Abstract
AbstractIn the context of Riordan arrays, the problem of determining the square root of a Bell matrix $$R={\mathcal {R}}(f(t)/t,\ f(t))$$
R
=
R
(
f
(
t
)
/
t
,
f
(
t
)
)
defined by a formal power series $$f(t)=\sum _{k \ge 0}f_kt^k$$
f
(
t
)
=
∑
k
≥
0
f
k
t
k
with $$f(0)=f_0=0$$
f
(
0
)
=
f
0
=
0
is presented. It is proved that if $$f^\prime (0)=1$$
f
′
(
0
)
=
1
and $$f^{\prime \prime }(0)\ne 0$$
f
″
(
0
)
≠
0
then there exists another Bell matrix $$H={\mathcal {R}}(h(t)/t,\ h(t))$$
H
=
R
(
h
(
t
)
/
t
,
h
(
t
)
)
such that $$H*H=R;$$
H
∗
H
=
R
;
in particular, function h(t) is univocally determined by a symbolic computational method which in many situations allows to find the function in closed form. Moreover, it is shown that function h(t) is related to the solution of Schröder’s equation. We also compute a Riordan involution related to this kind of matrices.
Funder
Università degli Studi di Firenze
Publisher
Springer Science and Business Media LLC
Subject
Applied Mathematics,Mathematics (miscellaneous)
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