COLORED GRAPHS, BURGERS EQUATION AND HESSIAN CONJECTURE

2007 ◽  
Vol 18 (07) ◽  
pp. 797-808
Author(s):  
I. V. ARTAMKIN
Keyword(s):  
Differential Equations ◽  
Generating Function ◽  
Generating Functions ◽  
Burgers Equation ◽  
Heat Equations ◽  
System Of Differential Equations ◽  
Colored Graphs ◽  
Initial Term ◽  
Generating Series ◽  
Genus Expansion

We prove that generating series for colored modular graphs satisfy some systems of partial differential equations generalizing Burgers or heat equations. The solution is obtained by genus expansion of the generating function. The initial term of this expansion is the corresponding generating function for trees. For this term the system of differential equations is equivalent to the inversion problem for the gradient mapping defined by the initial condition. This enables to state the Jacobian conjecture in the language of generating functions. The use of generating functions provides rather short and natural proofs of resent results of Zhao and of the well-known Bass–Connell–Wright tree inversion formula.

scholarly journals Labels distance in bucket recursive trees with variable capacities of buckets

2021 ◽  
Vol 13 (2) ◽  
pp. 413-426
Author(s):  
S. Naderi ◽  
R. Kazemi ◽  
M. H. Behzadi
Keyword(s):  
Partial Differential Equations ◽  
Probability Distribution ◽  
Differential Equations ◽  
Generating Function ◽  
Generating Functions ◽  
Function Approach ◽  
Closed Formula ◽  
Tree Model ◽  
Partial Differential ◽  
Recursive Trees

Abstract The bucket recursive tree is a natural multivariate structure. In this paper, we apply a trivariate generating function approach for studying of the depth and distance quantities in this tree model with variable bucket capacities and give a closed formula for the probability distribution, the expectation and the variance. We show as j → ∞, lim-iting distributions are Gaussian. The results are obtained by presenting partial differential equations for moment generating functions and solving them.

scholarly journals Application of the Hori Method in the Theory of Nonlinear Oscillations

2012 ◽  
Vol 2012 ◽  
pp. 1-32
Author(s):  
Sandro da Silva Fernandes
Keyword(s):  
Differential Equations ◽  
Generating Function ◽  
Nonlinear Oscillations ◽  
System Of Differential Equations ◽  
Van Der Pol Equation ◽  
Van Der Pol ◽  
Identity Transformations ◽  
Hori Method ◽  
New System ◽  
Made In

Some remarks on the application of the Hori method in the theory of nonlinear oscillations are presented. Two simplified algorithms for determining the generating function and the new system of differential equations are derived from a general algorithm proposed by Sessin. The vector functions which define the generating function and the new system of differential equations are not uniquely determined, since the algorithms involve arbitrary functions of the constants of integration of the general solution of the new undisturbed system. Different choices of these arbitrary functions can be made in order to simplify the new system of differential equations and define appropriate near-identity transformations. These simplified algorithms are applied in determining second-order asymptotic solutions of two well-known equations in the theory of nonlinear oscillations: van der Pol equation and Duffing equation.

scholarly journals On the Modular Completion of Certain Generating Functions

2020 ◽  
Author(s):  
Kathrin Bringmann ◽  
Stephan Ehlen ◽  
Markus Schwagenscheidt
Keyword(s):  
Modular Form ◽  
Generating Function ◽  
Modular Forms ◽  
Generating Functions ◽  
Natural Family ◽  
Weakly Holomorphic Modular Forms ◽  
Holomorphic Modular Form ◽  
Singular Moduli ◽  
Generating Series ◽  
Holomorphic Modular Forms

Abstract We complete several generating functions to non-holomorphic modular forms in two variables. For instance, we consider the generating function of a natural family of meromorphic modular forms of weight two. We then show that this generating series can be completed to a smooth, non-holomorphic modular form of weights $\frac 32$ and two. Moreover, it turns out that the same function is also a modular completion of the generating function of weakly holomorphic modular forms of weight $\frac 32$, which prominently appear in work of Zagier [ 27] on traces of singular moduli.

scholarly journals Enumeration of Generalized $BCI$ Lambda-terms

10.37236/3051 ◽  
2013 ◽  
Vol 20 (4) ◽  
Author(s):  
Olivier Bodini ◽  
Danièle Gardy ◽  
Bernhard Gittenberger ◽  
Alice Jacquot
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Recurrence Relation ◽  
Generating Function ◽  
Generating Functions ◽  
Combinatory Logic ◽  
Efficient Computation ◽  
Asymptotic Number ◽  
Axiom Systems

