*International Journal of Quantum Chemistry*Algebraic Lq-norms and complexity-like properties of Jacobi
polynomials-Degree and parameter asymptotics

The Jacobi polynomials
$\hat{P}_n^{(\alpha,\beta)}(x)$
conform the canonical family of hypergeometric orthogonal polynomials
(HOPs) with the two-parameter weight function
$(1-x)^\alpha (1+x)^\beta,
\alpha,\beta>-1,$ on the
interval $[-1,+1]$. The spreading of its associated probability
density (i.e., the Rakhmanov density) over the orthogonality support has
been quantified, beyond the dispersion measures (moments around the
origin, variance), by the algebraic
$\mathfrak{L}_{q}$-norms (Shannon and
R\’enyi entropies) and the monotonic complexity-like
measures of Cram\’er-Rao, Fisher-Shannon and LMC
(L\’opez-Ruiz, Mancini and Calbet) types. These
quantities, however, have been often determined in an analytically
highbrow, non-handy way; specially when the degree or the parameters
$(\alpha,\beta)$ are large. In this
work, we determine in a simple, compact form the leading term of the
entropic and complexity-like properties of the Jacobi polynomials in the
two extreme situations: ($n\rightarrow
\infty$; fixed
$\alpha,\beta$) and
($\alpha\rightarrow
\infty$; fixed $n,\beta$). These two
asymptotics are relevant \textit{per se} and because
they control the physical entropy and complexity measures of the high
energy (Rydberg) and high dimensional (pseudoclassical) states of many
exactly, conditional exactly and quasi-exactly solvable
quantum-mechanical potentials which model numerous atomic and molecular
systems.

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