Appendix 2: Some useful special functions

Abstract Inequality theory provides a significant mechanism for managing symmetrical aspects in real-life circumstances. The renowned distinguishing feature of integral inequalities and fractional calculus has a solid possibility to regulate continuous issues with high proficiency. This manuscript contributes to a captivating association of fractional calculus, special functions and convex functions. The authors develop a novel approach for investigating a new class of convex functions which is known as an n-polynomial $\mathcal{P}$ P -convex function. Meanwhile, considering two identities via generalized fractional integrals, provide several generalizations of the Hermite–Hadamard and Ostrowski type inequalities by employing the better approaches of Hölder and power-mean inequalities. By this new strategy, using the concept of n-polynomial $\mathcal{P}$ P -convexity we can evaluate several other classes of n-polynomial harmonically convex, n-polynomial convex, classical harmonically convex and classical convex functions as particular cases. In order to investigate the efficiency and supremacy of the suggested scheme regarding the fractional calculus, special functions and n-polynomial $\mathcal{P}$ P -convexity, we present two applications for the modified Bessel function and $\mathfrak{q}$ q -digamma function. Finally, these outcomes can evaluate the possible symmetric roles of the criterion that express the real phenomena of the problem.

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Gibbs partitions, Riemann–Liouville fractional operators, Mittag–Leffler functions, and fragmentations derived from stable subordinators

Journal of Applied Probability ◽

10.1017/jpr.2020.93 ◽

2021 ◽

Vol 58 (2) ◽

pp. 314-334

Author(s):

Man-Wai Ho ◽

Lancelot F. James ◽

John W. Lau

Keyword(s):

Special Functions ◽

Dirichlet Distribution ◽

Random Trees ◽

Fractional Integrals ◽

Biased Sampling ◽

Scaling Limits ◽

Fractional Operators ◽

Random Partitions ◽

Two Parameter ◽

Potential Use

AbstractPitman (2003), and subsequently Gnedin and Pitman (2006), showed that a large class of random partitions of the integers derived from a stable subordinator of index $\alpha\in(0,1)$ have infinite Gibbs (product) structure as a characterizing feature. The most notable case are random partitions derived from the two-parameter Poisson–Dirichlet distribution, $\textrm{PD}(\alpha,\theta)$, whose corresponding $\alpha$-diversity/local time have generalized Mittag–Leffler distributions, denoted by $\textrm{ML}(\alpha,\theta)$. Our aim in this work is to provide indications on the utility of the wider class of Gibbs partitions as it relates to a study of Riemann–Liouville fractional integrals and size-biased sampling, and in decompositions of special functions, and its potential use in the understanding of various constructions of more exotic processes. We provide characterizations of general laws associated with nested families of $\textrm{PD}(\alpha,\theta)$ mass partitions that are constructed from fragmentation operations described in Dong et al. (2014). These operations are known to be related in distribution to various constructions of discrete random trees/graphs in [n], and their scaling limits. A centerpiece of our work is results related to Mittag–Leffler functions, which play a key role in fractional calculus and are otherwise Laplace transforms of the $\textrm{ML}(\alpha,\theta)$ variables. Notably, this leads to an interpretation within the context of $\textrm{PD}(\alpha,\theta)$ laws conditioned on Poisson point process counts over intervals of scaled lengths of the $\alpha$-diversity.

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

Mathematics ◽

10.3390/math9090984 ◽

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.

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A Korn-type inequality in SBD for functions with small jump sets

Mathematical Models and Methods in Applied Sciences ◽

10.1142/s021820251750049x ◽

2017 ◽

Vol 27 (13) ◽

pp. 2461-2484 ◽

Cited By ~ 10

Author(s):

Manuel Friedrich

Keyword(s):

Bounded Variation ◽

Elastic Strain ◽

Special Functions ◽

Type Inequality ◽

Rigid Motion ◽

Exceptional Set ◽

Finite Perimeter ◽

Functions Of Bounded Deformation ◽

Jump Set

We present a Korn-type inequality in a planar setting for special functions of bounded deformation. We prove that for each function in [Formula: see text] with a sufficiently small jump set the distance of the function and its derivative from an infinitesimal rigid motion can be controlled in terms of the linearized elastic strain outside of a small exceptional set of finite perimeter. Particularly, the result shows that each function in [Formula: see text] has bounded variation away from an arbitrarily small part of the domain.

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Crystallization of Random Matrix Orbits

International Mathematics Research Notices ◽

10.1093/imrn/rny052 ◽

2018 ◽

Vol 2020 (3) ◽

pp. 883-913 ◽

Cited By ~ 1

Author(s):

Vadim Gorin ◽

Adam W Marcus

Keyword(s):

Random Matrix ◽

Special Functions ◽

Free Field ◽

Gaussian Free Field ◽

Complex Quaternion ◽

Cutting Out ◽

Multiplicative Convolution ◽

Global Asymptotics ◽

Associated Special Functions ◽

Derivatives Of

Abstract Three operations on eigenvalues of real/complex/quaternion (corresponding to $\beta =1,2,4$) matrices, obtained from cutting out principal corners, adding, and multiplying matrices, can be extrapolated to general values of $\beta>0$ through associated special functions. We show that the $\beta \to \infty $ limit for these operations leads to the finite free projection, additive convolution, and multiplicative convolution, respectively. The limit is the most transparent for cutting out the corners, where the joint distribution of the eigenvalues of principal corners of a uniformly-random general $\beta $ self-adjoint matrix with fixed eigenvalues is known as the $\beta $-corners process. We show that as $\beta \to \infty $ these eigenvalues crystallize on an irregular lattice consisting of the roots of derivatives of a single polynomial. In the second order, we observe a version of the discrete Gaussian Free Field put on top of this lattice, which provides a new explanation as to why the (continuous) Gaussian Free Field governs the global asymptotics of random matrix ensembles.

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