Reciprocal series

Definitions

  • Chambers's Twentieth Century Dictionary
    • Reciprocal series a series each of whose terms is the reciprocal of the corresponding term of another series
    • ***

Etymology

Chambers's Twentieth Century Dictionary
L.,—serĕre, sertum, to join.

Usage


In science:

In particular, for each fixed λ 6= 0, lim supn→∞ (mn (λ))1/n is the reciprocal of the radius of convergence of the series expansion of eG(z , λ) at z = 0.
Large deviations for the leaves in some random trees
Variations on computing reciprocals of power series.
A simple and fast algorithm for computing exponentials of power series
E (~r , ω ), where the crystal’s spatially periodic dielectric permittivity can be represented as the Fourier series: ε(~r, ω ) = P ετ (ω )ei~τ r , ~τ is the reciprocal lattice vector.
Spontaneous and Induced Radiation by Relativistic Particles in Natural and Photonic Crystals. Crystal X-ray Lasers and Volume Free Electron Lasers (VFEL)
We show that the characteristic series for the greedy normal form of a Coxeter group is always a rational series, and prove a reciprocity formula for this series when the group is right-angled and the nerve is Eulerian.
Reciprocity and rationality for the greedy normal form of a Coxeter group
As corollaries we obtain many of the known rationality and reciprocity results for the growth series of Coxeter groups as well as some new ones.
Reciprocity and rationality for the greedy normal form of a Coxeter group
The ℓ2 -Euler characteristic and the reciprocity formulas above also have natural interpretations in terms of this multivariable series γ (t) (see [8, 9]).
Reciprocity and rationality for the greedy normal form of a Coxeter group
To make sense of reciprocity in the context of multivariate noncommutative power series, we work over the ring of formal Laurent series.
Reciprocity and rationality for the greedy normal form of a Coxeter group
In fact, in the generality we work, we also obtain a new reciprocity formula for the complete growth series of a right-angled Coxeter group.
Reciprocity and rationality for the greedy normal form of a Coxeter group
Theorem 2 fails to hold in general for non right-angled Coxeter groups with Eulerian nerve (even though the usual growth series does satisfy reciprocity ).
Reciprocity and rationality for the greedy normal form of a Coxeter group
Our reciprocity formulas require working with formal inverses of generators, so we work over the ring of formal noncommutative Laurent series.
Reciprocity and rationality for the greedy normal form of a Coxeter group
Finally, in Section 9, we apply our main theorems to obtain rationality and reciprocity results for growth series of Coxeter groups relative to both the standard generating set S and the larger generating set A = {wσ | σ ∈ A}.
Reciprocity and rationality for the greedy normal form of a Coxeter group
Since the series λ∗ (A, Q, B ) does not depend on the representation, we denote it simply by λ∗ and call it the reciprocal of λ.
Reciprocity and rationality for the greedy normal form of a Coxeter group
But the matrix J seems to be new, and not only yields new proofs of some of these known formulas, but also provides the key to reciprocity formulas for noncommutative power series.
Reciprocity and rationality for the greedy normal form of a Coxeter group
For an even simpler illustration of reciprocity, we can use the series χ(x, y ) obtained by the substitution σi = x, σij = y .
Reciprocity and rationality for the greedy normal form of a Coxeter group
On the other hand, the characteristic series χ for the greedy normal form does not satisfy the reciprocity formula χ∗ = χ.
Reciprocity and rationality for the greedy normal form of a Coxeter group
***