AbstractLet K be a field and put $${\mathcal {A}}:=\{(i,j,k,m)\in \mathbb {N}^{4}:\;i\le j\;\text{ and }\;m\le k\}$$
A
:
=
{
(
i
,
j
,
k
,
m
)
∈
N
4
:
i
≤
j
and
m
≤
k
}
. For any given $$A\in {\mathcal {A}}$$
A
∈
A
we consider the sequence of polynomials $$(r_{A,n}(x))_{n\in \mathbb {N}}$$
(
r
A
,
n
(
x
)
)
n
∈
N
defined by the recurrence $$\begin{aligned} r_{A,n}(x)=f_{n}(x)r_{A,n-1}(x)-v_{n}x^{m}r_{A,n-2}(x),\;n\ge 2, \end{aligned}$$
r
A
,
n
(
x
)
=
f
n
(
x
)
r
A
,
n
-
1
(
x
)
-
v
n
x
m
r
A
,
n
-
2
(
x
)
,
n
≥
2
,
where the initial polynomials $$r_{A,0}, r_{A,1}\in K[x]$$
r
A
,
0
,
r
A
,
1
∈
K
[
x
]
are of degree i, j respectively and $$f_{n}\in K[x], n\ge 2$$
f
n
∈
K
[
x
]
,
n
≥
2
, is of degree k with variable coefficients. The aim of the paper is to prove the formula for the resultant $${\text {Res}}(r_{A,n}(x),r_{A,n-1}(x))$$
Res
(
r
A
,
n
(
x
)
,
r
A
,
n
-
1
(
x
)
)
. Our result is an extension of the classical Schur formula which is obtained for $$A=(0,1,1,0)$$
A
=
(
0
,
1
,
1
,
0
)
. As an application we get the formula for the resultant $${\text {Res}}(r_{A,n},r_{A,n-2})$$
Res
(
r
A
,
n
,
r
A
,
n
-
2
)
, where the sequence $$(r_{A,n})_{n\in \mathbb {N}}$$
(
r
A
,
n
)
n
∈
N
is the sequence of orthogonal polynomials corresponding to a moment functional which is symmetric.
On a generalization of Schur theorem concerning resultants
