On differentially algebraic generating series for walks in the quarter plane

نویسندگان

چکیده

We refine necessary and sufficient conditions for the generating series of a weighted model quarter plane walk to be differentially algebraic. In addition, we give algorithms based on theory Mordell–Weil lattices, that, each model, yield polynomial weights determining this property associated series.

برای دانلود رایگان متن کامل این مقاله و بیش از 32 میلیون مقاله دیگر ابتدا ثبت نام کنید

اگر عضو سایت هستید لطفا وارد حساب کاربری خود شوید

منابع مشابه

Walks in the quarter plane: Kreweras’ algebraic model

We consider planar lattice walks that start from (0, 0), remain in the first quadrant i, j ≥ 0, and are made of three types of steps: North-East, West and South. These walks are known to have remarkable enumerative and probabilistic properties: – they are counted by nice numbers (Kreweras 1965), – the generating function of these numbers is algebraic (Gessel 1986), – the stationary distribution...

متن کامل

On the Holonomy or Algebraicity of Generating Functions Counting Lattice Walks in the Quarter-Plane

In two recent works [2, 1], it has been shown that the counting generating functions (CGF) for the 23 walks with small steps confined in a quarter-plane and associated with a finite group of birational transformations are holonomic, and even algebraic in 4 cases – in particular for the so-called Gessel’s walk. It turns out that the type of functional equations satisfied by these CGF appeared in...

متن کامل

Hypergeometric Expressions for Generating Functions of Walks with Small Steps in the Quarter Plane

We study nearest-neighbors walks on the two-dimensional square lattice, that is, models of walks on Z 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 N, counted by their length and by the position of their ending point. B...

متن کامل

Walks in the Quarter Plane: Analytic Approach and Applications

In this survey we present an analytic approach to solve problems concerning (deterministic or random) walks in the quarter plane. We illustrate the recent breakthroughs in that domain with two examples. The first one is about the combinatorics of walks confined to the quarter plane, and more precisely about the numbers of walks evolving in the quarter plane and having given length, starting and...

متن کامل

Two non-holonomic lattice walks in the quarter plane

We present two classes of random walks restricted to the quarter plane whose generating function is not holonomic. The non-holonomy is established using the iterated kernel method, a recent variant of the kernel method. This adds evidence to a recent conjecture on combinatorial properties of walks with holonomic generating functions. The method also yields an asymptotic expression for the numbe...

متن کامل

ذخیره در منابع من


  با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید

ژورنال

عنوان ژورنال: Selecta Mathematica-new Series

سال: 2021

ISSN: ['1022-1824', '1420-9020']

DOI: https://doi.org/10.1007/s00029-021-00703-9