q-Differential equations for q-classical polynomials and q-Jacobi-Stirling numbers

We introduce, characterise and provide a combinatorial interpretation for the so-called q-Jacobi–Stirling numbers. This study is motivated by their key role in the (reciprocal) expansion of any power of a second order q-differential operator having the q-classical polynomials as eigenfunctions in terms of other even order operators, which we explicitly construct in this work. The results here obtained can be viewed as the q-version of those given by Everitt et al. and by the first author, whilst the combinatorics of this new set of numbers is a q-version of the Jacobi–Stirling numbers given by Gelineau and the second author

Cape Verde: The Case for Euroization

After 10 years of a fixed exchange rate against the euro and a deepening integration with the European Union (EU), the authorities of Cape Verde maintain a strong commitment to nominal stability and are now considering the official euroization of the country. Compared to the current pegging, euroization could be costly if the economic conditions of Cape Verde were to require control over the interest rates and the exchange rate. Given the strong economic and financial integration between Cape Verde and Europe, and the fact that Cape Verde records inflation rates at levels that are similar to those of the European Monetary Union (EMU), the relevant issue is whether the European Central Bank (ECB) monetary policy fits the needs of Cape Verde. In order to answer this question, we empirically assess the synchronization between the business cycle of Cape Verde and the business cycle of the EMU. For that purpose, we compute output gaps and then use conventional correlation measures as well as other indicators recently suggested in the literature. Replicating the methodology for each of the current 27 EU members, our results show that Cape Verde ranks better than several EU countries and even better than some EMU countries. We thus argue that there is a strong case for the euroization of Cape Verde. Euroization would secure the benefits already attained with the pegging to the euro and would warrant additional benefits, most likely with no relevant costs stemming from inappropriate ECB monetary policies.Africa, Cape Verde, European Monetary Union, Euroization, Business Cycles

Unique positive solution for an alternative discrete Painlevé I equation

We show that the alternative discrete Painleve I equation has a unique solution which remains positive for all n >0. Furthermore, we identify this positive solution in terms of a special solution of the second Painleve equation involving the Airy function Ai(t). The special-function solutions of the second Painleve equation involving only the Airy function Ai(t) therefore have the property that they remain positive for all n>0 and all t>0, which is a new characterization of these special solutions of the second Painlevé equation and the alternative discrete Painlevé I equation

Spectral approximation of convolution operator

We develop a unified framework for constructing matrix approximations for the convolution operator of Volterra type defined by functions that are approximated using classical orthogonal polynomials on [?1, 1]. The numerically stable algorithms we propose exploit recurrence relations and symmetric properties satisfied by the entries of these convolution matrices. Laguerrebased convolution matrices that approximate Volterra convolution operator defined by functions on [0, ?] are also discussed for the sake of completeness

Multiple orthogonal polynomials with respect to Gauss' hypergeometric function

A new set of multiple orthogonal polynomials of both type I and type II with respect to two weight functions involving Gauss' hypergeometric function on the interval (0,1) is studied. This type of polynomials has direct applications in the investigation of singular values of products of Ginibre random matrices and are connected with branched continued fractions and total-positivity problems in combinatorics. The pair of orthogonality measures is shown to be a Nikishin system and to satisfy a matrix Pearson-type differential equation. The focus is on the polynomials whose indices lie on the step-line, for which it is shown that the differentiation gives a shift in the parameters, therefore satisfying Hahn's property. We obtain Rodrigues-type formulas for type I polynomials and functions, while a more detailed characterization is given for the type II polynomials (aka 2-orthogonal polynomials) that include an explicit expression as a terminating hypergeometric series, a third-order differential equation, and a third-order recurrence relation. The asymptotic behavior of their recurrence coefficients mimics those of Jacobi–Piñeiro polynomials, based on which their asymptotic zero distribution and a Mehler–Heine asymptotic formula near the origin are given. Particular choices of the parameters and confluence relations give some known systems such as special cases of the Jacobi–Piñeiro polynomials, Jacobi-type 2-orthogonal polynomials, components of the cubic decomposition of threefold symmetric Hahn-classical polynomials, and multiple orthogonal polynomials with respect to confluent hypergeometric functions of the second kind

Volterra-type convolution of classical polynomials

We present a general framework for calculating the Volterra-type convolution of polynomials from an arbitrary polynomial sequence {Pk(x)}k?0 with degPk(x)=k. Based on this framework, series representations for the convolutions of classical orthogonal polynomials, including Jacobi and Laguerre families, are derived, along with some relevant results pertaining to these new formulas

Three-fold symmetric Hahn-classical multiple orthogonal polynomials

We characterize all the multiple orthogonal three-fold symmetric polynomial sequences whose sequence of derivatives is also multiple orthogonal. Such a property is commonly called the Hahn property and it is an extension of the concept of classical polynomials to the context of multiple orthogonality. The emphasis is on the polynomials whose indices lie on the step line, also known as 2-orthogonal polynomials. We explain the relation of the asymptotic behavior of the recurrence coefficients to that of the largest zero (in absolute value) of the polynomial set. We provide a full characterization of the Hahn-classical orthogonality measures supported on a 3-star in the complex plane containing all the zeros of the polynomials. There are essentially three distinct families, one of them 2-orthogonal with respect to two confluent functions of the second kind. This paper complements earlier research of Douak and Maroni

Keratin-based peptide: biological evaluation and strengthening properties on relaxed hair

A peptide based on a fragment of hair keratin type II cuticular protein, keratin peptide (KP), was studied as a possible strengthening agent for weakened relaxed hair. The peptide was prepared both in aqueous water formulation (WF) and organic solvent formulations (OF), to determine the effect of organic solvents on peptide interaction with hair and the differences in hair recovery. Both peptide formulations were shown to improve mechanical and thermal properties of weakened hair with peptide in OF showing the stronger effect. As a potential new hair care product, and so would necessitate contact with skin, the cytotoxicity and genotoxicity of the peptide were also evaluated through different methodologies (Alamar Blue assay, 2′-7′-dichlorofluorescein probe, cell morphology and growth and evaluation of DNA damage by an alkaline version of the comet assay) in skin fibroblasts. These tests are indicators of the potential of peptide to cause irritation on skin or to be carcinogenic, respectively. The peptide in WF did not cause cytotoxicity or genotoxicity in any of the concentrations tested. The presence of OF, however, induced a 20% decrease in cell viability in all of the range of concentrations used after 72-h incubation. Moreover, OF inhibited cell growth and was considered genotoxic at first contact with cells. The peptide was therefore considered a promising strengthening agent for hair and was shown to be innocuous when applied in WF.The work was supported by a PhD Grant Fellowship (scholarship SFRH/BD/38363/2007) from FCT "Fundacao para a Ciencia e Tecnologia", Portugal. The authors have declared no conflict of interest