Franz Lehner message

Dear Doron,

sorry I was away and did not reply earlier.
A few more references came to my mind, because they are close
in spirit to your approach, which is can be seen as the method
of orthogonal polynomials.
There are a few papers by T.Pytlik on the algebra of radial functions,
the closest one in J reine angew Math 326 (1981) 124-135,
and
J M Cohen
Operator norms on free groups"
Boll. Un. Mat. Ital. B (6) vol 1 nr 3 (1982) 1055--1065

It is all written in group algebra language of the free groups,
whose Cayley graphs are the Cayley trees of even order.
But it can be adapted to arbitrary Cayley trees, if
one considers the free product of n copies of the cyclic group,
whose Cayley graph is the Cayley tree of order n.

They compute the polynomials P_n such that
$$
P_n(\sum x_i + x_i^{-1}) = \sum \text{all words of length $n$}
$$
and these polynomials satisfy the recursion you found in your paper.

Other trees are considered in the same spirit bey
P Gerl
Continued fraction methods for random walks on N and on trees
Springer Lecture Notes in Mathematics 1064 (1984) 131--146

> I should have guessed that something so simple (and natural) is
> as old as the hills,
well the tree is the popular model for many things,
because its recursive structure allows exact enumeration,
contrary to virtually all other infinite graphs.

best regards,
Franz