A Note on the (p, q)-Derivative Operator
In this note, for the ( p , q )-derivative operator D p , q defined by D p , q f ( x ) = f ( p x ) - f ( q x ) ( p - q ) x if x ≠ 0 ; f ′ ( 0 ) , if x = 0 . , where p ≠ q , we investigate the following property: ( D p , q n f ) ( 0 ) = lim x → 0 D p , q n f ( x ) = f ( n ) ( 0 ) ( ( p , q ) ; ( p ,...
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| Veröffentlicht in: | International journal of applied and computational mathematics 2024, Vol.10 (6), Article 172 |
|---|---|
| 1. Verfasser: | |
| Format: | Artikel |
| Sprache: | eng |
| Schlagworte: | |
| Online-Zugang: | Volltext |
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| Zusammenfassung: | In this note, for the (
p
,
q
)-derivative operator
D
p
,
q
defined by
D
p
,
q
f
(
x
)
=
f
(
p
x
)
-
f
(
q
x
)
(
p
-
q
)
x
if
x
≠
0
;
f
′
(
0
)
,
if
x
=
0
.
,
where
p
≠
q
,
we investigate the following property:
(
D
p
,
q
n
f
)
(
0
)
=
lim
x
→
0
D
p
,
q
n
f
(
x
)
=
f
(
n
)
(
0
)
(
(
p
,
q
)
;
(
p
,
q
)
)
n
n
!
(
p
-
q
)
n
=
f
(
n
)
(
0
)
[
n
]
p
,
q
!
n
!
,
n
∈
{
1
,
2
,
…
}
,
for which
f
(
n
)
(
0
)
exists. The real variable and the complex variable cases are considered. Specializing to the case
p
=
1
, we recover the property for the
q
-derivative operator as obtained by J. Koekoek and R. Koekoek in 1993, in related paper. Furthermore, some applications were discussed. |
|---|---|
| ISSN: | 2349-5103 2199-5796 |
| DOI: | 10.1007/s40819-024-01800-x |