At the Crossroads of Continuous and Discrete Hypercomplex Analysis: A Sturm-liouville Equation

The subject of our talk is the treatment of generalized Cauchy-Riemann equations, [1], by methods of hypercomplex analysis. 20 years ago, it led us to a special class of multivariate polynomials with coefficients that in the case of complex C-R equations are identical with rational numbers encountered in 1958 by L. Vietoris, see [2], in connection with the positivity of trigonometric sums.

Some elementary tools for passing from continuous to discrete and back to continuous structures in the sense of D. Knuth's CONCRETE ANALYSIS (see [3]) were sufficient to detect a Sturm-Liouville type ODE on the crossroads between continuous and discrete hypercomplex analysis.

As we will show, the unorthodox use of several hypercomplex variables, see [4], opens the eyes to those new insights in CONCRETE ANALYSIS. Very recently, 2019 and 2025, the obtained generalized Vietoris numbers have been detected as Wigner Numbers in quantum mechanics and in arithmetical structures appearing in quantum scattering problems.

[1] E. M. Stein, G. Weiss: Generalization of the Cauchy-Riemann Equations and Representations of the Rotation Group. Amer. J. of Mathematics 90 (1), 163 - 196 (1968)

[2] L. Vietoris: Über das Vorzeichen gewisser trigonometrischer Summen. Sitzungsber. Österr Akad. Wiss. 1958; 167:125-135.

[3] Graham RL, Knuth DE, Patashnik O. Concrete Mathematics: A Foundation for Computer Science. Reading, Mass: Addison-Wesley; 1994

[4] H. R. Malonek et al.: Harmonic Analysis and Hypercomplex Function Theory in Co-dimension One, in: Modern Methods in Operator Theory and Harmonic Analysis. OTHA 2018, eds. A. Karapetyants, V. Kravchenko, E. Liflyand, Springer, (2019), 93 - 115