+1 =
!
(nb)n−j xj .
n−1
j−1
!
(nb)n−j Qj ,
whence
(αn,Q pk )(x) =
m
X
Ã
j=1
n−1
j−1
!
k!
(nb)n−j pk−j (x)
(k − j)!
with m = min(k, n).
Theorem 2 If (qm ) is a basic sequence for the delta operator αn,D , then
for n > 0
1
qm =
mn−m
(nb)
Γ(mn − m)
(3)
Z∞
0
µ
¶m−1
1
e−t tmn−m−1 x x − t
dt.
nb
Proof. We have
αn,D = D(D + nbI)n−1
and
qm = x(D + nbI)−mn+m em−1 ,
qm (x) = x
I
xm−1 .
mn−m
(D + nbI)
Hence
1
1
qm (x) =
·
(nb)mn−m Γ(mn − m)
µ
Z∞
−t mn−m−1
e t
0
1
x x− t
nb
¶m−1
dt.
Note on a class of delta operators
51
2
Ler R be a delta operator with two Sheffer sequences (rn ) and (e
rn ).
We note
T = eax Dm e−ax = (D − aI)m
and
tn (x) = S(T E b )n ren−1 (x)
where S is the linear operator defined by
Se
rn = rn+1 , n = 0, 1, 2, ...
Theorem 3 If m is natural number, a, b ∈ R, a 6= 0 and RS0 ,
Q
PS0 ∈ ∗t where
P −1 = T E b
then
tn (x) = tn (a, b, m; ren ; x) = Seax Dnm e−ax ren−1 (x + nb)
is a set of Sheffer polynomials.
Proof. It is used the theorem of [3].
For m = 0, we find
tn (a, b, 0; ren ; x) = Se
rn−1 (x + nb)
and for a = m = 1, b = 0,
tn (1, 0, 1; ren ; x) = S(D − I)n ren−1 (x).
Now for ren = rn = xn we have S = X, (XP )(x) = xp(x), and
tn (a, b, 0; xn ; x) = x(x + nb)n−1 ,
tn (1, 0, 1; xn , x) = x(D − I)n xn−1 .
Hence from
tn (x) = tn (a, b, m; ren ; x) = S(D − aI)mn ren−1 (x + nb)
we obtain the linear operator
tn,D = tn (D)
whence we find Rn,α and αn,D operators, (see [1], [2]).
52
Popa C. Emil
References
[1] E.C.Popa, A certain class of linear operators, ICAOR, Cluj-Napoca,
1996, Revue D‘Analyse N umérique et de T héorie de l‘Approximation,
Tome 26, No. 1-2, 1997.
[2] E.C.Popa, Linear operators of Abel type, Bull.Math. de la Soc. des Sci.
Math. de Roum., Vol.40, 1997.
[3] E.C.Popa, Note on Sheffer polynomials, Octogon, Mathematical Magazine, Vol.5, Nr.2, oct. 1997, 56-57.
[4] E.C.Popa, Note on a delta operators, General Mathematics, Vol.8,
Nr.1-2, oct. 2000, 79-82.
[5] S.Roman, G-C.Rota, The umbral calculus, Adv. in Math. 27, 1978,
96-188.
[6] G-C.Rota, D.Kahaner, A.Odlyzko,
J.Math.Anal.Appl. 42, 1973, 685-760.
Department of Mathematics
Faculty of Sciences
“Lucian Blaga” University of Sibiu
2400 Sibiu, Romania
E-mail: emil.popa@ulbsibiu.ro
Finite
operator
calculus,