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On a new generalization of Jacobsthal quaternions and several identities involving these numbers

100%

Bród D. , Szynal-Liana A.

nr 1-2

In this paper we generalize Jacobsthal quaternions to $(s,p)$\dywiz Jacobsthal quaternions. We give some of their properties, among others the Binet formula, the generating function and the matrix representation of these quaternions. We will show how a~graph interpretation can be used in proving some identities for quaternions.

Linearization of Arbitrary products of classical orthogonal polynomials

88%

Hounkonnou M. N. , Belmehdi S. , Ronveaux A.

tom 27

nr 2

187-196

A procedure is proposed in order to expand $w=\prod^N_{j=1} P_{i_j}(x)=\sum^M_{k=0} L_ k P_ k(x)$ where $P_i(x)$ belongs to aclassical orthogonal polynomial sequence (Jacobi, Bessel, Laguerre and Hermite) ($M=\sum^N_{j=1} i_j$). We first derive a linear differential equation of order $2^N$ satisfied by w, fromwhich we deduce a recurrence relation in k for the linearizationcoefficients $L_k$. We develop in detail the two cases $[P_i(x)]^N$, $P_ i(x)P_ j(x)P_ k(x)$ and give the recurrencerelation in some cases (N=3,4), when the polynomials $P_i(x)$are monic Hermite orthogonal polynomials.

Recurrence relations for a multi-channel closed queueing system with Erlangian service times of second order

75%

Zhernovyi Y. , Kopytko B.

tom Vol. 16, nr 3

123--128

We propose a method for determining the steady-state characteristics of a multichannel closed queueing system with exponential distribution of the time generation of service requests and the second order Erlang distributions of the service times. Recurrence relations to compute the steady-state distribution of the number of customers are obtained. The developed algorithms are tested on examples using simulation models constructed with the assistance of the GPSS World tools.

Steady-state characteristics of three-channel queueing systems with Erlangian service times

63%

Kopytko B. , Zhernovyi K.

tom Vol. 15, nr 3

75--87

We propose a method of study the M/E2/3/∞ queueing systems: standard system and systems with the threshold and hysteretic strategies of the random dropping of customers in order to control the input flow. Recurrence relations to compute the stationary distribution of the number of customers and the steady-state characteristics are obtained. The developed algorithms are tested on examples using simulation models constructed with the assistance of the GPSS World tools.