scholarly journals Counting quadrant walks via Tutte's invariant method (extended abstract)

2020 ◽  
Vol DMTCS Proceedings, 28th... ◽  
Author(s):  
Olivier Bernardi ◽  
Mireille Bousquet-Mélou ◽  
Kilian Raschel
Keyword(s):  
Differential Equation ◽  
Functional Equation ◽  
Linear Differential Equation ◽  
Generating Function ◽  
Algebraic Approach ◽  
Rational Functions ◽  
The Past ◽  
Lattice Walks ◽  
International Audience ◽  
Linear Differential

Extended abstract presented at the conference FPSAC 2016, Vancouver. International audience In the 1970s, Tutte developed a clever algebraic approach, based on certain " invariants " , to solve a functional equation that arises in the enumeration of properly colored triangulations. The enumeration of plane lattice walks confined to the first quadrant is governed by similar equations, and has led in the past decade to a rich collection of attractive results dealing with the nature (algebraic, D-finite or not) of the associated generating function, depending on the set of allowed steps. We first adapt Tutte's approach to prove (or reprove) the algebraicity of all quadrant models known or conjectured to be algebraic (with one small exception). This includes Gessel's famous model, and the first proof ever found for one model with weighted steps. To be applicable, the method requires the existence of two rational functions called invariant and decoupling function respectively. When they exist, algebraicity comes out (almost) automatically. Then, we move to an analytic viewpoint which has already proved very powerful, leading in particular to integral expressions of the generating function in the non-D-finite cases, as well as to proofs of non-D-finiteness. We develop in this context a weaker notion of invariant. Now all quadrant models have invariants, and for those that have in addition a decoupling function, we obtain integral-free expressions of the generating function, and a proof that this series is differentially algebraic (that is, satisfies a non-linear differential equation).

scholarly journals Identity-Type Functions and Polynomials

2007 ◽  
Vol 2007 ◽  
pp. 1-10
Author(s):  
M. Aslam Chaudhry ◽  
Asghar Qadir
Keyword(s):  
Differential Equation ◽  
Functional Equation ◽  
Boundary Conditions ◽  
Linear Differential Equation ◽  
Closed Form ◽  
Order Linear Differential Equation ◽  
Specific Function ◽  
Type Function ◽  
Form Representation ◽  
Linear Differential

For a noncommuting product of functions, similar to convolutions, an “identity-type function” leaving a specific function invariant is defined. It is evaluated for any choice of function on which it acts by solving a functional equation. A closed-form representation for the identity-type function of(1+t)−b(b>0)is obtained, which is a solution of a second-order linear differential equation with given boundary conditions. It yields orthogonal polynomials whose graphs are also given. The relevance for solution of boundary value problems by a series and convergence of the series are briefly discussed.

scholarly journals X. On linear differential equations.—No. V

1871 ◽  
Vol 19 (123-129) ◽  
pp. 526-528
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Linear Differential Equation ◽  
Rational Functions ◽  
Linear Differential Equations ◽  
Linear Differential

Let us now endeavour to ascertain under what circumstance a linear differential equation admits a solution of the form P log e Q, where P and Q are rational functions of ( x ).

III. On linear differential equations.—No. VI

1873 ◽  
Vol 21 (139-147) ◽  
pp. 14-19
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Linear Differential Equation ◽  
Explicit Solution ◽  
Rational Functions ◽  
Linear Differential Equations ◽  
Linear Differential

We now consider linear differential equations which are satisfied by the roots of an algebraical equation admitting of explicit solution. To determine in what cases the linear differential equation P d n y / dx n + Q d n -1 y / dx n -1 + . ..+ R y = 0 is satisfied by assuming y = m √X + r √Y + s √Z + ... when X, Y, Z are rational functions of ( x ).

