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A mixed Lagrange–Bernoulli tensor product polynomial operator for bivariate functions of class ( m , n ) with only boundary data on the rectangle is presented. Properties of interpolation and error bounds are studied. Application to functions approximation, solution of Poisson equations with Dirichelet’s conditions and numerical cubature are proposed. Numerical results are also given.

For polynomial sequence of binomial type we propose a new recurrence formula equivalent to a determinantal form of the sequence, from which known and unknown properties are deducted. Relationships whit $${\delta}$$ –functionals, $${\delta}$$ –operators and generating function are pointed out. Finally, some classical examples are studied, highlighting interesting determinantal identities.

A systematic exposition of Sheffer polynomial sequences via determinantal form is given. A general linear interpolation problem related to Sheffer sequences is considered. It generalizes many known cases of linear interpolation. Numerical examples and conclusions close the paper.

Background Polypharmacy presents a serious and significant public health challenge as average lifespans increase. The objective of this study was to identify patterns of polypharmacy before dementia diagnosis, their prevalence, and possible complications. Methods We used electronic health records in the Welsh general practitioners’ database from 1990 to 2015, via the Secure Anonymised Information Linkage Databank. Cohort selection, based on a previously validated phenotype, was done with the use of Read Codes to identify any dementia diagnosis within the dataset. No age restriction was applied, but complete data for each patient was required. Analysis was stratified by sex and age, but not dementia type. Medications were identified by Read Codes, and split into four 5-year sub-periods. Factor analysis was used for patients taking at least three medicines in each period. Findings We identified 33 451 patients with a dementia diagnosis from more than 42 million rows of health records. 22 011 (65·8%) patients in the cohort were female, with mean age at dementia diagnosis of 72·75 years (SD 13·59). In sub-period 1 (0–5 years pre-diagnosis), 22 157 (82·2%) of 26 968 patients were taking three or more medications, and three clusters were identified: medicines for urinary or respiratory infections, arthritis or rheumatism, and heart disease (17 947 [66·5%] patients); medicines for urinary or respiratory infections, arthritis or rheumatism, heart disease, and depression (5938 [22·0%]); and medicines for osteoarthritis (701 [2·6%]). By contrast, in sub-period 4 (15–20 years pre-diagnosis), 421 (5·5%) of 7660 patients were taking three or more medications, and five patterns emerged: medicines for infections, arthropathy, and cardiovascular disease (421 [5·5%] patients); medicines for anxiety and acute respiratory infection (184 [2·4%]); medicines for acute respiratory infection and cardiovascular disease (161 [2·1%]); medicines for dermatological disease (23 [0·3%]); and medicines for asthma and other acute respiratory infections (four [0·1%]). 23 891 (71·4%) patients in all periods took medications for arthritis, heart disease, and infections, with or without medications for mental health. Interpretation The earlier before dementia diagnosis, the greater the number of medicine clusters are observed, with lower prevalence of each cluster. Understanding how dementia medicines interact and cause complications with other groups of medicines is fundamental. These patterns can inform safe prescribing practices before diagnosis in selecting medicines with a low anticholinergic burden to minimise their effect on cognitive impairments. Further analysis of these types of polypharmacy among people without dementia will be done in the future. Funding Health Data Research UK (NIWA1).

A matrix approach to Sheffer polynomial sequences is proposed; in particular, two different determinantal forms of Sheffer sequences are given, the one as the function of a polynomial sequence of binomial type and the other as the function of the canonical base xi. The equivalence with the classical definitions of Sheffer and Roman and Rota is proven. Then, elementary matrix algebra tools are employed to reveal the known and unknown properties of Sheffer polynomials. Finally, classical and non-classical examples are also considered.

A general linear interpolation problem is considered. We will call it the Appell interpolation problem because the solution can be expressed by a basis of Appell polynomials. Some classical and non-classical examples are also considered. Finally, numerical calculations are given.

A new definition by means of a determinantal form for Appell (1880) [1] polynomials is given. General properties, some of them new, are proved by using elementary linear algebra tools. Finally classic and non-classic examples are considered and the coefficients, calculated by an ad hoc Mathematica code, for particular sequences of Appell polynomials are given.

A Birkhoff interpolation problem is studied. The explicit solution and estimation of remainder is given. The cases of equidistant and Chebyshev nodes are considered. This problem is of interest in collocation methods of boundary value problem (Costabile and Napoli in Far East J. Appl. Math. 25(3):289–305, 2006; Costabile and Longo in Applied and Industrial Mathematics in Italy III, 2009, in press). In addition an application to quadrature is given.

Sequences of polynomials, verifying the (▭), nowadays called Appell polynomials, have been well studied because of their remarkable applications not only in different branches of mathematics ([2], [3]) but also in theoretical physics and chemistry ([4], [5]). In 1936 an initial bibliography was provided by Davis (p. 25[6]). In 1939 Sheffer ([7]) introduced a new class of polynomials which extends the class of Appell polynomials; he called these polynomials of type zero, but nowadays they are called Sheffer polynomials. Sheffer also noticed the sim‐ ilarities between Appell polynomials and the umbral calculus, introduced in the second half of the 19th century with the work of such mathematicians as Sylvester, Cayley and Blissard (for examples, see [8]). The Sheffer theory is mainly based on formal power series. In 1941 Steffensen ([9]) published a theory on Sheffer polynomials based on formal power series too. However, these theories were not suitable as they did not provide sufficient computational tools. Afterwards Mullin, Roman and Rota ([10], [11], [12]), using operators method, gave a beautiful theory of umbral calculus, including Sheffer polynomials. Recently, Di Bucchianico and Loeb ([13]) summarized and documented more than five hundred old and new findings related to Appell polynomial sequences. In last years attention has centered on finding a