Classifying (almost)-Belyi maps with five exceptional points

Hoeij, Mark van; Kunwar, Vijay Jung

doi:10.1016/j.indag.2018.09.003

Cited by 2 publications

(1 citation statement)

References 17 publications

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“…This surprisingly general feature, reminiscent of Dwork's conjecture mentioned in [6, §3.2], begs for a combinatorial explanation. See also [48,Section 8] for a similar discussion.…”

Section: Introductionmentioning

confidence: 96%

Hypergeometric expressions for generating functions of walks with small steps in the quarter plane

Bostan

Chyzak

Hoeij

et al. 2017

European Journal of Combinatorics

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Abstract. We study nearest-neighbors walks on the two-dimensional square lattice, that is, models of walks on Z 2 defined by a fixed step set that is a subset of the non-zero vectors with coordinates 0, 1 or −1. We concern ourselves with the enumeration of such walks starting at the origin and constrained to remain in the quarter plane , pp. 201-215, 2009]. We give here the first proof that these equations are indeed satisfied by the corresponding generating functions. As a first corollary, we prove that all these 19 generating functions can be expressed in terms of Gauss' hypergeometric functions that are intimately related to elliptic integrals. As a second corollary, we show that all the 19 generating functions are transcendental, and that among their 19 × 4 combinatorially meaningful specializations only four are algebraic functions.

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“…This surprisingly general feature, reminiscent of Dwork's conjecture mentioned in [6, §3.2], begs for a combinatorial explanation. See also [48,Section 8] for a similar discussion.…”

Section: Introductionmentioning

confidence: 96%

Hypergeometric expressions for generating functions of walks with small steps in the quarter plane

Bostan

Chyzak

Hoeij

et al. 2017

European Journal of Combinatorics

Self Cite

View full text Add to dashboard Cite

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Generalized Thomas-Fermi equation: existence, uniqueness, and analytic approximation solutions

Bougoffa

Bougouffa

Khanfer

2023

MATH

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<abstract>The existence and uniqueness theorem for the generalized boundary value problem of the Thomas-Fermi equation: <disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{eqnarray*} \left\{ \begin{array}{l} y''+f(x, y) = 0, \ 0<x <\infty, \\ y(0) = 1, \ y(\infty) = 0, \end{array} \right. \end{eqnarray*} $\end{document} </tex-math></disp-formula> where <disp-formula> <label/> <tex-math id="FE2"> \begin{document}$ \begin{equation*} \label{6}f(x, y) = -y \left(\frac{y}{x}\right)^{\frac{p}{p+1}}, \ p>0, \ 0<x <\infty, \end{equation*} $\end{document} </tex-math></disp-formula> is proved. Also, highly accurate approximate solutions are obtained explicitly for this new boundary value problem which arises in particular studies of many-electron systems (atoms, ions, molecules, metals, crystals). To the best of our knowledge, the results obtained here are new and provide the lower and upper bounds approximate solutions for the generalized Thomas-Fermi problem.</abstract>

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Classifying (almost)-Belyi maps with five exceptional points

Cited by 2 publications

References 17 publications

Hypergeometric expressions for generating functions of walks with small steps in the quarter plane

Hypergeometric expressions for generating functions of walks with small steps in the quarter plane

Generalized Thomas-Fermi equation: existence, uniqueness, and analytic approximation solutions

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