[seqfan] A seemingly pathological result related to A057059.

[Ed Jeffery]

Seqfans,

Let n in {1,2,...} and N=2*n+1. For N>3, let G_N be the n X n tridiagonal
matrix defined by

G_N=[0,1,0,...,0; 1,0,1,0,...,0; 0,1,0,1,0,...,0; ...; 0,...,0,1,0,1;
0,...,0,1,1].

For example, for N=9, and ignoring the dots,

G_9=
.0.1.0.0.
.1.0.1.0.
.0.1.0.1.
.0.0.1.1.

Define the sequence of matrix polynomials

A_(N,0)=[G_N]^0=I_n (the n X n identity),
A_(N,1)=G_N,
A_(N,r)=G_N*A_(N,r-1)-A_(N,r-2), r=2,3,...,n-1.

Define A_(3,0)=[1], taken as the 1 X 1 identity matrix, for the sake of
completeness. The A_(N,k) are called unit-primitive matrices (see
https://oeis.org/wiki/User:L._Edson_Jeffery/UnitPrimitiveMatrix).

Let j=1,2,...,n and x_j=(-1)^(j-1)*cos(j*Pi/N). Let {U_r(t)} be a sequence
of Chebyshev polynomials of the second kind, defined by

U_0(t)=1,
U_1(t)=2*t
U_r(t)=2*t*U_(r-1)(t)-U_(r-2)(t)  (r>1).

The ordered spectrum of G_N, and therefore of A_(N,1), is given by the
column vector

Spec(G_N)=[U_1(x_1), U_1(x_2), ..., U_1(x_n)]^T,

where M^T denotes the transpose of matrix M; and, generally, the ordered
spectrum of the unit-primitive matrix A_(N,k), k in {0,1,...,n-1}, is given
by the column vector

Spec(A_(N,k))=[U_k(x_1), U_k(x_2), ..., U_k(x_n)]^T.

Let

(1) S_k=Spec(A_(N,k)),

and let

E_N=[S_0, S_1, ..., S_(n-1)]

be the n X n matrix of ordered eigenvalues of the unit-primitive matrices
formed from the components of the vectors S_k. (I hope I haven't made any
mistakes with these definitions.) E_N can be modified in a certain way to
produce a matrix containing ordered column vectors which are ordered
eigenvectors of the unit-primitives, but I won't give those details here.

THEOREM 1. Let J_1=[1]. For all (odd) N>3, the n X n matrix J_N=[E_N]^T*E_N
is integral and positive.

THEOREM 2. Let M_(i,j) denote the entry in row i and column j of matrix M,
i,j in {1,...,n}. (i)

(2) J_N=a_(N,1)*A_(N,0)+a_(N,2)*A_(N,1)+...+a_(N,n)*A_(N,n-1),

where a_(N,1),...,a_(N,n) > 0 are integers, that is, J_N has representation
as an integral linear combination of unit-primitives; (ii)

(3) a_(N,j)=[J_N]_(1,j), j=1,...,n,

that is, the representation is determined by the entries in the first row
of J_N; (iii) moreover, the representation (2) is unique.

CONJECTURE 3. Using the notation of Theorem 2 and the identities (3), by
the association of n with N, let C(1,1)=a_(3,1)=1 and C(n,j)=a_(N,j), for
N>3 (i.e., for n>1), j=1,...,n. Then

C(1,1)=1;
C(2,1)=2, C(2,2)=1;
C(3,1)=3, C(3,2)=1, C(3,3)=2;
etc.,

with the triangle (read by rows) written out beginning as

{C(n,j)}=
{1};
{2,1};
{3,1,2};
{4,1,3,2};
etc.,

from which arises the sequence A057059={1,2,1,3,1,2,4,1,3,2,...}.

CONJECTURE 4. Spec(J_N)=sum[j=1..n, [J_N]_(1,j)*S_(j-1)], where S_k is as
defined in (1).

This last result is a consequence of a more general one from the theory of
unit-primitive matrices, that if

M in {b_1*A_(N,0)+...+b_n*A_(N,n-1) : b_j is an integer},

then Spec(M)=sum[j=1..n, b_j*S_(j-1)], with M_(1,j)=b_j as before.

CONJECTURE 5. J_N has characteristic polynomial of the form
p_N(x)=sum[v=0..n, c_(n,v)*t^(n-v)], where the c_(n,v) are given by the
triangle beginning as

{c_(n,v)}=
{1};
{1,-1};
{1,-5,5};
{1,-14,49,-49};
{1,-30,243,-729,729};
etc.,

with the absolute values of the outer diagonal terms yielding the sequence,
for r=0,1,...,

A52750={(2*r+1)^(r-1)/(r!*2^r)}={1,1,5,49,729,...}.

The triangle of Conjecture 5 has many interesting properties as well and
which I am still investigating. Taking the (1,1) entries of successive
powers [(J_N)^r]_(1,1) yields sequences which I am currently studying (some
of which are already known), and the resulting array will be submitted to
OEIS soon. The generating functions for the absolute values of the columns
of the above triangle have numerators whose coefficients form another
triangle equally interesting. I suppose that at some point I'll run out of
luck with determining their column generating functions.

Regards,

Ed Jeffery