The raising and lowering operators and for a sequence of functions , with and , defined by
have the commutator relation
with respect to action on the space spanned by this sequence of functions.
If for any particular natural number
Since this holds for , the relation holds for all natural numbers, and formally for a function analytic about the origin (or a formal power series or exponential generating function)
The reader should be able to modify the argument to show that also
“The many avatars of a simple algebra” by Coutinho
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Tagged Bernoulli numbers, Chebyshev polynomials, Combinatorics, Compositional inverse, Differential operators, Elliptic curves, Elliptic formal group law, Elliptic genera, Enumerative combinatorics, Euler numbers, Eulerian numbers, Evolution equations, Faber polynomials, Geodesics of Virasoro-Bott group, Grassmann polynomials, Grassmannian, Hyperbolic tangent, Infinigens, Infinitesimal generators, KdV, Korteweg-de Vries equation, L-genus, Lie algebra, Lucas (Cardan) polynomials, Refined Eulerian numbers, Riccati, Schwarzian derivative, SL2, Solitons, Swiss knife polynomials, Viscous Burgers equation
The Euler-Bernoulli numbers: what they count and associations to algebraic geometry, elliptic curves, and differential ops. Coming soon.