Fractional generalizations of Rodrigues-type formulas for Laguerre functions in function spaces

Generalized Laguerre polynomials, L(a) n, 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. © 2021 by the authors. Licensee MDPI, Basel, Switzerland

New Developments in Geometric Function Theory

The book contains papers published in a Special Issue of Axioms, entitled "New Developments in Geometric Function Theory". An Editorial describes the 14 papers devoted to the study of complex-valued functions which present new outcomes related to special classes of univalent and bi-univalent functions, new operators and special functions associated with differential subordination and superordination theories, fractional calculus, and certain applications in geometric function theory

On the Generalized Confluent Hypergeometric Function and Its Application

2000 Mathematics Subject Classification: 26A33, 33C20This paper is devoted to further development of important case of Wright’s hypergeometric function and its applications to the generalization of Γ-, B-, ψ-, ζ-, Volterra functions

Condensed vortex ground states of rotating Bose-Einstein condensate in harmonic atomic trap

We study a system of $N$ Bose atoms trapped by a symmetric harmonic potential, interacting via weak central forces. Considering the ground state of the rotating system as a function of the two conserved quantities, the total angular momentum and its collective component, we develop an algebraic approach to derive exact wave functions and energies of these ground states. We describe a broad class of the interactions for which these results are valid. This universality class is defined by simple integral condition on the potential. Most of the potentials of practical interest which have pronounced repulsive component belong to this universality class.Comment: 34 pages, 10 ps figures, minor revisions, to be publ. in Ann. Phy

Special functions arising from discrete Painlevé equations: A survey

AbstractThis article is a survey on recent studies on special solutions of the discrete Painlevé equations, especially on hypergeometric solutions of the q-Painlevé equations. The main part of this survey is based on the joint work [K. Kajiwara, T. Masuda, M. Noumi, Y. Ohta, Y. Yamada, Hypergeometric solutions to the q-Painlevé equations, IMRN 2004 47 (2004) 2497–2521, K. Kajiwara, T. Masuda, M. Noumi, Y. Ohta, Y. Yamada, Construction of hypergeometric solutions to the q-Painlevé equations, IMRN 2005 24 (2005) 1439–1463] with Kajiwara, Masuda, Ohta and Yamada. After recalling some basic facts concerning Painlevé equations for comparison, we give an overview of the present status of studies on difference (discrete) Painlevé equations as a source of special functions

On the entire functions from the Laguerre--P\'olya class having monotonic second quotients of Taylor coefficients

We investigate the famous Laguerre–Pólya class of entire functions and its subclass, the Laguerre–Pólya class of type I. The functions from these classes can be expressed in terms of the Hadamard Canonical Factorization (see Chapter 1, Definition 1.2 and 1.3). The prominent theorem by E. Laguerre and G. Pólya gives a complete description of the Laguerre–Pólya class and the Laguerre–Pólya class of type I, showing that these classes are the respective closures in the topology of uniform convergence on compact sets of the set of real polynomials having only real zeros (that is, the set of so-called hyperbolic polynomials) and the set of real polynomials having only real negative zeros. Both the Laguerre–Pólya class and the Laguerre–Pólya class of type I play an essential role in complex analysis. For the properties and characterizations of these classes, see, for example, [31] by A. Eremenko, [40] by I.I. Hirschman and D.V. Widder, [43] by S. Karlin, [57] by B.Ja. Levin, [66, Chapter 2] by N. Obreschkov, and [74] by G. Pólya and G. Szegö. In the thesis, we study entire functions with positive coefficients and with the monotonic sequence of their second quotients of Taylor coefficients. We find necessary and sufficient conditions under which such functions belong to the Laguerre–Pólya class (or the Laguerre–Pólya class of type I).:List of symbols Introduction 1 Background of research 1 1.1 The Laguerre–Pólya class .................... 1 1.2 The quotients of Taylor coefficients ............... 3 1.3 Hutchinson’s constant ...................... 4 1.4 Multiplier sequences ....................... 4 1.5 Apolar polynomials........................ 8 1.6 The partial theta function .................... 10 1.7 Decreasing second quotients ................... 13 1.8 Increasing second quotients ................... 14 2 A necessary condition for an entire function with the increasing second quotients of Taylor coefficients to belong to the Laguerre–Pólya class 15 2.1 Proof of Theorem 2.1....................... 16 2.2 The q-Kummer function ..................... 29 2.3 Proof of Theorem 2.10 ...................... 31 2.4 Proof of Theorem 2.11 ...................... 43 3 Closest to zero roots and the second quotients of Taylor coefficients of entire functions from the Laguerre–Pólya I class 49 3.1 Proof of Statement 3.1 ...................... 50 3.2 Proof of Theorem 3.2....................... 53 3.3 Proof of Theorem 3.4....................... 61 3.4 Proof of Theorem 3.6....................... 66 4 Entire functions from the Laguerre–Pólya I class having the increasing second quotients of Taylor coefficients 69 4.1 Proof of Theorem 4.1....................... 70 4.2 Proof of Theorem 4.3....................... 76 5 Number of real zeros of real entire functions with a non-decreasing sequence of the second quotients of Taylor coefficients 81 5.1 Proof of Theorem 5.1....................... 82 5.2 Proof of Corollary 5.2....................... 88 5.3 Proof of Theorem 5.4....................... 88 6 Further questions 95 Acknowledgements 97 Selbständigkeitserklärung 101 Curriculum Vitae 103 Bibliography 10

A Guide to Special Functions in Fractional Calculus

Dedicated to the memory of Professor Richard Askey (1933-2019) and to pay tribute to the Bateman Project. Harry Bateman planned his project (accomplished after his death as Higher Transcendental Functions, Vols. 1-3, 1953-1955, under the editorship by A. Erdelyi) as a "Guide to the Functions". This inspired the author to use the modified title of the present survey. Most of the standard (classical) Special Functions are representable in terms of the Meijer G-function and, specially, of the generalized hypergeometric functions pFq. These appeared as solutions of differential equations in mathematical physics and other applied sciences that are of integer order, usually of second order. However, recently, mathematical models of fractional order are preferred because they reflect more adequately the nature and various social events, and these needs attracted attention to "new" classes of special functions as their solutions, the so-called Special Functions of Fractional Calculus (SF of FC). Generally, under this notion, we have in mind the Fox H-functions, their most widely used cases of the Wright generalized hypergeometric functions pΨq and, in particular, the Mittag-Leffler type functions, among them the "Queen function of fractional calculus", the Mittag-Leffler function. These fractional indices/parameters extensions of the classical special functions became an unavoidable tool when fractalized models of phenomena and events are treated. Here, we try to review some of the basic results on the theory of the SF of FC, obtained in the author's works for more than 30 years, and support the wide spreading and important role of these functions by several examples