The concept of number: from quaternions to monads and by Benno Artmann

By Benno Artmann

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QZ+ql-Z)(q+1)-1, and let mr=X'(Q) be finitely additive the measure such that f(x) = f PZ(x,w) dm(w. Then m(S2)=f(o). if x' is the vertex at distance 1 Moreover, from x in the chain [o,x], then m(S2(x)) = (qZ-q Z)-iq Zd(o>x')(f(x) - gZf(x' )). PROOF. The fact that m(f2)=f(o) follows directly from the fact that P(o,x)=1. Observe that PZ(x,w)=q ZPZ(x',w) if woS2(x), and PZ(x,w)=gzPZ(x',w) if wecZ(x). Therefore f(x)= I PZ(x,w)dm= 2 qZ PZ(x,w)dm + S2J\(x) PZ(x',w)dm + = L"WX) J(x ) PZ(x,w)dm c2 W qZ PZ(',w)dm.

Therefore f(x)= I PZ(x,w)dm= 2 qZ PZ(x,w)dm + S2J\(x) PZ(x',w)dm + = L"WX) J(x ) PZ(x,w)dm c2 W qZ PZ(',w)dm. S2(x) On the other hand PZ(x',w) is equal to gZd(X',o) for every weS2(x). , As before let oe3E be a fixed vertex. 2. Spherical functions. We shall call a function defined on X radial (with respect to the vertex o) vertex o. if That it depends only on the distance from the f is, radial is if d(o,x)=d(o,y) implies f(x)=f(y). We study in this section the radial eigenfunctions of L. 2), we write again for simplicity of notation P(o,x,w)=P(x,w).

Of ¢z A radial function is then a function of lxl alone. Since lxl takes only integral values, function on we may think of a spherical function as a nonnegative the Oz(n)=Oz(x), when lxl=n. elements having distance integers. We therefore write Observe that the value of ¢Z on the one from o exactly is µ(z). By definition qz(o)=1. 3) LEMMA. If 0 z by induction. is a spherical function, then, for n>--1, z z (n+1) = q+1 q 0 (1)¢ (n) z z q1 z (n-1). Spherical functions 43 If lxl=n, the equality Loz()=µ()oz(x) implies PROOF.

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