Infinitesimal Generators, the Pascal Pyramid, and the Witt and Virasoro Algebras: A short set of notes sketching some relationships among infinitesimal generators represented as differential operators and infinite-dimensional matrices, the Pascal triangle / pyramid, conformal transformations, the Witt and Virasoro algebras, the Hermite and generalized Laguerre polynomials of order , the Dedekind eta function, and combinatorics of the underlying integer matrices, inspired by Alexander Givental’s presentation of Virasoro operators in “Gromov-Witten invariants and quantization of quadratic Hamiltonians.” Some potential connections to knots, flows, dynamics, and Poincare-Maass series are also mentioned.
(Revised Dec. 11. 2013.)
Added 2/2014: For some discussion of the relations between the formal group laws and local Lie groups introduced in this paper, see Applications of Lie Groups to Differential Equations (2nd Ed., pg. 18) by Peter Olver.
(8/2016) The Fourier and Laplace transforms are associated with the derivative ; the Mellin transform, with ; and the Meijer transform, with (see Prudnikov, “On the continuation of the ideas of Heaviside and Mikusinki in operational calculus”).
(7/2018) Pg. 3 contains the infinitesimal generator related to general Moebius transformations that also occurs on pg. 24 of “Modern Developments in the Theory and Applications of Moving Frames” by Olver.
On page 1 in Equation 6, .