The Structure of Bivariate Rational Hypergeometric Functions

Cattani, Eduardo; Dickenstein, Alicia; Villegas, Fernando Rodríguez

doi:10.1093/imrn/rnq168

Cited by 4 publications

(5 citation statements)

References 31 publications

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“…The previous result was stated in Display ( 7) of [23], in the special case when I A is homogeneous. Its proof for the case when β ∈ Z d appeared in Proposition 4.2 of [8]; we generalize that argument here.…”

Section: Logarithm-free Nilsson Seriesmentioning

confidence: 80%

Nilsson solutions for irregular $A$-hypergeometric systems

Dickenstein¹,

Martínez²,

Matusevich³

2012

Rev. Mat. Iberoam.

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Abstract.We study the solutions of irregular A-hypergeometric systems that are constructed from Gröbner degenerations with respect to generic positive weight vectors. These are formal logarithmic Puiseux series that belong to explicitly described Nilsson rings, and are therefore called (formal) Nilsson series. When the weight vector is a perturbation of (1, . . . , 1), these series converge and provide a basis for the (multivalued) holomorphic hypergeometric functions in a specific open subset of C n . Our results are more explicit when the parameters are generic or when the solutions studied are logarithm-free. We also give an alternative proof of a result of Schulze and Walther that inhomogeneous A-hypergeometric systems have irregular singularities.

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Section: Logarithm-free Nilsson Seriesmentioning

confidence: 80%

Nilsson solutions for irregular $A$-hypergeometric systems

Dickenstein¹,

Martínez²,

Matusevich³

2012

Rev. Mat. Iberoam.

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“…As a consequence it is also a solution to a generalized hypergeometric system of partial differential equations in the sense of Gel'fand, Graev, Kapranov and Zelevinskiȋ (GGKZ, commonly shortened to GKZ) [29,[60][61][62][63]. The singularities of these hypergeometric systems are described by the principal A-determinant, see [64, §3] and [65, section 9] or [66,Theorem 1.36] or [48, §3]. We will define the principal A-determinant in section 2.2 below; in the context of Feynman integrals the zero set of the principal A-determinant contains all kinematic points where the Feynman integral fails to be an analytic function.…”

Section: Jhep10(2023)161mentioning

confidence: 99%

Symbol alphabets from the Landau singular locus

Dlapa,

Helmer,

Papathanasiou

et al. 2023

J. High Energ. Phys.

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We provide evidence through two loops, that rational letters of polylogarithmic Feynman integrals are captured by the Landau equations, when the latter are recast as a polynomial of the kinematic variables of the integral, known as the principal A-determinant. Focusing on one loop, we further show that all square-root letters may also be obtained, by re-factorizing the principal A-determinant with the help of Jacobi identities. We verify our findings by explicitly constructing canonical differential equations for the one-loop integrals in both odd and even dimensions of loop momenta, also finding agreement with earlier results in the literature for the latter case. We provide a computer implementation of our results for the principal A-determinants, symbol alphabets and canonical differential equations in an accompanying Mathematica file. Finally, we study the question of when a one-loop integral satisfies the Cohen-Macaulay property and show that for almost all choices of kinematics the Cohen-Macaulay property holds. Throughout, in our approach to Feynman integrals, we make extensive use of the Gel’fand, Graev, Kapranov and Zelevinskiĭ theory on what are now commonly called GKZ-hypergeometric systems whose singularities are described by the principal A-determinant.

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“…in the studies by Korn and Korn (1983, p. 269 ff. ) or Cattani (2006). Applications of Gaussian hypergeometric functions in the modelling of economic growth processes are discussed in the studies by Ruiz-Tamarit (2004, 2008) or Zawadzki (2015) for the Uzawa-Lucas model, by Guerrini (2006) for the Solow model and by Krawiec and Szydłowski (2002) for the Mankiw-Romer-Weil model.…”

Section: Notesmentioning

confidence: 99%

The Solow Model of Economic Growth

Dykas¹,

Tokarski²,

Wisła³

2022

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List of figures xi List of tables xiii Author biographies xv). (a) Scenario I, (b) scenario V, and (c) scenario IX * , k P * , y R * and y P * in the several variants relative to Variant I (Variant I = 100) 6.10 Ratios of average estimated k R * , k P * , y R * and y P * in the several variants relative to Variant I, assuming that either economy invests 10% of its savings in the other economy (Variant I = 100) 6.11 Selected numerical simulation results in Variant A (ω = 20%) 6.12 Selected numerical simulation results in Variant B (ω = 40%) 6.13 Selected numerical simulation results in Variant C (ω = 50%) 6.14 Selected simulation results in Variant D (ω = 60%) 6.15 Selected simulation results in Variant E (ω = 80%) 6.16. Selected results of numerical simulations in Variant F (s R = 25% and s P = 15%) 6.17 Selected results of numerical simulations in Variant F (s R = 15% and s P = 25%) 6.18 Ratios of average estimated k R * , k P * , y R * and y P * in the several variants relative to Variant I (Variant I = 100) 8.1 Simulations of labour productivity at standard (S), logistic (L) and post-Malthusian (PM) trajectories of the number of workers and at δ = 0.07, n = 0.01, α ≈ 0.68261 and m = e 9.1 Simulation of labour productivity at ω = 3 Tables xiv Tables 9.2 Simulation of labour productivity at ω = 5 9.3 Simulation of labour productivity at ω = 10 9.4 Simulation of labour productivity at ω = 25 9.5 Simulation of labour productivity at ω = 50 10.1 Scenarios of epidemic development 10.2 Epidemiological indicators in consecutive scenarios 10.3 Economic indicators in consecutive scenarios at K1/K * = 0.4 (a poorly developed economy) 10.4 Economic indicators in consecutive scenarios at K1/K * = 0.9 (a strongly developed economy)Author Biographies Paweł Dykas is an Associate Professor at the Department of Mathematical Economics of the Jagiellonian University in Krakow (Poland) and is also the author and co-author of over 50 academic publications such as The Neoclassical Model of Economic Growth and Its Ability to Account for Demographic Forecasts (2018), Demographic Forecasts and Volatility of Investment Rates vs. Labour Productivity Trajectories (2019) and An Impact of the Variable Technological Progress Rate on the Trajectory of Labour Productivity (2020). Recently, he has co-edited and contributed to The Socioeconomic Impact of COVID-19 on Eastern European Countries (Routledge, 2022). His main interests include the mathematical theory of economic growth, an analysis of the spatial diversity of economic development and an analysis of regional differences in the labour market. Tomasz Tokarski is Full Professor of Economics at the Department of Mathematical Economics of the Jagiellonian University in Krakow (Poland) and is also the author and co-author of over 200 academic publications such as Wybrane Modele Podażowych Czynników Wzrostu Gospodarczego (Selected Supply Models of Economic Growth) (2005), Ekonomia Matematyczna Modele Mikroekonomiczne (Microeconomic models of Mathematical Economics) (2011), Ekonomia Matematycz...

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The Structure of Bivariate Rational Hypergeometric Functions

Cited by 4 publications

References 31 publications

Nilsson solutions for irregular $A$-hypergeometric systems

Nilsson solutions for irregular $A$-hypergeometric systems

Symbol alphabets from the Landau singular locus

The Solow Model of Economic Growth

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