Author:
Avitabile Marina,Mattarei Sandro
Abstract
AbstractFor an odd prime p, let $${{\,\textrm{E}\,}}_{p}(X)=\sum _{n=0}^{\infty } a_{n}X^{n}\in {\mathbb {F}}_p[[X]]$$
E
p
(
X
)
=
∑
n
=
0
∞
a
n
X
n
∈
F
p
[
[
X
]
]
denote the reduction modulo p of the Artin-Hasse exponential series. It is known that there exists a series $$G(X^p)\in {\mathbb {F}}_{p}[[X]]$$
G
(
X
p
)
∈
F
p
[
[
X
]
]
, such that $$L_{p-1}^{(-T(X))}(X)={{\,\textrm{E}\,}}_{p}(X)\cdot G(X^p)$$
L
p
-
1
(
-
T
(
X
)
)
(
X
)
=
E
p
(
X
)
·
G
(
X
p
)
, where $$T(X)=\sum _{i=1}^{\infty }X^{p^{i}}$$
T
(
X
)
=
∑
i
=
1
∞
X
p
i
and $$L_{p-1}^{(\alpha )}(X)$$
L
p
-
1
(
α
)
(
X
)
denotes the (generalized) Laguerre polynomial of degree $$p-1$$
p
-
1
. We prove that $$G(X^p)=\sum _{n=0}^{\infty }(-1)^n a_{np}X^{np}$$
G
(
X
p
)
=
∑
n
=
0
∞
(
-
1
)
n
a
np
X
np
, and show that it satisfies $$ G(X^p)\,G(-X^p)\,T(X)=X^p. $$
G
(
X
p
)
G
(
-
X
p
)
T
(
X
)
=
X
p
.
Funder
Università degli Studi di Milano - Bicocca
Publisher
Springer Science and Business Media LLC
Subject
Applied Mathematics,Discrete Mathematics and Combinatorics,General Mathematics
Reference10 articles.
1. Avitabile, M., Mattarei, S.: A generalized truncated logarithm. Aequationes Math. 93(4), 711–734 (2019)
2. Avitabile, M., Mattarei, S.: Grading switching for modular non-associative algebras. Lie algebras and related topics. Contemp. Math. Am. Math. Soc. 625, 1–14 (2015)
3. Avitabile, Marina, Mattarei, Sandro: Laguerre polynomials of derivations. Israel J. Math. 205(1), 109–126 (2015)
4. Avitabile, Marina, Mattarei, Sandro: Generalized finite polylogarithms. Glasgow Math. J. 63(1), 66–80 (2021)
5. Block, R.E., Wilson, R.L.: The simple Lie $$p$$-algebras of rank two. Ann. of Math. 115(1), 93–168 (1982)
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