q-Binomial Convolution and Transformations of q-Appell Polynomials

In this paper, binomial convolution in the frame of quantum calculus is studied for the set Aq of q-Appell sequences. It has been shown that the set Aq of q-Appell sequences forms an Abelian group under the operation of binomial convolution. Several properties for this Abelian group structure Aq have been studied. A new definition of the q-Appell polynomials associated with a random variable is proposed. Scale transformation as well as transformation based on expectation with respect to a random variable is used to present the determinantal form of q-Appell sequences.

An algebraic exposition of umbral calculus with application to general linear interpolation problem: A survey

A systematic exposition of Sheffer polynomial sequences via determinantal form is given. A general linear interpolation problem related to Sheffer sequences is considered. It generalizes many known cases of linear interpolation. Numerical examples and conclusions close the paper.

PROOF OF THE DETERMINANTAL FORM OF THE SPONTANEOUS MAGNETIZATION OF THE SUPERINTEGRABLE CHIRAL POTTS MODEL

The superintegrable chiral Potts model has many resemblances to the Ising model, so it is natural to look for algebraic properties similar to those found for the Ising model by Onsager, Kaufman and Yang. The spontaneous magnetization ℳr can be written in terms of a sum over the elements of a matrix Sr. The author conjectured the form of the elements, and this conjecture has been verified by Iorgov et al. The author also conjectured in 2008 that this sum could be expressed as a determinant, and has recently evaluated the determinant to obtain the known result for ℳr. Here we prove that the sum and the determinant are indeed identical expressions. Since the order parameters of the superintegrable chiral Potts model are also those of the more general solvable chiral Potts model, this completes the algebraic calculation of ℳr for the general model.

"VECTAL" REPRESENTATION OF THE WAVE FUNCTION AND ITS PHYSICAL MEANING

A new form of one-electron wave function, "vectal," is introduced. It is shown that an arbitrary CI geminal and a certain class of many-electron wave functions can be represented in a single-determinantal form in terms of the vectal. Eigenvalue equations for the vectal, similar to that of the Hartree–Fock theory, are derived and the vectal representation is shown to enable a formal interpretation of the CI theory in the Hartree–Fock manner. The eigenvalue, vectal energy, is interpreted as the negative of an ionization potential, in Koop-man's sense, of the system described by the CI wave function. It is also shown that the expectation value of any one-electron operator, [Formula: see text], where Ψ is the CI wave function, is expressible in terms of the expectation values of the same operator with respect to the vectals. The vectals are interpreted as the one-electron wave function in the CI space.A possible application of the vectal representation is briefly described, and the relationship between the vectal representation and the "scalar representation" is discussed.

A nomogram for calculating extended terms

The simplest form of nomogram is a graphical device for representing a functional relationship between three variables in a manner which is often more convenient for practical reference than that of plotting a series of contours for chosen values of one of the three variables on a Cartesian graph. We develop the basis of such a nomogram by means of analytical geometry in this section.The condition that three points (ξ1, η1), (ξ2, η2) and (ξ3, η3) shall be collinear is commonly expressed by means of the determinantIf the relationship between three variablesa, bandcwhich we wish to represent nomographically isF (a,b,c) = o, (2)and we can express (2) in the determinantal form [similar to (1)]wheref1f2(a),f2(b),f3(c),g1(a),g2(b) andg3(c) represent (generally) different functions ofa, bandc, it is apparent that we can associateany three particular values ofa, bandcsatisfying (2) with the pointson a Cartesian graph; and they will be collinear because (3) is of the same form as (1).