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The distributions $P(M_L,M_S)$ of the total magnetic quantum numbers $M_L$ and $M_S$ for $N$ electrons of angular momentum $\ell$, as well as the enumeration of $LS$ spectroscopic terms and spectral lines, are crucial for the calculation of atomic structure and spectra, in particular for the modeling of emission or absorption properties of hot plasmas. However, no explicit formula for $P(M_L,M_S)$ is known yet. In the present work, we show that the generating function for the cumulants, which characterize the distribution, obeys a recurrence relation, similar to the Newton-Girard identities relating elementary symmetric polynomials to power sums. This enables us to provide an explicit formula for the generating function. We also analyze the possibility of representing the $P(M_L,M_S)$ distribution by a bi-variate Gram-Charlier series, which coefficients are obtained from the knowledge of the exact moments of $P(M_L,M_S)$. It is shown that a simple approximation is obtained by truncating this series to the first few terms, though it is not convergent., Comment: submitted to J. Phys. B: At. Mol. Opt. Phys

We give new explicit representations as well as new generating functions for the associated Meixner, Charlier, Laguerre, and Krawtchouk polynomials. The obtained results are then used to derive new generating functions and convolution-type formulas of the corresponding classical polynomials. Some consequences of our results are also mentioned., Comment: 14 pages. arXiv admin note: substantial text overlap with arXiv:2103.08367

We analyze the asymptotic distribution of roots of Charlier polynomials with negative parameter depending linearly on the index. The roots cluster on curves in the complex plane. We determine implicit equations for these curves and deduce the limiting density of the root distribution supported on these curves. The proof is based on a determination of the limiting Cauchy transform in a specific region and a careful application of the saddle point method. The obtained result represents a solvable example of a more general open problem., Comment: 26 pages

In this paper we tackle the asymptotic behavior of a family of orthogonal polynomials with respect to a nonstandard inner product involving the forward operator {\Delta}. Concretely, we treat the generalized Charlier weights in the framework of {\Delta}--Sobolev orthogonality. We obtain an asymptotic expansion for this orthogonal polynomials where the falling factorial polynomials play an important role., Comment: 18 pages

In this paper we consider two examples of certain recurrence relations, or nonlinear discrete dynamical systems, that appear in the theory of orthogonal polynomials, from the point of view of Sakai's geometric theory of Painlev\'e equations. On one hand, this gives us new examples of the appearance of discrete Painlev\'e equations in the theory of orthogonal polynomials. On the other hand, it serves as a good illustration of the effectiveness of a recently proposed procedure on how to reduce such recurrences to some canonical discrete Painlev\'e equations. Of particular interest is the fact that both recurrences are regularized on the same family of rational algebraic surfaces, but at the same time their dynamics are non-equivalent., Comment: 28 pages

The Cholesky factorization of the moment matrix is considered for the generalized Charlier, generalized Meixner, and Gauss hypergeometric discrete orthogonal polynomials. For the generalized Charlier, we present an alternative derivation of the Laguerre-Freud relations found by Smet and Van Assche. Third-order and second-order nonlinear ordinary differential equations are found for the recursion coefficient $\gamma_n$, that happen to be forms of the Painlev\'e $\text{deg-P}_{\text V}$ in disguise. Laguerre-Freud relations are also found for the generalized Meixner case, which are compared with those of Smet and Van Assche. Finally, the Gauss hypergeometric discrete orthogonal polynomials, also known as generalized Hahn of type I, are also studied. Laguerre-Freud equations are found, and the differences with the equations found by Dominici and by Filipuk and Van Assche are provided., Comment: Some minor typos corrected In the new version the bibliography has been updated and the text improved. Typo in Theorem 8 corrected. arXiv admin note: substantial text overlap with arXiv:2107.01747

We derive uniform and non-uniform asymptotics of the Charlier polynomials by using difference equation methods alone. The Charlier polynomials are special in that they do not fit into the framework of the turning point theory, despite the fact that they are crucial in the Askey scheme. In this paper, asymptotic approximations are obtained respectively in the outside region, an intermediate region, and near the turning points. In particular, we obtain uniform asymptotic approximation at a pair of coalescing turning points with the aid of a local transformation. We also give a uniform approximation at the origin by applying the method of dominant balance and several matching techniques., Comment: 31 pages, 1 figure (in two separated files)

Strong stability preserving (SSP) coefficients govern the maximally allowable step-size at which positivity or contractivity preservation of integration methods for initial value problems is guaranteed. In this paper, we show that the task of computing linear SSP coefficients of explicit one-step methods is, to a certain extent, equivalent to the problem of characterizing positive quadratures with integer nodes with respect to Poisson-Charlier measures. Using this equivalence, we provide sharp upper and lower bounds for the optimal linear SSP coefficients in terms of the zeros of generalized Laguerre orthogonal polynomials. This in particular provides us with a sharp upper bound for the optimal SSP coefficients of explicit Runge-Kutta methods. Also based on this equivalence, we propose a highly efficient and stable algorithm for computing these coefficients, and their associated optimal linear SSP methods, based on adaptive spectral transformations of Poisson-Charlier measures. The algorithm possesses the remarkable property that its complexity depends only on the order of the method and thus is independent of the number of stages. Our results are achieved by adapting and extending an ingenious technique by Bernstein in his seminal work on absolutely monotonic functions. Moreover, the techniques introduced in this work can be adapted to solve the integer quadrature problem for any positive discrete multi-parametric measure supported on $\mathbb{N}$ under some mild conditions on the zeros of the associated orthogonal polynomials., Comment: 50 pages, 5 figures