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Modular macdonald functions and generalized Newton's identity

JOURNAL OF ALGEBRA 2015年 442卷 124-136页

作者： Cai, Tommy Wuxing Jing, Naihuan Zhang, Jian S China Univ Technol Sch Math Sci Guangzhou 510640 Guangdong Peoples R China MIT Dept Math Cambridge MA 02139 USA N Carolina State Univ Dept Math Raleigh NC 27695 USA

关键词： Newton's identities macdonald functions Modular Hall-Littlewood polynomials Vertex operators

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Comment on the orthogonality of the macdonald functions of imaginary order

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JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS 2010年第1期365卷 195-197页

作者： Szmytkowski, Radoslaw Bielski, Sebastian Gdansk Univ Technol Atom Phys Div Dept Atom Phys & Luminescence Fac Appl Phys & Math PL-80233 Gdansk Poland

Recently, Yakubovich [Opuscula Math. 26 (2006) 161-172] and Passian et al. [J. Math. Anal. Appl. 360 (2009) 380-390] have presented alternative proofs of an orthogonality relation obeyed by the macdonald functions of imaginary order. In this note, we show that the validity of that relation may be also proven in a simpler way by applying a technique occasionally used in mathematical physics to normalize scattering wave functions to the Dirac delta distribution. (c) 2009 Elsevier Inc. All rights reserved.

关键词： macdonald functions Orthogonal functions Dirac delta distribution

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Shuffle algebras, lattice paths and macdonald functions

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ADVANCES IN MATHEMATICS 2024年 458卷

作者： Garbali, Alexandr Gunna, Ajeeth Univ Melbourne Sch Math & Stat Parkville Vic 3010 Australia

We consider partition functions on the N x N square lattice with the local Boltzmann weights given by the R-matrix s l n +1 | m ) quantum algebra. We identify boundary states such that the square lattice can be viewed on a conic surface. The partition function ZN N on this lattice computes the weighted sum over all possible closed coloured lattice paths with n +m m different colours: n "bosonic" colours and m "fermionic" colours. Each bosonic (fermionic) path of colour i contributes a factor of zi(wi) i ( w i ) to the weight of the configuration. We show the following: i) ZN N is a symmetric function in the spectral parameters x 1 ... x N and generates basis elements of the commutative trigonometric Feigin-Odesskii shuffle algebra. The generating function of ZN N admits a shuffle-exponential formula analogous to the macdonald Cauchy kernel. ii) ZN N is a symmetric function in two alphabets (z1 z 1 ... z n ) and (w1 w 1 ... w m ). When x 1 ... xN N are set to be equal to the box content of a skew Young diagram mu/ with N boxes the partition function ZN N reproduces the skew macdonald function P mu/nu [w w - z ]. (c) 2024 The Authors. Published by Elsevier Inc. This is an open access article under the CC BY license (http:// creativecommons .org /licenses /by /4 .0/). of the U t (

关键词： macdonald functions Shuffle algebra Exactly solvable lattice models

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macdonald symmetric functions of rectangular shapes

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JOURNAL OF COMBINATORIAL THEORY SERIES A 2014年第1期128卷 162-179页

作者： Cai, Tommy Wuxing S China Univ Technol Sch Sci Guangzhou 510640 Guangdong Peoples R China

Using vertex operators we study macdonald symmetric functions of rectangular shapes and their connection with the q-Dyson Laurent polynomial. We find a vertex operator realization of macdonald functions and give a generalized Frobenius formula for them. As byproducts of the realization, we find a q-Dyson constant term orthogonality relation which generalizes a conjecture due to Kadell (2000), and we generalize Matsumoto's hyperdeterminant formula for rectangular Jack functions to macdonald functions. (C) 2014 Elsevier Inc. All rights reserved.

关键词： macdonald functions Vertex operators q-Dyson constant term

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macdonald-positive specializations of the algebra of symmetric functions: Proof of the Kerov conjecture

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ANNALS OF MATHEMATICS 2019年第1期189卷 277-316页

作者： Matveev, Konstantin Brandeis Univ Waltham MA 02453 USA

We prove the classification of homomorphisms from the algebra of sym- metric functions to R with non-negative values on macdonald symmetric functions P-lambda, which was conjectured by S. V. Kerov in 1992.

