Home » publications » Preprint “Special functions of hypercomplex variable on the lattice based on SU(1,1)”

Preprint “Special functions of hypercomplex variable on the lattice based on SU(1,1)”

Based on the representation of a set of canonical operators on the lattice h\mathbb{Z}^n, which are Clifford-vector-valued, we will introduce new families of special functions of hypercomplex variable possessing \mathfrak{su}(1,1) symmetries. The Fourier decomposition of the space of Clifford-vector-valued polynomials with respect to the SO(n)\times \mathfrak{su}(1,1)– module gives rise to the construction of new families of polynomial sequences as eigenfunctions of a coupled system involving forward/backward discretizations E_h^{\pm} of the Euler operator E=\sum_{j=1}^nx_j \partial_{x_j}.

Moreover, the interpretation of the one-parameter representation \mathbb{E}_h(t)=\exp(tE_h^--tE_h^+) of the Lie group SU(1,1) as a semigroup \left(\mathbb{E}_h(t)\right)_{t\geq 0} will allows us to describe the polynomial solutions of a homogeneous Cauchy problem on [0,\infty)\times h\mathbb{Z}^n involving the differencial-difference operator \partial_t+E_h^+-E_h^-.

arXiv preprint: http://arxiv.org/abs/1304.7191v2

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2 Comments

  1. David Horgan says:

    Hi Nelson, really like your blog. I’m doing research on discrete spacetimes – that’s how I found your blog. Alot of the stuff here looks useful and interesting – I can see I’m going to need to review the whole blog. Please feel free to visit my research blog at http://quantumtetrahedron.wordpress.com/
    – any feedback or comments are welcome.

    • nelson says:

      Hi David,

      Many thanks for your comment. Currently I am start working on Feynman diagrams in interrelationship with quantization of discrete spacetimes. Maybe we must keep in touch by e-mail.
      Soon,
      Nelson

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