Arithmetical Investigations : Representation Theory Orthogonal Polynomials and Quantum Interpolations Introduction: Motivations from Geometry Gamma and Beta Measures Markov Chains Real Beta Chain and q Interpolation Ladder Structure q Interpolation of Local Tate Thesis Pure Basis and Semi Group Higher Dimensional Theory Real Grassmann Manifold p Adic Grassmann Manifold q Grassmann Manifold Quantum Group Uq(su(1 1)) and the q Hahn Basis. In this volume the author further develops his philosophy of quantum interpolation between the real numbers and the p adic numbers. The p adic numbers contain the p adic integers Zp which are the inverse limit of the finite rings Z pn. This gives rise to a tree and probability measures w on Zp correspond to Markov chains on this tree. From the tree structure one obtains special basis for the Hilbert space L2(Zp w). The real analogue of the p adic integers is the interval [ 1 1] and a probability measure w on it gives rise to a special basis for L2([ 1 1] w) the orthogonal polynomials and to a Markov chain on finite approximations of [ 1 1]. For special (gamma and beta) measures there is a quantum or q analogue Markov chain and a special basis that within certain limits yield the real and the p adic theories. This idea can be generalized variously. In representation theory it is the quantum general linear group GLn(q)that interpolates between the p adic group GLn(Zp) and between its real (and complex) analogue the orthogonal On (and unitary Un )groups. There is a similar quantum interpolation between the real and p adic Fourier transform and between the real and p adic (local unramified part of) Tate thesis and Weil explicit sums.