Conference abstracts

Session C7 - Special Functions and Orthogonal Polynomials

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Linearization and Krein-like functionals of hypergeometric orthogonal polynomials by means of Lauricella functions

Irene Valero Toranzo

Departamento de Física Atómica, Molecular y Nuclear, Universidad de Granada, Spain   -   ivtoranzo@ugr.es

The Krein-like functionals of the hypergeometric orthogonal polynomials $\{p_{n}(x) \}$ with kernel of the form $x^{s}[\omega(x)]^{\beta}p_{m_{1}}(x)\ldots p_{m_{r}}(x)$, being $\omega(x)$ the weight function on the interval $\Delta\in\mathbb{R}$, are considered and analytically computed by means of the Srivastava linearization method. The particular ($r=2$)-functionals of the form $x^{s}[\omega(x)]^{\beta}p_{n}(x)p_{m}(x)$, which are particularly relevant in quantum physics, are explicitly given in terms of the degrees and the characteristic parameters of the polynomials. They include the well-known power moments $\langle x^{s}\rangle_{n}$ and the novel Krein-like moments $\langle [\omega(x)]^{k}\rangle_{n}$, where $\langle f(x)\rangle_{n} :=\int_{\Delta} f(x)\rho_{n}(x)\, dx$ and $\rho_{n}(x) = \omega(x)p_{n}^{2}(x)$ is the Rakhmanov density of the polynomials. Moreover, various related types of exponential and logarithmic functionals of the form $\langle x^{k}e^{-\alpha x}\rangle_{n}$, $\langle (\log x)^{k}\rangle_{n}$ and $\langle [\omega(x)]^{k}\log \omega(x)\rangle_{n}$, are also investigated.

References

J. S Dehesa, J. J. Moreno-Balcázar, I. V. Toranzo, Linearization and Krein-like functionals of hypergeometric orthogonal polynomials by means of Lauricella functions. Preprint in preparation 2017.

P. Sánchez-Moreno, J. S. Dehesa, A. Zarzo, A. Guerrero, Rényi entropies, $L_{q}$ norms and linearization of powers of hypergeometric orthogonal polynomials by means of multivariate special functions, Applied Mathematics and Computation 223 (2013) 25–33.

H. M. Srivastava, A. W. Niukkanen, Some Clebsch-Gordan Type Linearization Relations and Associated Families of Dirichlet Integrals, Math. Comput. Model 37 (2003) 245-250.

E. A. Rakhmanov, On the asymptotics of the ratio of orthogonal polynomials, Math. USSR Sb. 32 (1977) 199-213.

M. Krein, N. I. Ahiezer, Some questions in the Theory of Moments (AMS, Providence, 1962).

F. G. Parr and W. Yang, Density Functional Theory of Atoms and Molecules (Oxford University Press, New York, 1989).

Joint work with J. S. Dehesa (Departamento de Física Atómica, Molecular y Nuclear, Universidad de Granada, Granada 18071, Spain; Instituto Carlos I de Física Teórica y Computacional, Universidad de Granada, Granada 18071, Spain) and J. J. Moreno-Balcázar (Departamento de Matemáticas, Universidad de Almería, Almería 04120, Spain; Instituto Carlos I de Física Teórica y Computacional, Universidad de Granada, Granada 18071, Spain).

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