Operational formula for the generalized Bessel matrix polynomials

The main idea of the present paper is to compute the spectrum and the fine spectrum of the generalized difference operator over the sequence spaces . The operator denotes a triangular sequential band matrix defined by with for , where or , ; the set nonnegative integers and is either a constant or strictly decreasing sequence of positive real numbers satisfying certain conditions. Finally, we obtain the spectrum, the point spectrum, the residual spectrum, and the continuous spectrum of the operator over the sequence spaces and . These results are more general and comprehensive than the spectrum of the difference operators , , , , and and include some other special cases such as the spectrum of the operators , , and over the sequence spaces or .

Download Full-text

A general inverse matrix series relation and associated polynomials – II

Mathematica Slovaca ◽

10.1515/ms-2017-0469 ◽

2021 ◽

Vol 71 (2) ◽

pp. 301-316

Author(s):

Reshma Sanjhira

Keyword(s):

Matrix Polynomial ◽

Combinatorial Identities ◽

Inverse Matrix ◽

Matrix Polynomials ◽

Special Cases ◽

Series Representations ◽

The Matrix ◽

Associated Polynomials ◽

Matrix Pair ◽

Matrix Series

Abstract We propose a matrix analogue of a general inverse series relation with an objective to introduce the generalized Humbert matrix polynomial, Wilson matrix polynomial, and the Rach matrix polynomial together with their inverse series representations. The matrix polynomials of Kiney, Pincherle, Gegenbauer, Hahn, Meixner-Pollaczek etc. occur as the special cases. It is also shown that the general inverse matrix pair provides the extension to several inverse pairs due to John Riordan [An Introduction to Combinatorial Identities, Wiley, 1968].

Download Full-text

Refinements of the Lower Bounds of the Jensen Functional

Abstract and Applied Analysis ◽

10.1155/2011/924319 ◽

2011 ◽

Vol 2011 ◽

pp. 1-13 ◽

Cited By ~ 1

Author(s):

Iva Franjić ◽

Sadia Khalid ◽

Josip Pečarić

Keyword(s):

Lower Bounds ◽

Jensen Inequality ◽

Left Hand ◽

Right Hand ◽

Special Cases ◽

Jensen Functional ◽

The Difference ◽

The Right

The lower bounds of the functional defined as the difference of the right-hand and the left-hand side of the Jensen inequality are studied. Refinements of some previously known results are given by applying results from the theory of majorization. Furthermore, some interesting special cases are considered.

Download Full-text

Degenerate Sklyanin algebras

Open Physics ◽

10.2478/s11534-009-0142-5 ◽

2010 ◽

Vol 8 (4) ◽

Cited By ~ 5

Author(s):

Andrey Smirnov

Keyword(s):

Difference Operators ◽

Baxter Equation ◽

Quantum Algebras ◽

Rational Solutions ◽

R Matrix ◽

Gauge Transformations ◽

Sklyanin Algebras ◽

Sklyanin Algebra ◽

The Difference

AbstractNew trigonometric and rational solutions of the quantum Yang-Baxter equation (QYBE) are obtained by applying some singular gauge transformations to the known Belavin-Drinfeld elliptic R-matrix for sl(2;?). These solutions are shown to be related to the standard ones by the quasi-Hopf twist. We demonstrate that the quantum algebras arising from these new R-matrices can be obtained as special limits of the Sklyanin algebra. A representation for these algebras by the difference operators is found. The sl(N;?)-case is discussed.

Download Full-text

Comment on "Quantitative performance metrics for stratospheric-resolving chemistry-climate models" by Waugh and Eyring (2008)

Atmospheric Chemistry and Physics ◽

10.5194/acp-9-9101-2009 ◽

2009 ◽

Vol 9 (23) ◽

pp. 9101-9110 ◽

Cited By ~ 9

Author(s):

V. Grewe ◽

R. Sausen

Keyword(s):