关键词： Kerov conjecture symmetric functions macdonald functions asymptotic Rep-resentation Theory total positivity boundaries of branching graphs

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Algorithms and codes for the macdonald function: recent progress and comparisons

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JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS 2003年第1期161卷 179-192页

作者： Fabijonas, BR Lozier, DW Rappoport, JM So Methodist Univ Dept Math Dallas TX 75275 USA NIST Informat Technol Lab Gaithersburg MD 20899 USA Russian Acad Sci Moscow 117335 Russia

The modified Bessel function K-iv(x), also known as the macdonald function, finds application in the Kontorovich-Lebedev integral transform when x and v are real and positive. In this paper, a comparison of three codes for computing this function is made. These codes differ in algorithmic approach, timing, and regions of validity. One of them can be tested independent of the other two through Wronskian checks, and therefore is used as a standard against which the others are compared.

关键词： macdonald functions Bessel functions computation of special functions

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Changing the preferred direction of the refined topological vertex

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JOURNAL OF GEOMETRY AND PHYSICS 2013年第1期64卷 91-110页

作者： Awata, Hidetoshi Kanno, Hiroaki Nagoya Univ Grad Sch Math Nagoya Aichi 4648602 Japan

We consider the issue of the slice invariance of refined topological string amplitudes, which means that they are independent of the choice of the preferred direction of the refined topological vertex. We work out two examples. The first example is a geometric engineering of five-dimensional U(1) gauge theory with adjoint matter. The slice invariance follows from a highly non-trivial combinatorial identity which equates two known ways of computing the chi(y) genus of the Hilbert scheme of points on C-2. The second example is concerned with the proposal that the superpolynomials of the colored Hopf link are obtained from a refinement of topological open string amplitudes. We provide a closed formula for the superpolynomial, which confirms the slice invariance when the Hopf link is colored with totally anti-symmetric representations. However, we observe a breakdown of the slice invariance for other representations. (C) 2012 Elsevier B.V. All rights reserved.

关键词： Refined topological string Superpolynomials of the colored Hopf link macdonald functions Instanton counting

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Higher order macdonald operators with stability properties and their applications

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LETTERS IN MATHEMATICAL PHYSICS 2006年第2期77卷 169-181页

作者： Yamamoto, Takashi Univ Adelaide Special Res Ctr Subatom Struct Matter Adelaide SA 5005 Australia

The higher order macdonald operators which have compatibility with the restriction homomorphism are defined. As an application of these operators the new derivation of the transition coefficients between the Hall-Littlewood functions P ((r))(x;t (+/- k)) indexed by one part partitions and macdonald functions is given. A family of q,t-identities is also obtained from the specialization of the resultant expansion formulae.

关键词： macdonald functions higher order macdonald moperators Hall-Littlewood functions transition matrices q,t-identities

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ADDITIONAL CONSIDERATION OF THE SPECTRAL PROPERTIES OF SYNCHROTRON LIGHT

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RUSSIAN PHYSICS JOURNAL 2021年第9期63卷 1479-1482页

作者： Shishanin, O. E. Moscow State Univ Food Prod Moscow Russia

With the help of an identified small parameter, asymptotic of the Bessel function are found here for the first time out to the third order of accuracy. On the basis of these asymptotic, spectral expressions for synchr... 详细信息

关键词： Bessel functions with high index macdonald functions asymptotic spectral characteristics

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Index integral representations for connection between cartesian, cylindrical, and spheroidal systems

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INTEGRAL TRANSFORMS AND SPECIAL functions 2011年第8期22卷 549-560页

作者： Passian, A. Koucheckian, S. Yakubovich, S. Oak Ridge Natl Lab Oak Ridge TN 37831 USA Univ Tennessee Dept Phys Knoxville TN 37996 USA Univ S Florida Dept Math & Stat Tampa FL 33620 USA Univ Porto Dept Math Fac Sci P-4169007 Oporto Portugal

In this paper, we present two new index integral representations for connection between cartesian, cylindrical, and spheroidal coordinate systems in terms of Bessel, macdonald, and conical functions. Our result is mainly motivated by solution of the boundary value problems in domains composed of both cartesian and hyperboloidal boundaries, and the need for new integral representations that facilitate the transformation between these coordinates. As a by-product, the special cases of our results will produce new proofs to known index integrals and provide some new integral identities.

关键词： Kontorovich-Lebedev and Mellin transforms spheroidal systems conical functions macdonald functions Bessel functions

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