Orthogonality relations for the primitives of Legendre polynomials and their applications to some spectral problems for differential operators (Q2666619)

Combining the method of integration by parts, [Eq. (2), [\textit{T. A. Garmanova}, Math. Notes 109, No. 4, 527--533 (2021; Zbl 1479.46039); translation from Mat. Zametki 109, No. 4, 500--507 (2021)]] and the definition of hypergeometric function \(_{3}F_{2}\) the authors provide orthogonality relations for the primitives of Legendre polynomials on the interval \([0,1]\), (cf. Theorem 1). Next, the authors use such orthogonality relations to establish a relation between the spectral problem for differential operators and generalized Jacobi matrices (see Section 4 and [\textit{K. V. Kholshevnikov} and \textit{V. Sh. Shaidulin}, Vestn. St. Petersbg. Univ., Math. 47, No. 1, 28--38 (2014; Zbl 1302.33010); translation from Vestn. St-Peterbg. Univ., Ser. I, Mat. Mekh. Astron. 1(59), No. 1, 55--66 (2014)]). To conclude, they show that these Jacobi matrices can be simplified when studying the embedding in the Sobolev spaces. In addition, a discussion of some aspects of approximate calculations of embedding constants is done (see Section 5 and [\textit{T. A. Garmanova} and \textit{I. A. Sheipak}, Funct. Anal. Appl. 55, No. 1, 34--44 (2021; Zbl 1485.46040); translation from Funkts. Anal. Prilozh. 55, No. 1, 43--55 (2021)]).