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On a Relation Between Classical and Free Infinitely Divisible Transforms

Jurek Zbigniew J.

Probability and Mathematical Statistics

2020

Vol. 40, Fasc. 2

349--367

We study two ways (two levels) of finding free-probability analogues of classical infinitely divisible measures. More precisely, we identify their Voiculescu transforms on the imaginary axis. For free-selfdecomposable measures we find a formula (a differential equation) for their background driving transforms. It is different from the one known for classical selfdecomposable measures. We illustrate our methods on hyperbolic characteristic functions. Our approach may produce new formulas for definite integrals of some special functions.

Approximation by Stancu-Chlodowsky type lambda-Bernstein operators

Mursaleen M., Al-Abied A. A. H., Salman M. A.

Journal of Applied Analysis

2020

Vol. 26, nr 1

97--110

In this paper,we give some approximation properties by Stancu-Chlodowsky type λ-Bernstein operators in the polynomial weighted space and obtain the convergence properties of these operators by using Korovkin’s theorem. We also establish the direct result and the Voronovskaja type asymptotic formula.

On the Euler function on differences between the coordinates of points on modular hyperbolas

Shparlinski I. E.

Bulletin of the Polish Academy of Sciences. Mathematics

2008

Vol. 56, no 1

1-7

For a prime p > 2, an integer a with gcd(a,p) = 1 and real 1 ≤ X,Y < p, we consider the set of points on the modular hyperbola Ηa,p(X,Y) = {(x,y) : x,y ≡ a (mod p), 1 ≤ x ≤ X, 1 ≤ y ≤ Y}. We give asymptotic formulas for the average values (x,y)∈ ... [wzór] with the Euler function φ(k) on the difference between the components of points of Ηa,p(X,Y).

Fibonacci numbers with the Lehmer property

Luca F.

Bulletin of the Polish Academy of Sciences. Mathematics

2007

Vol. 55, no 1

7-15

We show that if m > 1 is a Fibonacci number such that φ(m) | m - 1, where φ is the Euler function, then m is prime.

Average value of the Euler function on binary palindromes

Banks W. D., Shparlinski I. E.

Bulletin of the Polish Academy of Sciences. Mathematics

2006

Vol. 54, no 2

95-101

We study values of the Euler function φ(n) taken on binary palindromes of even length. In particular, if B2l denotes the set of binary palindromes with precisely 2l binary digits, we derive an asymptotic formula for the average value of the Euler function on B2l.