scholarly journals Permutable functions concerning differential equations

2007 ◽  
Vol 83 (3) ◽  
pp. 369-384 ◽  
Author(s):  
X. Hua ◽  
R. Vaillancourt ◽  
X. L. Wang
Keyword(s):  
Differential Equation ◽  
Linear Differential Equation ◽  
Entire Functions ◽  
Julia Set ◽  
Rational Functions ◽  
Exponential Functions ◽  
Polynomial Coefficients ◽  
Transcendental Entire Functions ◽  
Linear Differential ◽  
Open Question

AbstractLet f and g be two permutable transcendental entire functions. Assume that f is a solution of a linear differential equation with polynomial coefficients. We prove that, under some restrictions on the coefficients and the growth of f and g, there exist two non-constant rational functions R1 and R2 such that R1 (f) = R(g). As a corollary, we show that f and g have the same Julia set: J(f) = J(g). As an application, we study a function f which is a combination of exponential functions with polynomial coefficients. This research addresses an open question due to Baker.

Asymptotic behavior of solutions of half-linear differential equations and generalized Karamata functions

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Kusano Takaŝi ◽  
Jelena V. Manojlović
Keyword(s):  
Differential Equation ◽  
Asymptotic Behavior ◽  
Continuous Function ◽  
Differential Equations ◽  
Linear Differential Equation ◽  
Regular Variation ◽  
Asymptotic Formulas ◽  
Asymptotic Behavior Of Solutions ◽  
Positive Continuous Function ◽  
Linear Differential

AbstractWe study the asymptotic behavior of eventually positive solutions of the second-order half-linear differential equation(p(t)\lvert x^{\prime}\rvert^{\alpha}\operatorname{sgn}x^{\prime})^{\prime}+q(% t)\lvert x\rvert^{\alpha}\operatorname{sgn}x=0,where q is a continuous function which may take both positive and negative values in any neighborhood of infinity and p is a positive continuous function satisfying one of the conditions\int_{a}^{\infty}\frac{ds}{p(s)^{1/\alpha}}=\infty\quad\text{or}\quad\int_{a}^% {\infty}\frac{ds}{p(s)^{1/\alpha}}<\infty.The asymptotic formulas for generalized regularly varying solutions are established using the Karamata theory of regular variation.

scholarly journals Steady Motion of Ice Sheets

1980 ◽  
Vol 25 (92) ◽  
pp. 229-246 ◽  
Author(s):  
L. W. Morland ◽  
I. R. Johnson
Keyword(s):  
Differential Equation ◽  
Linear Differential Equation ◽  
Power Law ◽  
Perturbation Expansion ◽  
Ice Sheet ◽  
Leading Order ◽  
Plane Flow ◽  
General Law ◽  
Non Linear ◽  
Linear Differential

AbstractSteady plane flow under gravity of a symmetric ice sheet resting on a horizontal rigid bed, subject to surface accumulation and ablation, basal drainage, and basal sliding according to a shear-traction-velocity power law, is treated. The surface accumulation is taken to depend on height, and the drainage and sliding coefficient also depend on the height of overlying ice. The ice is described as a general non-linearly viscous incompressible fluid, with illustrations presented for Glen’s power law, the polynomial law of Colbeck and Evans, and a Newtonian fluid. Uniform temperature is assumed so that effects of a realistic temperature distribution on the ice response are not taken into account. In dimensionless variables a small paramter ν occurs, but the ν = 0 solution corresponds to an unbounded sheet of uniform depth. To obtain a bounded sheet, a horizontal coordinate scaling by a small factor ε(ν) is required, so that the aspect ratio ε of a steady ice sheet is determined by the ice properties, accumulation magnitude, and the magnitude of the central thickness. A perturbation expansion in ε gives simple leading-order terms for the stress and velocity components, and generates a first order non-linear differential equation for the free-surface slope, which is then integrated to determine the profile. The non-linear differential equation can be solved explicitly for a linear sliding law in the Newtonian case. For the general law it is shown that the leading-order approximation is valid both at the margin and in the central zone provided that the power and coefficient in the sliding law satisfy certain restrictions.

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