We investigate the use of generating functions in the analysis of discrete Markov chains. Generating functions are introduced as power series whose coefficients are certain hitting probabilities. Being able to compute such functions implies that the calculation of a number of quantities of interest, including absorption probabilities, expected hitting time and number of visits, and variances th...

We calculate accurate bound states and resonances of two interesting perturbed Coulomb models by means of the Riccati-Padé method. This approach is based on a rational approximation to a modified logarithmic derivative of the eigenfunction and produces sequences of roots of Hankel determinants that converge towards the eigenvalues of the equation.

We study the construction of a quadrature rule which allows the simultaneous integration of a given function with respect to different weights. This construction is built on the basis of simultaneous Padé approximation of a Nikishin system of functions. The properties of these approximants are used in the proof of convergence of the quadratures and positivity of the corresponding quadrature coe...

In this paper, a further investigation for the number of Derangements and Bell numbers is performed, and some new recursion formulae for the number of Derangements and Bell numbers are established by applying the generating function methods and Padé approximation techniques. Illustrative special cases of the main results are also presented.

Using the Padé approximation of the exponential function, we obtain recurrence relations between Apostol-Bernoulli and between Apostol-Euler polynomials. As applications, we derive some new lacunary recurrence relations for Bernoulli and Euler polynomials with gap of length 4 and lacunary relations for Bernoulli and Euler numbers with gap of length 6.

Using the Padé approximation of the exponential function, we obtain a general recurrence relation for values of the zeta function which contains, as particular cases, many of relations already proved. Applications to Bernoulli polynomials are given. At last, we derive some new recurrence relations with gap of length 4 for zeta numbers. MSC: 11B68 41A21

The scaling and squaring method is the most widely used method for computing the matrix exponential, not least because it is the method implemented in MATLAB’s expm function. The method scales the matrix by a power of 2 to reduce the norm to order 1, computes a Padé approximant to the matrix exponential, and then repeatedly squares to undo the effect of the scaling. We give a new backward error...

In this paper the properties of Rédei rational functions are used to derive rational approximations for square roots and both Newton and Padé approximations are given as particular cases. As a consequence, such approximations can be derived directly by power matrices. Moreover, Rédei rational functions are introduced as convergents of particular periodic continued fractions and are applied for ...

In this paper, a technique to approximate a class of passive delay systems by a bond graph model is investigated. This method is designed to preserve the passivity of the initial model. Through interconnections of passive elementary blocks (passivity is stable under interconnections), a finite dimensional passive approximant is constructed. This finite dimensional model, if need be, is reduced ...