Log-convexity and log-concavity of hypergeometric-like functions

We find sufficient conditions for log-convexity and log-concavity for the functions of the forms $a\mapsto\sum{f_k}(a)_kx^k$, $a\mapsto\sum{f_k}\Gamma(a+k)x^k$ and $a\mapsto\sum{f_k}x^k/(a)_k$. The most useful examples of such functions are generalized hypergeometric functions. In particular, we generalize the Tur\'{a}n inequality for the confluent hypergeometric function recently proved by Barnard, Gordy and Richards and log-convexity results for the same function recently proved by Baricz. Besides, we establish a reverse inequality which complements naturally the inequality of Barnard, Gordy and Richards. Similar results are established for the Gauss and the generalized hypergeometric functions. A conjecture about monotonicity of a quotient of products of confluent hypergeometric functions is made.Comment: 13 pages, no figure

On a two variable class of Bernstein-Szego measures

The one variable Bernstein-Szego theory for orthogonal polynomials on the real line is extended to a class of two variable measures. The polynomials orthonormal in the total degree ordering and the lexicographical ordering are constructed and their recurrence coefficients discussed.Comment: minor change

Preliminary notes on dual relevance of ITS sequences and pigments in Hygrocybe taxonomy

The relationships based on ITS sequences of 48 Hygrocybe s.l. specimens were studied and compared with previously described taxonomic groups. Our specimens formed two well separated genetic groups. The first one includes the species characterized by vivid yellow and red colours, while species belonging to other clades were pallid or pale brown, and in most cases with pink or olive tones. This separation is supported by the presence of muscaflavin pigments among some species referred to Hygrocybe (Bresinsky & Kronawitter 1986). The subgenera distinguished by morphological features can be relatively well recognized on phylogenetic trees, however, the majority of sections were not supported. Variability in the ITS region of Hygrocybe species is unusually high. In some cases sequences differed by more than 25 %, and the lengths of ITS regions also showed large differences. Taxa that were considered as closely related, e.g. the H. conica aggregate, were found to have identical or highly similar sequences. Our results seem to confirm the taxonomic concept of Bresinsky (2008) who proposed the division of the genus Hygrocybe. Hence H. calyptriformis and all examined members of subg. Gliophorus (H. irrigata, H. laeta, H. nitrata, H. psittacina) and subg. Cuphophyllus could be excluded from the genus Hygrocybe s.str. Based on these results further research using DNA markers at the intergeneric level is suggested to revaluate the taxonomy of former Hygrocybe species

An Isoperimetric Inequality for Fundamental Tones of Free Plates

We establish an isoperimetric inequality for the fundamental tone (first nonzero eigenvalue) of the free plate of a given area, proving the ball is maximal. Given $\tau>0$, the free plate eigenvalues $\omega$ and eigenfunctions $u$ are determined by the equation $\Delta\Delta u-\tau\Delta u = \omega u$ together with certain natural boundary conditions. The boundary conditions are complicated but arise naturally from the plate Rayleigh quotient, which contains a Hessian squared term $|D^2u|^2$. We adapt Weinberger's method from the corresponding free membrane problem, taking the fundamental modes of the unit ball as trial functions. These solutions are a linear combination of Bessel and modified Bessel functions.Comment: PhD thesis. Papers are in preparatio

On Kaluza's sign criterion for reciprocal power series

T. Kaluza has given a criterion for the signs of the power series of a function that is the reciprocal of another power series. In this note the sharpness of this condition is explored and various examples in terms of the Gaussian hypergeometric series are given. A criterion for the monotonicity of the quotient of two power series due to M. Biernacki and J. Krzy\.z is applied.Comment: 13 page

Some Orthogonal Polynomials Arising from Coherent States

We explore in this paper some orthogonal polynomials which are naturally associated to certain families of coherent states, often referred to as nonlinear coherent states in the quantum optics literature. Some examples turn out to be known orthogonal polynomials but in many cases we encounter a general class of new orthogonal polynomials for which we establish several qualitative results.Comment: 21 page

A conjecture on Exceptional Orthogonal Polynomials

Exceptional orthogonal polynomial systems (X-OPS) arise as eigenfunctions of Sturm-Liouville problems and generalize in this sense the classical families of Hermite, Laguerre and Jacobi. They also generalize the family of CPRS orthogonal polynomials. We formulate the following conjecture: every exceptional orthogonal polynomial system is related to a classical system by a Darboux-Crum transformation. We give a proof of this conjecture for codimension 2 exceptional orthogonal polynomials (X2-OPs). As a by-product of this analysis, we prove a Bochner-type theorem classifying all possible X2-OPS. The classification includes all cases known to date plus some new examples of X2-Laguerre and X2-Jacobi polynomials

Roots of the derivative of the Riemann zeta function and of characteristic polynomials

We investigate the horizontal distribution of zeros of the derivative of the Riemann zeta function and compare this to the radial distribution of zeros of the derivative of the characteristic polynomial of a random unitary matrix. Both cases show a surprising bimodal distribution which has yet to be explained. We show by example that the bimodality is a general phenomenon. For the unitary matrix case we prove a conjecture of Mezzadri concerning the leading order behavior, and we show that the same follows from the random matrix conjectures for the zeros of the zeta function.Comment: 24 pages, 6 figure

Multipoint Schur algorithm and orthogonal rational functions: convergence properties, I

Classical Schur analysis is intimately connected to the theory of orthogonal polynomials on the circle [Simon, 2005]. We investigate here the connection between multipoint Schur analysis and orthogonal rational functions. Specifically, we study the convergence of the Wall rational functions via the development of a rational analogue to the Szeg\H o theory, in the case where the interpolation points may accumulate on the unit circle. This leads us to generalize results from [Khrushchev,2001], [Bultheel et al., 1999], and yields asymptotics of a novel type.Comment: a preliminary version, 39 pages; some changes in the Introduction, Section 5 (Szeg\H o type asymptotics) is extende

Idebenone and Resveratrol Extend Lifespan and Improve Motor Function of HtrA2 Knockout Mice

Heterozygous loss-of-function mutation of the human gene for the mitochondrial protease HtrA2 has been associated with increased risk to develop mitochondrial dysfunction, a process known to contribute to neurodegenerative disorders such as Huntington's disease (HD) and Parkinson's disease (PD). Knockout of HtrA2 in mice also leads to mitochondrial dysfunction and to phenotypes that resemble those found in neurodegenerative disorders and, ultimately, lead to death of animals around postnatal day 30. Here, we show that Idebenone, a synthetic antioxidant of the coenzyme Q family, and Resveratrol, a bioactive compound extracted from grapes, are both able to ameliorate this phenotype. Feeding HtrA2 knockout mice with either compound extends lifespan and delays worsening of the motor phenotype. Experiments conducted in cell culture and on brain tissue of mice revealed that each compound has a different mechanism of action. While Idebenone acts by downregulating the integrated stress response, Resveratrol acts by attenuating apoptosis at the level of Bax. These activities can account for the delay in neuronal degeneration in the striata of these mice and illustrate the potential of these compounds as effective therapeutic approaches against neurodegenerative disorders such as HD or PD