Determinantal Forms Research Articles

Several closed and determinantal forms for convolved Fibonacci numbers

Several closed and determinantal forms for convolved Fibonacci numbers

Determinantal forms and recursive relations of the Delannoy two-functional sequence

In the paper, the authors establish closed forms for the Delannoy two-functional sequence and its difference in terms of the Hessenberg determinants, derive recursive relations for the Delannoy two-functional sequence and its difference, and deduce closed forms, in terms of the Hessenberg determinants, and recursive relations for the Delannoy one-functional sequence, the Delannoy numbers, and central Delannoy numbers. In the paper, the authors establish closed forms for the Delannoy two-functional sequence and its difference in terms of the Hessenberg determinants, derive recursive relations for the Delannoy two-functional sequence and its difference, and deduce closed forms, in terms of the Hessenberg determinants, and recursive relations for the Delannoy one-functional sequence, the Delannoy numbers, and central Delannoy numbers.

An algebraic approach to Sheffer polynomial sequences

A matrix approach to Sheffer polynomial sequences is proposed; in particular, two different determinantal forms of Sheffer sequences are given, the one as the function of a polynomial sequence of binomial type and the other as the function of the canonical base xi. The equivalence with the classical definitions of Sheffer and Roman and Rota is proven. Then, elementary matrix algebra tools are employed to reveal the known and unknown properties of Sheffer polynomials. Finally, classical and non-classical examples are also considered.

First Hitting Time of the Boundary of the Weyl Chamber by Radial Dunkl Processes

We provide two equivalent approaches for computing the tail distribution of the first hitting time of the boundary of the Weyl chamber by a radial Dunkl process. The first approach is based on a spectral problem with initial value. The second one expresses the tail distribution by means of the $W$-invariant Dunkl-Hermite polynomials. Illustrative examples are given by the irreducible root systems of types $A$, $B$, $D$. The paper ends with an interest in the case of Brownian motions for which our formulae take determinantal forms.

Symmetric functions, tableaux decompositions and the fermion-boson correspondence

An extended fermion-boson correspondence is introduced for skew Schur functions. Certain members of a general class of recently-developed determinantal forms, based on outer strip decompositions of skew shape tableaux, are described in the context of Wick's theorem for ‘composite’ operators.

Determinantal solution of density matrix equations in time-dependent quantum mechanics. II. Harmonic perturbation

The determinantal formalism previously built up in the evolution operator problem is extended to the density matrix equation in the case, here, of a Heaviside step external perturbation, and in the presence of internal collision potential. Reduced determinantal forms, matched to those first derived for the evolution operator, are obtained and compared with both the iterative expansion and the initial Fredholm-Laplace solution. They not only exhibit the multiple transition structure of the physical response, along with the associated transition width upon expansion, but also the trace conservation requirement is shown to be satisfied at any order and all times. An illustration is given in a simplified physical system.

On extensions of Cramer's rule for solutions of restricted linear systems1

For the unique solution of a special consistent restricted linear system Ax=bx∊M we derive two different determinantal forms, which both reduce to Cramer's classical rule if A is nonsingular. The representation for the minimum Euclidean-norm solution of Ax=b given recently by Ben-Israel [2] is reobtained in our first approach as special case. A determinantal formula for any {1, 2}-generalized inverse is also given.

Determinantal solution of density matrix equations in time-dependent quantum mechanics. I. Constant perturbation

The determinantal formalism previously built up in the evolution operator problem is extended to the density matrix equation in the case, here, of a Heaviside step external perturbation, and in the presence of internal collision potential. Reduced determinantal forms, matched to those first derived for the evolution operator, are obtained and compared with both the iterative expansion and the initial Fredholm-Laplace solution. They not only exhibit the multiple transition structure of the physical response, along with the associated transition width upon expansion, but also the trace conservation requirement is shown to be satisfied at any order and all times. An illustration is given in a simplified physical system.

Reduced determinantal forms for characters of the classical Lie groups

The reduced determinantal forms for the character of all the irreducible representations of the classical Lie groups are derived from Weyl's character formulae. The derivations use a lemma due to Frobenius and the results are expressed in terms of characters of representations specified by hook diagrams. The results are appropriate to all irreducible representations whether specified by ordinary or composite Young diagrams, and whether tensor or spinor in nature.

A Further Extension of the Routh-Hurwitz Stability Criteria for Quintic and Sextic Equations

Recently, Duncan ' 4 applied these test functions to the sextic and octic equations in determining their stability. I t is immediately apparent that after reducing the complexity of the stability problem, there still remains a tedious amount of necessary work in solving higher order equations. In a further effort at simplification, Higgins and Hogan employ the Hurwitz determinants in conjunction with algebraic manipulation to reduce the necessary number of parameters for a quintic equation to four. Even then, an innumerable number of graphs must be plotted in order to determine the stability of the quintic equations. Duncan derived simple relations between certain expressed polynomials and determinantal forms and tended to further minimize the number of effective parameters. Since there are no general parameters which hold for all degrees of polynomials, the stability determination becomes more difficult as the power of the polynomial increases. In Eq. (1), let

Conditions of analogy between the propagation of electromagnetic waves and the trajectories of particles of same spin with application to rectifying magnetrons

The object of this article is the study of the biunivocal correspondence established by the Pauli principle between the internal energy of a particle and the frequency of the associated wave in media which are the seat of strong coupling between the particles and when their spins are in a favored direction. In Section I, the determinantal forms of antisymmetrical wave functions are investigated, these being valid both for crystals and for electronic plasmas. It is shown that, starting from this determinant, two complementary series of wave functions can be constructed. Depending on the internal energy, two types of complementary particles are thus obtained: (1) free electrons associated with real waves, and (2) holes associated with evanescent waves. In Section II, a study is made of the mathematical analogy between the Schrodinger equation and the tropospheric propagation equation. It is shown that the potential energy can be assimilated to the refraction modulus and that the group velocity of the propagation around the earth can be assimilated to the group velocity of the complementary particle. By a very simple correspondence, the real modes of propagation predict the formation of holes while the imaginary modes of propagation predict the formation of free electrons. A special study is made of the analogy between the index barriers of the inversion layers and the potential barriers of the barrier layers. This analogy enables the existence to be predicted of purely electronic barrier layers without the need for any material support. In Section III, the rectification and photoconduction properties of these electronic barrier layers in magnetrons and in traveling wave magnetron detectors are considered. Their analogies with and differences from the barrier layers of p-n junctions are examined. Finally, in the conclusion, the advantages and description of radar detection arrangements devised, on these principles, by the Compagnie Generale de T.S.F. in Paris, are set out.

NOTE ON TEST FUNCTIONS FOR STABILITY

Simple relations are established between the determinantal forms of the test functions of a given polynomial and of certain related polynomials. These relations are applied to the computation of the test functions and, in particular, of the critical test determinant. It is shown that when the characteristic equation can be factorized, as when certain couplings are absent, the critical test determinant contains an irrelevant factor which is the eliminant of a pair of algebraic equations.