We investigate the asymptotic number of elements of size $n$ in a particular class of closed lambda-terms (so-called $BCI(p)$-terms) which are related to axiom systems of combinatory logic. By deriving a differential equation for the generating function of the counting sequence we obtain a recurrence relation which can be solved asymptotically. We derive differential equations for the generating functions of the counting sequences of other more general classes of terms as well: the class of $BCK(p)$-terms and that of closed lambda-terms. Using elementary arguments we obtain upper and lower estimates for the number of closed lambda-terms of size $n$. Moreover, a recurrence relation is derived which allows an efficient computation of the counting sequence. $BCK(p)$-terms are discussed briefly.

scholarly journals Explicit Identities for 3-Variable Degenerate Hermite Kampé de Fériet Polynomials and Differential Equation Derived from Generating Function

Symmetry ◽  
2020 ◽  
Vol 13 (1) ◽  
pp. 7
Author(s):  
Kyung-Won Hwang ◽  
Young-Soo Seol ◽  
Cheon-Seoung Ryoo
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Generating Function ◽  
Generating Functions ◽  
Symmetric Identities

We get the 3-variable degenerate Hermite Kampé de Fériet polynomials and get symmetric identities for 3-variable degenerate Hermite Kampé de Fériet polynomials. We make differential equations coming from the generating functions of degenerate Hermite Kampé de Fériet polynomials to get some identities for 3-variable degenerate Hermite Kampé de Fériet polynomials,. Finally, we study the structure and symmetry of pattern about the zeros of the 3-variable degenerate Hermite Kampé de Fériet equations.

scholarly journals Counting Permutations by Alternating Descents

10.37236/4624 ◽  
2014 ◽  
Vol 21 (4) ◽  
Author(s):  
Ira M. Gessel ◽  
Yan Zhuang
Keyword(s):  
Differential Equations ◽  
Generating Function ◽  
Symmetric Functions ◽  
System Of Differential Equations ◽  
Euler Numbers ◽  
Noncommutative Symmetric Functions ◽  
Exponential Generating Function ◽  
Permutation Enumeration

We find the exponential generating function for permutations with all valleys even and all peaks odd, and use it to determine the asymptotics for its coefficients, answering a question posed by Liviu Nicolaescu. The generating function can be expressed as the reciprocal of a sum involving Euler numbers: \[\left(1-E_1x+E_{3}\frac{x^{3}}{3!}-E_{4}\frac{x^{4}}{4!}+E_{6}\frac{x^{6}}{6!}-E_{7}\frac{x^{7}}{7!}+\cdots\right)^{-1},\tag{$*$}\]where $\sum_{n=0}^\infty E_n x^n\!/n! = \sec x + \tan x$. We give two proofs of this formula. The first uses a system of differential equations whose solution gives the generating function\begin{equation*}\frac{3\sin\left(\frac{1}{2}x\right)+3\cosh\left(\frac{1}{2}\sqrt{3}x\right)}{3\cos\left(\frac{1}{2}x\right)-\sqrt{3}\sinh\left(\frac{1}{2}\sqrt{3}x\right)},\end{equation*} which we then show is equal to $(*)$. The second proof derives $(*)$ directly from general permutation enumeration techniques, using noncommutative symmetric functions. The generating function $(*)$ is an "alternating" analogue of David and Barton's generating function \[\left(1-x+\frac{x^{3}}{3!}-\frac{x^{4}}{4!}+\frac{x^{6}}{6!}-\frac{x^{7}}{7!}+\cdots\right)^{-1},\]for permutations with no increasing runs of length 3 or more. Our general results give further alternating analogues of permutation enumeration formulas, including results of Chebikin and Remmel.

scholarly journals A General Family of q-Hypergeometric Polynomials and Associated Generating Functions

Mathematics ◽  
2021 ◽  
Vol 9 (11) ◽  
pp. 1161
Author(s):  
Hari Mohan Srivastava ◽  
Sama Arjika
Keyword(s):  
Generating Function ◽  
Generating Functions ◽  
Hypergeometric Functions ◽  
Additional Parameter ◽  
Physical Sciences ◽  
General Family ◽  
Hypergeometric Polynomials

Basic (or q-) series and basic (or q-) polynomials, especially the basic (or q-) hypergeometric functions and the basic (or q-) hypergeometric polynomials are studied extensively and widely due mainly to their potential for applications in many areas of mathematical and physical sciences. Here, in this paper, we introduce a general family of q-hypergeometric polynomials and investigate several q-series identities such as an extended generating function and a Srivastava-Agarwal type bilinear generating function for this family of q-hypergeometric polynomials. We give a transformational identity involving generating functions for the generalized q-hypergeometric polynomials which we have introduced here. We also point out relevant connections of the various q-results, which we investigate here, with those in several related earlier works on this subject. We conclude this paper by remarking that it will be a rather trivial and inconsequential exercise to give the so-called (p,q)-variations of the q-results, which we have investigated here, because the additional parameter p is obviously redundant.

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