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The authors propose a rather elementary method to compute a family of integrals on the half line, involving positive powers of sin x and negative powers of x, depending on the integer parameters n≥q≥1.

Combinatorics, sine and cosine integral functions.

The authors prove an explicit formula to evaluate sinc-type integrals.

The proof is not present in the current literature, and it could be of interest for a large audience.

This bibliometric study provides an overview of research related to digital transformation (DT) in the tourism industry from 2013 to 2022. The goals of the research are as follows: (1) to identify the development of academic papers related to DT in the tourism industry, (2) to analyze dominant research topics and the development of research interest and research impact over time and (3) to analyze the change in research topics during the pandemic.

In this study, the authors processed 3,683 papers retrieved from the Web of Science and Scopus. The authors performed different types of bibliometric analyses to identify the development of papers related to DT in the tourism industry. To reveal latent topics, the authors implemented topic modeling based on latent Dirichlet allocation with Gibbs sampling.

The authors identified eight topics related to DT in the tourism industry: City and urban planning, Social media, Data analytics, Sustainable and economic development, Technology-based experience and interaction, Cultural heritage, Digital destination marketing and Smart tourism management. The authors also identified seven topics related to DT in the tourism industry during the Covid-19 pandemic; the largest ones are smart analytics, marketing strategies and sustainability.

To identify research topics and their development over time, the authors applied a novel methodological approach – a smart literature review. This machine learning approach is able to analyze a huge amount of documents. At the same time, it can also identify topics that would remain unrevealed by a standard bibliometric analysis.

This study aims to provide sufficient conditions for the existence of periodic solutions of the fifth-order differential equation.

The authors shall use the averaging theory, more precisely Theorem $6$.

The main results on the periodic solutions of the fifth-order differential equation (equation (1)) are given in the statement of Theorem 1 and 2.

In this article, the authors provide sufficient conditions for the existence of periodic solutions of the fifth-order differential equation.