On discrete orthogonal U-Bernoulli Korobov-type polynomials

Abstract

The primary objective of this paper is to introduce and examine the new class of discrete orthogonal polynomials called $U$-Bernoulli Korobov-type polynomials. Furthermore, we derive essential recurrence relations and explicit representations for this polynomial class. Most of the results are proven through the utilization of generating function methods. Lastly, we place particular emphasis on investigating the orthogonality relation associated with these polynomials.

Keywords

• Bernoulli polynomials
• U–Bernoulli Korobov
• discrete orthogonal polynomials

References

1. F. Avram, M.S. Taqqu: Noncentral limit theorems and Appell polynomials, Ann. Probab., 15 (1987), 767–775.
2. J. Babini: Polinomios generalizados de Bernoulli y sus correlativos, Rev. Mat. Hisp.-Am., 10 (4) (1935), 23–25.
3. L. Carlitz: A note on Bernoulli and Euler polynomials of the second kind, Scr. Math., 25 (1961), 323–330.
4. L. Carlitz: Degenerate Stirling, Beroulli and Eulerian numbers, Utilitas Math., 15 (1979), 51–88.
5. C. V. L. Charlier: Über die darstellung willkurlicher funktionen, Arkiv för Matematik, Astronomi och Fysik., 25 (3) (1970), 1–11.
6. A.G. Asensi, E. Labarga, E. J.M. Ceniceros and J., Varona: Boole-Dunklpolynomialsandgeneralizations, Rev.RealAcad.Cienc.ExactasFis.Nat.Ser.A-Mat., 118 (2024), Article ID: 16.
7. I. Gavrea, M. Ivan: Approximation properties related to the Bell polynomials, Constr. Math. Anal., 4 (2) (2021), 253–259.
8. R.L. Graham, D.E. Knuth, O: Patashnik, Concrete Mathematics, A Foundation for Computer Science, 2nd ed., AddisonWesley, Reading, MA (1994).

9. C. Jordan: Sur les polynomes analogues aux polynomes de Bernoulli et sur des formules de sommation analogues à celle de MacLaurin-Euler, Acta Szeged, 4 (1929), 130–150.
10. V. Kostov: The disconnectedness of certain sets defined after uni-variate polynomials, Constr. Math. Anal., 5 (3) (2022), 119–133.
11. D.E. Knuth: Two notes on notation, Am. Math. Mon., 99 (1992), 403–422.
12. Korobov: Special polynomials and their applications diophantine approximations, Math. Notes, 2 ( 1996), 77–89.
13. J. Meixner: Orthogonale polynomsysteme mit einem besonderen Gestalt der erzeugenden funktion, J. London Math. Soc., 9 (1934), 6–13.
14. Y. Quintana, W. Ramírez and A. Urieles: Euler matrices and their algebraic properties revisited, Appl. Math. Inf. Sci., 14 (4) (2020), 583–596.
15. W. Ramírez, D. Bedoya, A. Urieles, C. Cesarano and M. Ortega: New U-Bernoulli, U-Euler and U-Genocchi Polynomials and Their Matrices, Carpathian Math. Publ., 15 (2) (2023), 449–467.
16. J. Rey Pastor: Polinomios correlativos de los de Bernoulli, Bol. Semin. Mat. Argent., 1 (3) (1929) 1–10.
17. S. Zagorodnyuk: On a Family of Hypergeometric Sobolev Orthogonal Polynomials on the Unit Circle, Constr. Math. Anal., 3 (2) (2020), 75–84.