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An lnfinite Wronskian Matrix, Binomial Sheffer Polynomials, and the Lagrange Reversion Theorem
Form the infinite Wronskian matrix with elements . A generating function for this matrix is with . If , then also , where is a binomial Sheffer sequence of polynomials. Then in this particular case, and so is the product … Continue reading →
The KervaireMilnor Formula
The KM formula and its ingredients are presented in 1) Bernoulli numbers and the unity of mathematics by Barry Mazur, p.14, Secs. 4, 5, and 6 2) Differential topology fortysix years later by Milnor 3) Homotopy group of spheres Wikipedia … Continue reading →
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Tagged Alternating permutations, Appell sequences, Bernoulli numbers, cycle graph, Dirichlet eta function, Euler numbers, Exotic sphere, exponential generating function, FermiDirac distribution, homotopy group, hypersphere, integer sequences, Kervaire Milnor formula, nsphere, parallelizable manifold, Riemann zeta function

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Mellin Interpolation of Differential Ops and Associated Infinigens and Appell Polynomials: The Ordered, Laguerre, and ScherkWittLie Diff Ops
Interpolations of the derivative operator the fundamental ordered op the Laguerre op the shifted Laguerre op and the generalized ScherkWitt Lie ops to the fractional operators and are consistently achieved using the Mellin transform of the negated e.g.f.s of the … Continue reading →
Posted in Math

Tagged Associated Laguerre polynomials, Compositional inverse, Differential operator interpolation, exponential generating function, Falling factorials, Infinigens, Infinitesimal generators, Integral curves, Inverse Mellin transform, Lie Witt algebra, Mellin transform interpolation, Rising factorials, Vector fields

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The Refined Lives of the F and HVectors of Associahedra
The compositional inversions noted in the Oct. 9th entry “Flipping Functions with Permutohedra” have counterparts with respect to the associahedra, or Stasheff polytopes. The hpolynomials of the simplicial complexes dual to the associahedra (see the Narayana number triangle OEISA001263) can … Continue reading →