SIGMA 9 (2013), 065, 18 pages      arXiv:1304.7191      https://doi.org/10.3842/SIGMA.2013.065

Special Functions of Hypercomplex Variable on the Lattice Based on SU(1,1)

Nelson Faustino
Departamento de Matemática Aplicada, IMECC-Unicamp, CEP 13083-859, Campinas, SP, Brasil

Received May 06, 2013, in final form October 28, 2013; Published online November 05, 2013

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

Key words: Clifford algebras; finite difference operators; Lie algebras.

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References

1. Cartier P., Mathemagics (a tribute to L. Euler and R. Feynman), in Noise, Oscillators and Algebraic Randomness (Chapelle des Bois, 1999), Lecture Notes in Phys., Vol. 550, Springer, Berlin, 2000, 6-67.
2. De Ridder H., De Schepper H., Kähler U., Sommen F., Discrete function theory based on skew Weyl relations, Proc. Amer. Math. Soc. 138 (2010), 3241-3256.
3. De Ridder H., De Schepper H., Sommen F., Fueter polynomials in discrete Clifford analysis, Math. Z. 272 (2012), 253-268.
4. Delanghe R., Sommen F., Soucek V., Clifford algebra and spinor-valued functions. A function theory for the Dirac operator, Mathematics and its Applications, Vol. 53, Kluwer Academic Publishers Group, Dordrecht, 1992.
5. Di Bucchianico A., Loeb D.E., Rota G.C., Umbral calculus in Hilbert space, in Mathematical Essays in Honor of Gian-Carlo Rota (Cambridge, MA, 1996), Progr. Math., Vol. 161, Birkhäuser Boston, Boston, MA, 1998, 213-238.
6. Dimakis A., Müller-Hoissen F., Striker T., Umbral calculus, discretization, and quantum mechanics on a lattice, J. Phys. A: Math. Gen. 29 (1996), 6861-6876, quant-ph/9509014.
7. Eelbode D., Monogenic Appell sets as representations of the Heisenberg algebra, Adv. Appl. Clifford Algebr. 22 (2012), 1009-1023.
8. Faustino N., Kähler U., Fischer decomposition for difference Dirac operators, Adv. Appl. Clifford Algebr. 17 (2007), 37-58, math.CV/0609823.
9. Faustino N., Ren G., (Discrete) Almansi type decompositions: an umbral calculus framework based on ${\mathfrak{osp}}(1|2)$ symmetries, Math. Methods Appl. Sci. 34 (2011), 1961-1979, arXiv:1102.5434.
10. Goodman R., Wallach N.R., Representations and invariants of the classical groups, Encyclopedia of Mathematics and its Applications, Vol. 68, Cambridge University Press, Cambridge, 1998.
11. Howe R., Remarks on classical invariant theory, Trans. Amer. Math. Soc. 313 (1989), 539-570.
12. Levi D., Tempesta P., Winternitz P., Umbral calculus, difference equations and the discrete Schrödinger equation, J. Math. Phys. 45 (2004), 4077-4105, nlin.SI/0305047.
13. Malonek H.R., Tomaz G., Bernoulli polynomials and Pascal matrices in the context of Clifford analysis, Discrete Appl. Math. 157 (2009), 838-847.
14. Odake S., Sasaki R., Discrete quantum mechanics, J. Phys. A: Math. Theor. 44 (2011), 353001, 47 pages, arXiv:1104.0473.
15. Sommen F., An algebra of abstract vector variables, Portugal. Math. 54 (1997), 287-310.
16. Tempesta P., On Appell sequences of polynomials of Bernoulli and Euler type, J. Math. Anal. Appl. 341 (2008), 1295-1310.
17. Vilenkin N.J., Klimyk A.U., Representation of Lie groups and special functions. Vol. 1. Simplest Lie groups, special functions and integral transforms, Mathematics and its Applications (Soviet Series), Vol. 72, Kluwer Academic Publishers Group, Dordrecht, 1991.