By Shai M. J. Haran

In this quantity the writer extra develops his philosophy of quantum interpolation among the genuine numbers and the p-adic numbers. The *p*-adic numbers include the *p*-adic integers *Z _{p}*which are the inverse restrict of the finite earrings

*Z/p*. this provides upward push to a tree, and chance measures w on

^{n}*Z*correspond to Markov chains in this tree. From the tree constitution one obtains distinct foundation for the Hilbert house

_{p}*L*(

_{2}*Z*). the true analogue of the

_{p},w*p*-adic integers is the period [-1,1], and a chance degree w on it offers upward push to a unique foundation for

*L*([-1,1],

_{2}*w*) - the orthogonal polynomials, and to a Markov chain on "finite approximations" of [-1,1]. For specific (gamma and beta) measures there's a "quantum" or "

*q*-analogue" Markov chain, and a different foundation, that inside of yes limits yield the genuine and the p-adic theories. this concept might be generalized variously. In illustration idea, it's the quantum common linear staff

*GL*(

_{n}*q*)that interpolates among the p-adic staff

*GL*(

_{n}*Z*), and among its actual (and complicated) analogue -the orthogonal

_{p}*O*(and unitary

_{n}*U*)groups. there's a comparable quantum interpolation among the genuine and p-adic Fourier rework and among the true and p-adic (local unramified a part of) Tate thesis, and Weil particular sums.

_{n}

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**Additional info for Arithmetical Investigations: Representation Theory, Orthogonal Polynomials, and Quantum Interpolations**

**Example text**

Our measure τpα,β is not invariant 0∗ under Z∗p Zp , but we can project it down to P1 (Qp )/Z∗p Zp , we denote the image (α)β measure by τp . Note that it is not symmetric in the two parameters α and β. Then we have a Markov chain on the tree P1 (Z/pn )/(Z/pn )∗ (Z/pn ), it is called P GL2 (Zp ) whose element are of the form n≥0 the p-adic β-chain. 1 Zp /Z∗p Every p-adic integer can be written as a power series in p and such a representation is unique. Namely we have Zp = lim Z/(pn ) = a0 + a1 p + a2 p2 + · · · 0 ≤ aj < p (j ≥ 0) .

This shows that the boundary is not totally disconnected, hence this is a non-tree (for any tree, the boundary is always totally disconnected). In this section we study the Markov chain on non-trees, which can have continuous boundary. 2 Harmonic Functions Let X = n≥0 Xn , X0 = {x0 } and Xn be a ﬁnite set for all n ≥ 0. We call X the state space. Let P : n≥0 Xn × Xn+1 → [0, 1] be a transition probability, that is, P satisﬁes P (x, x ) = 1 (x ∈ Xn ). 6) x ∈Xn+1 Then we says that we have a Markov chain.

2 Markov Chain on Non-Trees 41 Similarly we obtain the orthogonal decomposition of HN and H; β HN : = Hp(N ) = Cϕβp(N ),m , 0≤m≤N H := HZβp CϕβZp ,m , = m≥0 where ϕβp(N ),m (resp. ϕβZp ,m ) is the basis of HN (resp. H) deﬁned by ϕβp(N ),0 = 1, (1 − p−β )pβ if 0 < i ≤ N, −1 if i = 0, ⎧ −β βm ⎪ if m − 1 < i ≤ N , ⎨(1 − p )p β ϕp(N ),m (i, j) = −pβ(m−1) if i = m − 1, ⎪ ⎩ 0 if 0 ≤ i < m − 1, ϕβp(N ),1 (i, j) = (m ≥ 2). and ϕβZp ,0 = φZp , ϕβZp ,m = pβm φpm Zp − pβ(m−1) φpm−1 Zp (m ≥ 1). We call ϕβZp ,m the p-Laguerre basis, it is the analogue of the Laguerre polynomial.