On the Laplace integral representation of multivariate Mittag-Leffler functions in anomalous relaxation

2016 ◽  
Vol 39 (11) ◽  
pp. 2983-2992 ◽  
Author(s):  
Raoul R. Nigmatullin ◽  
Airat A. Khamzin ◽  
Dumitru Baleanu
Keyword(s):  
Integral Representation ◽  
Laplace Integral ◽  
Anomalous Relaxation ◽  
Laplace Integral Representation

POSITIVITY OF THE INTERTWINING OPERATOR AND HARMONIC ANALYSIS ASSOCIATED WITH THE JACOBI-DUNKL OPERATOR ON ${\mathbb R}$

2003 ◽  
Vol 01 (04) ◽  
pp. 387-412 ◽  
Author(s):  
F. CHOUCHANE ◽  
M. MILI ◽  
K. TRIMÈCHE
Keyword(s):  
Integral Representation ◽  
Harmonic Analysis ◽  
Difference Operator ◽  
Convolution Product ◽  
Dunkl Transform ◽  
Dunkl Operator ◽  
Intertwining Operator ◽  
Laplace Integral ◽  
Translation Operators ◽  
Laplace Integral Representation

We consider a differential-difference operator Λα,β, [Formula: see text], [Formula: see text] on [Formula: see text]. The eigenfunction of this operator equal to 1 at zero is called the Jacobi–Dunkl kernel. We give a Laplace integral representation for this function and we prove that for [Formula: see text], [Formula: see text], the kernel of this integral representation is positive. This result permits us to prove that the Jacobi–Dunkl intertwining operator and its dual are positive. Next we study the harmonic analysis associated with the operator Λα,β (Jacobi–Dunkl transform, Jacobi–Dunkl translation operators, Jacobi–Dunkl convolution product, Paley–Wiener and Plancherel theorems…).

The generation of surface waves over a sloping beach by an oscillating line-source. I. The general solution

1974 ◽  
Vol 76 (3) ◽  
pp. 545-554 ◽  
Author(s):  
Clare A. N. Morris
Keyword(s):  
General Solution ◽  
Integral Representation ◽  
Surface Waves ◽  
Line Source ◽  
Wave Train ◽  
Wave Generation ◽  
Outgoing Wave ◽  
Arbitrary Angle ◽  
Laplace Integral Representation ◽  
Sloping Beach

AbstractThe problem of wave generation by a line source of sinusoidally varying strength situated in water above a beach of arbitrary angle α(0 < α ≤ π) is solved by the use of a Laplace-integral representation of the solution. It is shown that a solution can be constructed which is regular at the shoreline and gives an outgoing wave-train at infinity.

scholarly journals General fractional calculus operators containing the generalized Mittag-Leffler functions applied to anomalous relaxation

2017 ◽  
Vol 21 (suppl. 1) ◽  
pp. 317-326 ◽  
Author(s):  
Xiaojun Yang
Keyword(s):  
Fractional Calculus ◽  
Fractional Order ◽  
Integral Transforms ◽  
Kernel Functions ◽  
Order Derivative ◽  
Fractional Order Derivative ◽  
Laplace Integral ◽  
Fractional Calculus Operators ◽  
Anomalous Relaxation ◽  
First Time

In this paper, we address a family of the general fractional calculus operators of Wiman and Prabhakar types for the first time. The general Mittag-Leffler function to structure the kernel functions of the fractional order derivative operators and their Laplace integral transforms are considered in detail. The formulations are as the mathematical tools proposed to investigate the anomalous relaxation.

scholarly journals Fractional Generalizations of Rodrigues-Type Formulas for Laguerre Functions in Function Spaces

Mathematics ◽  
2021 ◽  
Vol 9 (9) ◽  
pp. 984
Author(s):  
Pedro J. Miana ◽  
Natalia Romero
Keyword(s):  
Integral Representation ◽  
Special Functions ◽  
Laguerre Polynomials ◽  
Function Spaces ◽  
Laguerre Functions ◽  
Addition Formula ◽  
Lebesgue Spaces ◽  
Rodrigues Formula ◽  
Generalized Laguerre Polynomials ◽  
New Family

Generalized Laguerre polynomials, Ln(α), verify the well-known Rodrigues’ formula. Using Weyl and Riemann–Liouville fractional calculi, we present several fractional generalizations of Rodrigues’ formula for generalized Laguerre functions and polynomials. As a consequence, we give a new addition formula and an integral representation for these polynomials. Finally, we introduce a new family of fractional Lebesgue spaces and show that some of these special functions belong to them.

scholarly journals On a class of analytic functions related to Robertson’s formula and subordination

2021 ◽  
Vol 27 (1) ◽  
Author(s):  
Adam Lecko ◽  
Gangadharan Murugusundaramoorthy ◽  
Srikandan Sivasubramanian
Keyword(s):  
Integral Representation ◽  
Analytic Functions ◽  
Boundary Point ◽  
Unit Disc ◽  
Starlike Functions ◽  
Analytic Formula ◽  
Growth Theorem

AbstractIn this paper, we define and study a class of analytic functions in the unit disc by modification of the well-known Robertson’s analytic formula for starlike functions with respect to a boundary point combined with subordination. An integral representation and growth theorem are proved. Early coefficients and the Fekete–Szegö functional are also estimated.

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