scholarly journals Generating Function Associated with the Hankel Determinant Formula for the Solutions of the Painlevé IV Equation

2006 ◽  
Vol 49 (3) ◽  
pp. 451-468 ◽  
Author(s):  
Nalini Joshi ◽  
Kenji Kajiwara ◽  
Marta Mazzocco
Keyword(s):  
Generating Function ◽  
Hankel Determinant ◽  
Determinant Formula

scholarly journals Generating functions related to the Okamoto polynomials for the Painlevé IV equation

2005 ◽  
Vol 71 (3) ◽  
pp. 517-526 ◽  
Author(s):  
Hiromichi Goto ◽  
Kenji Kajiwara
Keyword(s):  
Asymptotic Expansions ◽  
Generating Functions ◽  
Hankel Determinant ◽  
Rational Solutions ◽  
Determinant Formula ◽  
Airy Equation

We construct generating functions for the entries of Hankel determinant formula for the Okamoto polynomials which characterise a class of rational solutions to the Painlevé IV equation. Generating functions are characterised as asymptotic expansions of log derivative of Ai and Bi, which are solutions of the Airy equation.

scholarly journals The Generating Function of Ternary Trees and Continued Fractions

10.37236/1079 ◽  
2006 ◽  
Vol 13 (1) ◽  
Author(s):  
Ira M. Gessel ◽  
Guoce Xin
Keyword(s):  
Generating Function ◽  
Continued Fraction ◽  
Hypergeometric Series ◽  
Combinatorial Proof ◽  
Hankel Determinant ◽  
Simple Method ◽  
Alternating Sign Matrices ◽  
Hankel Determinants ◽  
Fraction Method ◽  
Special Case

Michael Somos conjectured a relation between Hankel determinants whose entries ${1\over 2n+1}{3n\choose n}$ count ternary trees and the number of certain plane partitions and alternating sign matrices. Tamm evaluated these determinants by showing that the generating function for these entries has a continued fraction that is a special case of Gauss's continued fraction for a quotient of hypergeometric series. We give a systematic application of the continued fraction method to a number of similar Hankel determinants. We also describe a simple method for transforming determinants using the generating function for their entries. In this way we transform Somos's Hankel determinants to known determinants, and we obtain, up to a power of $3$, a Hankel determinant for the number of alternating sign matrices. We obtain a combinatorial proof, in terms of nonintersecting paths, of determinant identities involving the number of ternary trees and more general determinant identities involving the number of $r$-ary trees.

scholarly journals A remark on the Hankel determinant formula for solutions of the Toda equation

2007 ◽  
Vol 40 (42) ◽  
pp. 12661-12675 ◽  
Author(s):  
Kenji Kajiwara ◽  
Marta Mazzocco ◽  
Yasuhiro Ohta
Keyword(s):  
Hankel Determinant ◽  
Toda Equation ◽  
Determinant Formula

An upper bound to the nonlinear functional for certain subclasses of analytic functions associated with Hankel determinant

2014 ◽  
Vol 07 (02) ◽  
pp. 1350042
Author(s):  
D. Vamshee Krishna ◽  
T. Ramreddy
Keyword(s):  
Upper Bound ◽  
Convex Functions ◽  
Analytic Functions ◽  
Hankel Determinant ◽  
Toeplitz Determinants ◽  
Nonlinear Functional ◽  
Determinant Formula ◽  
Starlike And Convex Functions ◽  
Second Hankel Determinant ◽  
Strongly Starlike

The objective of this paper is to obtain an upper bound to the second Hankel determinant [Formula: see text] for the functions belonging to strongly starlike and convex functions of order α(0 < α ≤ 1). Further, we introduce a subclass of analytic functions and obtain the same coefficient inequality for the functions in this class, using Toeplitz determinants.

Research on the upper bound of delay based on the form of moment generating function

2002 ◽  
Vol 2 (1) ◽  
pp. 14-23
Author(s):  
Duan Pengfei
Keyword(s):  
Generating Function ◽  
Upper Bound ◽  
Moment Generating Function

System of thermodynamic quantities in the McMillan-Mayer solution theory

1985 ◽  
Vol 50 (4) ◽  
pp. 791-798 ◽  
Author(s):  
Vilém Kodýtek
Keyword(s):  
Free Energy ◽  
Generating Function ◽  
Simple Form ◽  
Unit Volume ◽  
Thermodynamic Quantities ◽  
Solution Theory ◽  
Pure Solvent ◽  
Second Derivatives ◽  
Definition Of ◽  
Derivatives Of

The McMillan-Mayer (MM) free energy per unit volume of solution AMM, is employed as a generating function of the MM system of thermodynamic quantities for solutions in the state of osmotic equilibrium with pure solvent. This system can be defined by replacing the quantities G, T, P, and m in the definition of the Lewis-Randall (LR) system by AMM, T, P0, and c (P0 being the pure solvent pressure). Following this way the LR to MM conversion relations for the first derivatives of the free energy are obtained in a simple form. New relations are derived for its second derivatives.

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.

On the absorption probabilities and mean time for absorption for discrete Markov chains

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Nikolaos Halidias
Keyword(s):  
Markov Chain ◽  
Random Walk ◽  
Markov Chains ◽  
Generating Function ◽  
Discrete Time ◽  
Probability Generating Function ◽  
The Mean ◽  
Mean Time ◽  
Absorption Probabilities

Abstract In this note we study the probability and the mean time for absorption for discrete time Markov chains. In particular, we are interested in estimating the mean time for absorption when absorption is not certain and connect it with some other known results. Computing a suitable probability generating function, we are able to estimate the mean time for absorption when absorption is not certain giving some applications concerning the random walk. Furthermore, we investigate the probability for a Markov chain to reach a set A before reach B generalizing this result for a sequence of sets A 1 , A 2 , … , A k {A_{1},A_{2},\dots,A_{k}} .

scholarly journals Coefficients problems for families of holomorphic functions related to hyperbola

2020 ◽  
Vol 70 (3) ◽  
pp. 605-616
Author(s):  
Stanisława Kanas ◽  
Vali Soltani Masih ◽  
Ali Ebadian
Keyword(s):  
Unit Disk ◽  
Holomorphic Functions ◽  
Hankel Determinant ◽  
Coefficient Problem

AbstractWe consider a family of analytic and normalized functions that are related to the domains ℍ(s), with a right branch of a hyperbolas H(s) as a boundary. The hyperbola H(s) is given by the relation $\begin{array}{} \frac{1}{\rho}=\left( 2\cos\frac{\varphi}{s}\right)^s\quad (0 \lt s\le 1,\, |\varphi| \lt (\pi s)/2). \end{array}$ We mainly study a coefficient problem of the families of functions for which zf′/f or 1 + zf″/f′ map the unit disk onto a subset of ℍ(s) . We find coefficients bounds, solve Fekete-Szegö problem and estimate the Hankel determinant.

The number of meetings in free Poisson traffic

1974 ◽  
Vol 11 (2) ◽  
pp. 320-331
Author(s):  
Hans D. Unkelbach ◽  
Helmut Wegmann
Keyword(s):  
Differential Equation ◽  
Generating Function ◽  
Practical Interest ◽  
Time Interval ◽  
Integro Differential Equation

Using Rényi's model of free Poisson traffic the distribution of the number of meetings of vehicles on a highway section during a given time interval is investigated. An integro-differential equation for the generating function of that variable is deduced and the first moments are calculated. The generating function is given explicitly in simple cases and approximately in cases of practical interest.

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