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 On Generating Functions

1930 ◽  
Vol 2 (2) ◽  
pp. 71-82 ◽  
Author(s):  
W. L. Ferrar
Keyword(s):  
Differential Equation ◽  
Recurrence Relation ◽  
Generating Function ◽  
Generating Functions ◽  
Order Differential Equation ◽  
Second Order ◽  
Second Order Differential Equation ◽  
Image Position ◽  
Three Term Recurrence Relation ◽  
Sequence Of Functions

It is well known that the polynomial in x,has the following properties:—(A) it is the coefficient of tn in the expansion of (1–2xt+t2)–½;(B) it satisfies the three-term recurrence relation(C) it is the solution of the second order differential equation(D) the sequence Pn(x) is orthogonal for the interval (— 1, 1),i.e. whenSeveral other familiar polynomials, e.g., those of Laguerre Hermite, Tschebyscheff, have properties similar to some or all of the above. The aim of the present paper is to examine whether, given a sequence of functions (polynomials or not) which has one of these properties, the others follow from it : in other words we propose to examine the inter-relation of the four properties. Actually we relate each property to the generating function.

Generating functions and semi-classical orthogonal polynomials

1994 ◽  
Vol 124 (5) ◽  
pp. 1003-1011 ◽  
Author(s):  
Pascal Maroni ◽  
Jeannette Van Iseghem
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Orthogonal Polynomials ◽  
Recurrence Relation ◽  
Generating Function ◽  
Generating Functions ◽  
Special Form ◽  
Classical Orthogonal Polynomials ◽  
Order Linear ◽  
First Order

An orthogonal family of polynomials is given, and a link is made between the special form of the coefficients of their recurrence relation and a first-order linear homogenous partial differential equation satisfied by the associated generating function. A study is also made of the semiclassical character of such families.

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 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 Differential Transform Algorithm for Functional Differential Equations with Time-Dependent Delays

Complexity ◽  
2020 ◽  
Vol 2020 ◽  
pp. 1-12 ◽  
Author(s):  
Josef Rebenda ◽  
Zuzana Pátíková
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Differential Equations ◽  
Recurrence Relation ◽  
Numerical Solutions ◽  
Functional Differential Equations ◽  
Neutral System ◽  
Initial Value ◽  
Differential Transformation ◽  
Functional Differential

An algorithm using the differential transformation which is convenient for finding numerical solutions to initial value problems for functional differential equations is proposed in this paper. We focus on retarded equations with delays which in general are functions of the independent variable. The delayed differential equation is turned into an ordinary differential equation using the method of steps. The ordinary differential equation is transformed into a recurrence relation in one variable using the differential transformation. Approximate solution has the form of a Taylor polynomial whose coefficients are determined by solving the recurrence relation. Practical implementation of the presented algorithm is demonstrated in an example of the initial value problem for a differential equation with nonlinear nonconstant delay. A two-dimensional neutral system of higher complexity with constant, nonconstant, and proportional delays has been chosen to show numerical performance of the algorithm. Results are compared against Matlab function DDENSD.

scholarly journals Generating Functions for Bessel Functions

1959 ◽  
Vol 11 ◽  
pp. 148-155 ◽  
Author(s):  
Louis Weisner
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Generating Function ◽  
Lie Group ◽  
Generating Functions ◽  
Bessel Functions ◽  
Partial Differential ◽  
Systematic Determination ◽  
Image Position

On replacing the parameter n in Bessel's differential equation1.1by the operator y(∂/∂y), the partial differential equation Lu = 0 is constructed, where1.2This operator annuls u(x, y) = v(x)yn if, and only if, v(x) satisfies (1.1) and hence is a cylindrical function of order n. Thus every generating function of a set of cylindrical functions is a solution of Lu = 0.It is shown in § 2 that the partial differential equation Lu = 0 is invariant under a three-parameter Lie group. This group is then applied to the systematic determination of generating functions for Bessel functions, following the methods employed in two previous papers (4; 5).

CORRELATION FUNCTIONS FOR TOPOLOGICAL LANDAU-GINZBURG MODELS WITH c≤3

1992 ◽  
Vol 07 (25) ◽  
pp. 6215-6244 ◽  
Author(s):  
ALBRECHT KLEMM ◽  
STEFAN THEISEN ◽  
MICHAEL G. SCHMIDT
Keyword(s):  
Field Theory ◽  
Differential Equation ◽  
Differential Equations ◽  
Generating Function ◽  
Correlation Functions ◽  
Coupling Constant ◽  
Superconformal Field Theory ◽  
Point Correlation ◽  
Marginal Deformation

We discuss c≤3 topological Landau-Ginzburg models. In particular we give the potential for the three exceptional models E6,7,8 in the constant metric coordinates of coupling constant space and derive the generating function F for correlation functions. For the c=3 torus cases with one marginal deformation and relevant perturbations, we derive and solve the differential equation resulting from flatness of coupling constant space. We perform the transformation to constant metric coordinates and calculate the generating function F. Comparing the three-point correlation functions with those of orbifold superconformal field theory, we find agreement. We finally demonstrate that the differential equations derived from flatness of coupling constant space are the same as the ones satisfied by the periods of the tori.

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 Differential equations for p, q-Touchard polynomials

2016 ◽  
Vol 14 (1) ◽  
pp. 908-912
Author(s):  
Taekyun Kim ◽  
Orli Herscovici ◽  
Toufik Mansour ◽  
Seog-Hoon Rim
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Generating Function

AbstractIn this paper, we present differential equation for the generating function of the p, q-Touchard polynomials. An application to ordered partitions of a set is investigated.

scholarly journals PROBABILISTIC SPACE FOR CALCULATION OF PROBABILISTIC CHARACTERISTICS OF A THREE-PARAMETER QUEUEING SYSTEM MODEL

2017 ◽  
pp. 100-109
Author(s):  
Valery Pavsky ◽  
Valery Pavsky ◽  
Kirill Pavsky ◽  
Kirill Pavsky ◽  
Svetlana Ivanova ◽  
...  
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Probability Distribution ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Generating Function ◽  
Queueing Theory ◽  
Probability Space ◽  
Queueing System ◽  
Probabilistic Space

A model of queueing theory is proposed that describes a queueing system with three parameters, which has important practical applications. The model is based on the continuous time Markov process with a discrete number of states. The model is formalized by a probabilistic space in which the space of elementary events is a set of inconsistent states of the queueing system; and the probabilistic measure is a probability distribution corresponding to a set of elementary events, that is, each elementary event is associated with the probability of the system staying in this state, for each fixed time moment. The model is represented by a system of ordinary differential equations, compiled by methods of queueing theory (Kolmogorov equations). To find the solution of the system of equations, the method of generating functions is used. For the generating function, a partial differential equation is obtained. Finding the generating function completes the construction of a probability space. The latter means that for any random variables and functions defined on the resulting probability space, one can find their probabilistic characteristics. In particular, analytical expressions of the moments (mathematical expectations and variances) of random functions that depend on time are obtained. The peculiarity of finding a solution is that it is obtained not from the probability distribution, but directly from the partial differential equation, which represents a system of ordinary differential equations. For the probability distribution, the solution was found by a combinatorial method, which made it possible to significantly reduce the computations. To apply the formulas in engineering calculations, we consider the stationary case, to which a considerable simplification of the calculations corresponds. A relationship between a system of differential equations and a polynomial distribution known in probability theory is shown. The results are used in the analysis of the reliability of the operation of scalable computing systems; graphical implementation is shown

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