[seqfan] Re: A269254, A034807 and Chebyshev polynomials

[Brad Klee]

After: http://list.seqfan.eu/pipermail/seqfan/2017-October/018052.html
And: http://list.seqfan.eu/pipermail/seqfan/2017-October/018058.html
And: https://ptpb.pw/txWp

Proof of composite property, Z_k = X_k*Y_k,  for cubic Chebyshev cases
===========================================
Using previous definition for T_k(j) from Andrew Hone, define coefficients:

c_x = T_3(j) = j^3 - 3*j ,
c_y =  1 + T_3(T_2(j)) = j^6 - 6*j^4 + 9*j^2 - 1 ,
c_z = T_3(T_3(j)) = j^9 - 9*j^7 + 27 * j^5 - 30 * j^3 + 9 * j.

Suppose Z_k = X_k*Y_k with recurrence relations,

0 = Z_k - c_z*Z_{k - 1} + Z_{k - 2} ,
0 = X_k - c_x*X_{k - 1} + X_{k - 2},
0 = Y_k - c_y*Y_{k - 1} + c_y*Y_{k - 2} - Y_{k - 3}

As before we state the recursion condition of the induction hypothesis as a
zero sum:

v_0 - c_x * v_1 + v_2 = 0.

In one particular basis, the coefficient vectors can be written explicitly
as:

v0 = {
-6 j + 2 j^3,
-3 j + 28 j^3 - 27 j^5 + 9 j^7 - j^9,
-3 j + j^3,
1 - 9 j^2 + 6 j^4 - j^6,
9 j^2 - 6 j^4 + j^6,
 -1
}

v1 = {
-1 + 18 j^2 - 12 j^4 + 2 j^6,
 9 j^2 - 87 j^4 + 109 j^6 - 54 j^8 + 12 j^10 - j^12,
1 + 9 j^2 - 6 j^4 + j^6,
-3 j + 28 j^3 - 27 j^5 + 9 j^7 - j^9,
9 j - 57 j^3 + 54 j^5 - 18 j^7 + 2 j^9,
6 j - 2 j^3
}

v2 = {
9 j - 57 j^3 + 54 j^5 - 18 j^7 + 2 j^9,
3 j - 55 j^3 + 297 j^5 - 423 j^7 + 272 j^9 - 90 j^11 + 15 j^13 - j^15,
-27 j^3 + 27 j^5 - 9 j^7 + j^9,
-1 + 18 j^2 - 93 j^4 + 110 j^6 - 54 j^8 + 12 j^10 - j^12,
-36 j^2 + 186 j^4 - 220 j^6 + 108 j^8 - 24 j^10 + 2 j^12,
1 - 18 j^2 + 12 j^4 - 2 j^6
}

It's easy to check that these vectors satisfy the zero-sum.

Consequently, Z_k =  X_k*Y_k implies Z_{k+1} =  X_{k+1}*Y_{k+1}.
===========================================

With this proof, verification of a(T_3(j)) = -1 only requires us to look
through a few of the initial conditions. This technique should generalize
to higher order Chebyshev cases. Comparing with the earlier table,
generalization will require the definition of more c-coefficients as
polynomial functions of index j>2.

Cheers,

Brad