Observational Data ◽

Confidence Intervals ◽

Climate Models ◽

Performance Metrics ◽

Statistical Tests ◽

Low Grade ◽

Perfect Model ◽

Special Cases ◽

Quantitative Performance ◽

The Difference

Abstract. This comment focuses on the statistical limitations of a model grading, as applied by D. Waugh and V. Eyring (2008) (WE08). The grade g is calculated for a specific diagnostic, which basically relates the difference of means of model and observational data to the standard deviation in the observational dataset. We performed Monte Carlo simulations, which show that this method has the potential to lead to large 95%-confidence intervals for the grade. Moreover, the difference between two model grades often has to be very large to become statistically significant. Since the confidence intervals were not considered in detail for all diagnostics, the grading in WE08 cannot be interpreted, without further analysis. The results of the statistical tests performed in WE08 agree with our findings. However, most of those tests are based on special cases, which implicitely assume that observations are available without any errors and that the interannual variability of the observational data and the model data are equal. Without these assumptions, the 95%-confidence intervals become even larger. Hence, the case, where we assumed perfect observations (ignored errors), provides a good estimate for an upper boundary of the threshold, below that a grade becomes statistically significant. Examples have shown that the 95%-confidence interval may even span the whole grading interval [0, 1]. Without considering confidence intervals, the grades presented in WE08 do not allow to decide whether a model result significantly deviates from reality. Neither in WE08 nor in our comment it is pointed out, which of the grades presented in WE08 inhibits such kind of significant deviation. However, our analysis of the grading method demonstrates the unacceptably high potential for these grades to be insignificant. This implies that the grades given by WE08 can not be interpreted by the reader. We further show that the inclusion of confidence intervals into the grading approach is necessary, since otherwise even a perfect model may get a low grade.

Download Full-text

Analytical Study of the Difference Equation xn+1 = xnxn-6 / xn-5 (±1 – xnxn-6)

Asian Research Journal of Mathematics ◽

10.9734/arjom/2019/v12i230081 ◽

2019 ◽

pp. 1-12

Author(s):

Abdualrazaq Sanbo ◽

Elsayed M. Elsayed ◽

Faris Alzahrani

Keyword(s):

Difference Equation ◽

Difference Equations ◽

Analytical Study ◽

Initial Conditions ◽

Specific Form ◽

Positive Real ◽

Rational Difference Equations ◽

Special Cases ◽

The Difference

This paper is devoted to find the form of the solutions of a rational difference equations with arbitrary positive real initial conditions. Specific form of the solutions of two special cases of this equation are given.

Download Full-text

On the 2-adic order of Stirling numbers of the second kind and their differences

Discrete Mathematics & Theoretical Computer Science ◽

10.46298/dmtcs.2694 ◽

2009 ◽

Vol DMTCS Proceedings vol. AK,... (Proceedings) ◽

Author(s):

Tamás Lengyel

Keyword(s):

Positive Integer ◽

Stirling Numbers ◽

Bell Polynomials ◽

Binary Representation ◽

Exact Order ◽

Special Cases ◽

International Audience ◽

The Difference ◽

Divisibility Properties ◽

Positive Integers

International audience Let $n$ and $k$ be positive integers, $d(k)$ and $\nu_2(k)$ denote the number of ones in the binary representation of $k$ and the highest power of two dividing $k$, respectively. De Wannemacker recently proved for the Stirling numbers of the second kind that $\nu_2(S(2^n,k))=d(k)-1, 1\leq k \leq 2^n$. Here we prove that $\nu_2(S(c2^n,k))=d(k)-1, 1\leq k \leq 2^n$, for any positive integer $c$. We improve and extend this statement in some special cases. For the difference, we obtain lower bounds on $\nu_2(S(c2^{n+1}+u,k)-S(c2^n+u,k))$ for any nonnegative integer $u$, make a conjecture on the exact order and, for $u=0$, prove part of it when $k \leq 6$, or $k \geq 5$ and $d(k) \leq 2$. The proofs rely on congruential identities for power series and polynomials related to the Stirling numbers and Bell polynomials, and some divisibility properties.

Download Full-text

Agmon estimates for the difference of exact and approximate Dirichlet eigenfunctions for difference operators

Asymptotic Analysis ◽

10.3233/asy-151343 ◽

2016 ◽

Vol 97 (1-2) ◽

pp. 61-89 ◽

Cited By ~ 2

Author(s):

Markus Klein ◽

Elke Rosenberger

Keyword(s):

Difference Operators ◽

The Difference

Download Full-text

Explicit asymptotic results for open times in ion channel models

Advances in Applied Probability ◽

10.1017/s0001867800047960 ◽

1997 ◽

Vol 29 (04) ◽

pp. 947-964

Author(s):

Valeri T. Stefanov ◽

Geoffrey F. Yeo

Keyword(s):

Single Channel ◽

Exponential Families ◽

Markov Renewal Process ◽

Actual Process ◽

Asymptotic Results ◽

Normal Distributions ◽

Special Cases ◽

The Difference ◽

Asymptotic Normal ◽

Receptor Channel

The dynamical aspects of single channel gating can be modelled by a Markov renewal process, with states aggregated into two classes corresponding to the receptor channel being open or closed, and with brief sojourns in either class not detected. This paper is concerned with the relation between the amount of time, for a given record, in which the channel appears to be open compared to the amount in which it is actually open and the difference in their proportions; this may be used to obtain information on the unobserved actual process from the observed one. Results, with extensions, on exponential families have been applied to obtain relevant generating functions and asymptotic normal distributions, including explicit forms for the parameters. Numerical results are given as illustration in special cases.

Download Full-text