A double inversion formula

A Fourier Inversion Formula for Evolutionary Trees

We establish a pair of identities, which will provide a useful tool in the reconstruction of evolutionary trees in Kimura's 3-parameter model. The starting point of this paper was an attempt for a better understanding and generalization of an Hadamard inverse pair of formulae, which was used in statistics by Cooper [l], in image processing by Andrews [2, Chapters 6,7], and in information theory...

Inversion Formula for Continuous Multifractals

In a previous paper MR the authors introduced the inverse measure y of a probability measure on It was argued that the respective multifractal spectra are linked by the inversion formula fy f Here the statements of MR are put in more mathematical terms and proofs are given for the inversion formula in the case of continuous measures Thereby f may stand for the Hausdor spectrum the packing spect...

The Lagrange Inversion Formula and Divisibility Properties

Wilf stated that the Lagrange inversion formula (LIF) is a remarkable tool for solving certain kinds of functional equations, and at its best it can give explicit formulas where other approaches run into stone walls. Here we present the LIF combinatorially in the form of lattice paths, and apply it to the divisibility property of the coefficients of a formal power series expansion. For the LIF,...

Multiplicative Functions and Möbius Inversion Formula

Examples. Let n ∈ N. Define functions τ, σ, π : N → N as follows: • τ(n) = the number of all natural divisors of n = #{d > 0 | d|n}; • σ(n) = the sum of all natural divisors of n = ∑ d|nd; • π(n) = the product of all natural divisors of n = ∏ d|nd. As we shall see below, τ and σ are multiplicative functions, while π is not. From now on we shall write the prime decomposition of n ∈ N as n = p1 1...

The Planar Tree Lagrange Inversion Formula