Search Results (6)

We extend previous research to derive three additional M-1-dimensional integral representations over the interval $[0,1]$. The prior version covered the interval $[0,\infty ]$. This extension applies to products of M Slater orbitals, since they (and wave functions derived from them) appear in quantum transition amplitudes. It enables the magnitudes of coordinate vector differences (square roots of polynomials) $|{\mathbf{x}}_1-{\mathbf{x}}_2|=\sqrt{x_1^2-2x_1{x}_{2}cos\theta +x_2^2}$ to be shifted from disjoint products of functions into a single quadratic form, allowing for the completion of its square. The M-1-dimensional integral representations of M Slater orbitals that both this extension and the prior version introduce provide alternatives to Fourier transforms and are much more compact. The latter introduce a 3M-dimensional momentum integral for M products of Slater orbitals (in M separate denominators), followed in many cases by another set of M-1-dimensional integral representations to combine those denominators into one denominator having a single (momentum) quadratic form. The current and prior methods are also slightly more compact than Gaussian transforms that introduce an M-dimensional integral for products of M Slater orbitals while simultaneously moving them into a single (spatial) quadratic form in a common exponential. One may also use addition theorems for extracting the angular variables or even direct integration at times. Each method has its strengths and weaknesses. We found that these M-1-dimensional integral representations over the interval $[0,1]$ are numerically stable, as was the prior version, having integrals running over the interval $[0,\infty ]$, and one does not need to test for a sufficiently large upper integration limit as one does for the latter approach. For analytical reductions of integrals arising from any of the three, however, there is the possible drawback for large M of there being fewer tabled integrals over $[0,1]$ than over $[0,\infty ]$. In particular, the results of both prior and current representations have integration variables residing within square roots asarguments of Macdonald functions. In a number of cases, these can be converted to Meijer G-functions whose arguments have the form $(a{x}^2+bx+c)/x$, for which a single tabled integral exists for the integrals from running over the interval $[0,\infty ]$ of the prior paper, and from which other forms can be found using the techniques given therein. This is not so for integral representations over the interval $[0,1]$. Finally, we introduce a fourth integral representation that is not easily generalizable to large M but may well provide a bridge for finding the requisite integrals for such Meijer G-functions over $[0,1]$. Full article

Named essentially after their close relationship with the modified Bessel function $K_{\nu }\left(z\right)$ of the second kind, which is known also as the Macdonald function (or, with a slightly different definition, the Basset function), the so-called Bessel polynomials $y_n\left(x\right)$ and the generalized Bessel polynomials $y_n(x;\alpha ,\beta )$ stemmed naturally in some systematic investigations of the classical wave equation in spherical polar coordinates. Our main purpose in this invited survey-cum-expository review article is to present an introductory overview of the Bessel polynomials $y_n\left(x\right)$ and the generalized Bessel polynomials $y_n(x;\alpha ,\beta )$ involving the asymmetric parameters $\alpha$ and $\beta$. Each of these polynomial systems, as well as their reversed forms ${\theta }_n\left(x\right)$ and ${\theta }_n(x;\alpha ,\beta )$, has been widely and extensively investigated and applied in the existing literature on the subject. We also briefly consider some recent developments based upon the basic (or quantum or q-) extensions of the Bessel polynomials. Several general families of hypergeometric polynomials, which are actually the truncated or terminating forms of the series representing the generalized hypergeometric function ${}_r{F}_s$ with r symmetric numerator parameters and s symmetric denominator parameters, are also investigated, together with the corresponding basic (or quantum or q-) hypergeometric functions and the basic (or quantum or q-) hypergeometric polynomials associated with ${}_r{\Phi }_s$ which also involves r symmetric numerator parameters and s symmetric denominator parameters. Full article

The affine Hecke algebra of type A has two parameters $\left(q,,,t\right)$ and acts on polynomials in N variables. There are two important pairwise commuting sets of elements in the algebra: the Cherednik operators and the Jucys–Murphy elements whose simultaneous eigenfunctions are the nonsymmetric Macdonald polynomials, and basis vectors of irreducible modules of the Hecke algebra, respectively. For certain parameter values, it is possible for special polynomials to be simultaneous eigenfunctions with equal corresponding eigenvalues of both sets of operators. These are called singular polynomials. The possible parameter values are of the form $q^m=t^{-n}$ with $2\le n\le N.$ For a fixed parameter, the singular polynomials span an irreducible module of the Hecke algebra. Colmenarejo and the author (SIGMA 16 (2020), 010) showed that there exist singular polynomials for each of these parameter values, they coincide with specializations of nonsymmetric Macdonald polynomials, and the isotype (a partition of N) of the Hecke algebra module is $\left(d,n,-,1,,,n,-,1,,,\dots ,,,n,-,1,,,r\right)$ for some $d\ge 1$. In the present paper, it is shown that there are no other singular polynomials. Full article

In a preceding paper the theory of nonsymmetric Macdonald polynomials taking values in modules of the Hecke algebra of type A (Dunkl and Luque SLC 2012) was applied to such modules consisting of polynomials in anti-commuting variables, to define nonsymmetric Macdonald superpolynomials. These polynomials depend on two parameters $\left(q,t\right)$ and are defined by means of a Yang–Baxter graph. The present paper determines the values of a subclass of the polynomials at the special points $\left(1,t,t^2,\dots \right)$ or $\left(1,t^{-1},t^{-2},\dots \right)$. The arguments use induction on the degree and computations with products of generators of the Hecke algebra. The resulting formulas involve $\left(q,t\right)$-hook products. Evaluations are also found for Macdonald superpolynomials having restricted symmetry and antisymmetry properties. Full article

For each partition $\tau$ of N, there are irreducible modules of the symmetric groups ${\mathcal{S}}_N$ and of the corresponding Hecke algebra ${\mathcal{H}}_N\left(t\right)$ whose bases consist of the reverse standard Young tableaux of shape $\tau$ . There are associated spaces of nonsymmetric Jack and Macdonald polynomials taking values in these modules. The Jack polynomials form a special case of the polynomials constructed by Griffeth for the infinite family $G\left(n,p,N\right)$ of complex reflection groups. The Macdonald polynomials were constructed by Luque and the author. For each of the groups ${\mathcal{S}}_N$ and the Hecke algebra ${\mathcal{H}}_N\left(t\right)$ , there is a commutative set of Dunkl operators. The Jack and the Macdonald polynomials are parametrized by $\kappa$ and $\left(q,t\right)$ , respectively. For certain values of these parameters (called singular values), there are polynomials annihilated by each Dunkl operator; these are called singular polynomials. This paper analyzes the singular polynomials whose leading term is $x_1^m\otimes S$ , where S is an arbitrary reverse standard Young tableau of shape $\tau$ . The singular values depend on the properties of the edge of the Ferrers diagram of $\tau$ . Full article