C-polynomials and LC-functions: towards a generalization of the Hurwitz zeta function
Let f ( t ) = ∑ n = 0 + ∞ C f , n n ! t n be an analytic function at 0, and let C f , n ( x ) = ∑ k = 0 n n k C f , k x n - k be the sequence of Appell polynomials, referred to as C-polynomials associated to f , constructed from the sequence of coefficients C f , n . We also define P f , n ( x ) as...
Gespeichert in:
| Veröffentlicht in: | The Ramanujan journal Jg. 65; H. 2; S. 821 - 855 |
|---|---|
| 1. Verfasser: | |
| Format: | Journal Article |
| Sprache: | Englisch |
| Veröffentlicht: |
New York
Springer US
01.10.2024
|
| Schlagworte: | |
| ISSN: | 1382-4090, 1572-9303 |
| Online-Zugang: | Volltext |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| Zusammenfassung: | Let
f
(
t
)
=
∑
n
=
0
+
∞
C
f
,
n
n
!
t
n
be an analytic function at 0, and let
C
f
,
n
(
x
)
=
∑
k
=
0
n
n
k
C
f
,
k
x
n
-
k
be the sequence of Appell polynomials, referred to as
C-polynomials associated to
f
, constructed from the sequence of coefficients
C
f
,
n
. We also define
P
f
,
n
(
x
)
as the sequence of C-polynomials associated to the function
p
f
(
t
)
=
f
(
t
)
(
e
t
-
1
)
/
t
, called
P-polynomials associated to
f
. This work investigates three main topics. Firstly, we examine the properties of C-polynomials and P-polynomials and the underlying features that connect them. Secondly, drawing inspiration from the definition of P-polynomials and subject to an additional condition on
f
, we introduce and study the bivariate complex function
P
f
(
s
,
z
)
=
∑
k
=
0
+
∞
z
k
P
f
,
k
s
z
-
k
, which generalizes the
s
z
function and is denoted by
s
(
z
,
f
)
. Thirdly, the paper’s main contribution is the generalization of the Hurwitz zeta function and its fundamental properties, most notably Hurwitz’s formula, by constructing a novel class of functions defined by
L
(
z
,
f
)
=
∑
n
=
n
f
+
∞
n
(
-
z
,
f
)
, which are intrinsically linked to C-polynomials and referred to as
LC-functions associated to
f
(the constant
n
f
is a positive integer dependent on the choice of
f
). |
|---|---|
| ISSN: | 1382-4090 1572-9303 |
| DOI: | 10.1007/s11139-024-00919-1 |