Kargın, Levent; Çay, Emre Semiorthogonality of geometric polynomials. (English) Zbl 1491.11030 Mediterr. J. Math. 19, No. 3, Paper No. 129, 9 p. (2022). Geometric polynomials are also termed as Fubini polynomials which is most commonly used in literature.Special polynomials and numbers have significant roles in various branches of mathematics, theoretical physics, and engineering. The problems arising in mathematical physics and engineering are framed in terms of differential equations. Most of these equations can only be treated by using various families of special polynomials which provide new means of mathematical analysis. They are widely used in computational models of scientific and engineering problems. In addition, these special polynomials allow the derivation of different useful identities in a straightforward way and help in introducing new families of special polynomials.The Fubini-type polynomials (or geometric-type polynomials) appear in combinatorial mathematics and play an important role in the theory and applications of mathematics, thus many number theory and combinatorics experts have extensively studied their properties and obtained series of interesting results. The Fubini-type numbers and polynomials are related Bernoulli numbers with diverse extensions and proven to be an effective tool in different topics in combinatorics and analysis.In the related paper, the authors have examined the semiorthogonality of Fubini and higher order Fubini polynomials. They showed that the integrals of products of the higher order Fubini polynomials can be evaluated in terms of Bernoulli numbers, which means that the higher order geometric polynomials are also semiorthogonal. As applications, they have given some new explicit formulas for Bernoulli and \(p\)-Bernoulli numbers. Reviewer: Uğur Duran (Iskenderun) MSC: 11B83 Special sequences and polynomials 11B68 Bernoulli and Euler numbers and polynomials 33C47 Other special orthogonal polynomials and functions Keywords:higher order geometric polynomials; Bernoulli numbers; \(p\)-Bernoulli numbers; semiorthogonal polynomials; Fubini polynomials; higher order Fubini polynomials PDFBibTeX XMLCite \textit{L. Kargın} and \textit{E. Çay}, Mediterr. J. Math. 19, No. 3, Paper No. 129, 9 p. (2022; Zbl 1491.11030) Full Text: DOI References: [1] Bell, ET, Exponential polynomials, Ann. Math., 35, 258-277 (1934) · Zbl 0009.21202 [2] Boyadzhiev, KN, A series transformation formula and related polynomials, Int. J. Math. Math. Sci., 23, 3849-3866 (2005) · Zbl 1086.05006 [3] Boyadzhiev, KN, Apostol-Bernoulli functions, derivative polynomials and Eulerian polynomials, Adv. Appl. 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