A remark on the distribution of products of independent normal random variables

We present a proof of the explicit formula of the probability density function of the product of normally distributed independent random variables using the multiplicative convolution formula for Meijer G functions.

Weighted Moving Averages for a Series of Fuzzy Numbers Based on Nonadditive Measures with σ − λ Rules and Choquet Integral of Fuzzy-Number-Valued Function

The aim of this study is to generalize moving average by means of Choquet integral. First, by employing nonadditive measures with δ − λ rules, the calculation of the moving average for a series of fuzzy numbers can be transformed into Choquet integration of fuzzy-number-valued function under discrete case. Meanwhile, the Choquet integral of fuzzy number and Choquet integral of fuzzy number vector are defined. Finally, some properties are investigated by means of convolution formula of Choquet integral. It shows that the results obtained in this paper extend the previous conclusions.

Spectrality of Moran measures with finite arithmetic digit sets

Let [Formula: see text] be a prime and [Formula: see text] be a sequence of finite arithmetic digit sets in [Formula: see text] with [Formula: see text] uniformly bounded, and let [Formula: see text] be the discrete probability measure on the finite set [Formula: see text] with equal distribution. For [Formula: see text], the infinite Bernoulli convolution [Formula: see text] converges weakly to a Borel probability measure (Moran measure). In this paper, we study the existence of exponential orthonormal basis for [Formula: see text].

Determining cuspforms from critical values of convolution L-functions and Rankin–Cohen brackets

We show that a holomorphic newform [Formula: see text] that is a Hecke eigenfunction can be uniquely determined by certain noncentral critical values of a family of convolution [Formula: see text]-functions [Formula: see text]. The methods are different and simpler than those that use the central value to determine cuspforms.

Some Convolution Formulae Related to the Second-Order Linear Recurrence Sequence

The main aim of this paper is that for any second-order linear recurrence sequence, the generating function of which is f ( t ) = 1 1 + a t + b t 2 , we can give the exact coefficient expression of the power series expansion of f x ( t ) for x ∈ R with elementary methods and symmetry properties. On the other hand, if we take some special values for a and b, not only can we obtain the convolution formula of some important polynomials, but also we can establish the relationship between polynomials and themselves. For example, we can find relationship between the Chebyshev polynomials and Legendre polynomials.

Identities for Dirichlet and Lambert-type series arising from the numbers of a certain special word

The goal of this paper is to give several new Dirichlet-type series associated with the Riemann zeta function, the polylogarithm function, and also the numbers of necklaces and Lyndon words. By applying Dirichlet convolution formula to number-theoretic functions related to these series, various novel identities and relations are derived. Moreover, some new formulas related to Bernoulli-type numbers and polynomials obtain from generating functions and these Dirichlet-type series. Finally, several relations among the Fourier expansion of Eisenstein series, the Lambert series and the number-theoretic functions are given.