Algebraic theory of degenerate general bivariate Appell polynomials and related interpolation hints

Year 2024, Volume: 53 Issue: 1, 1 - 21, 29.02.2024

Subuhi Khan Mehnaz Haneef Mumtaz Riyasat

https://doi.org/10.15672/hujms.1183047

Cited By: 1

Abstract

The algebraic study of polynomials based on determinant representations is important in many fields of mathematics, ranging from algebraic geometry to optimization. The motivation to introduce determinant expressions of special polynomials comes from the fact that they are useful in scientific computing in solving systems of equations effectively. It is critical for this application to have determinant representations not just for single valued polynomials but also for bivariate polynomials. In this article, a family of degenerate general bivariate Appell polynomials is introduced. Several different explicit representations, recurrence relations, and addition theorems are established for this family. With the aid of different recurrence relations, we establish the determinant expressions for the degenerate general bivariate Appell polynomials. We also establish determinant definitions for degenerate general polynomials. Several examples are framed as the applications of this family and their graphical representations are shown. As concluding remarks, we propose a linear interpolation problem for these polynomials and some hints are provided.

Keywords

degenerate general bivariate Appell polynomials, determinant expressions, degenerate bivariate Laguerre-Appell sequences, interpolation hints

References

• [1] P. Appell, Sur une classe de polynˆomes, Ann. Sci. ´E cole. Norm. Sup. 9, 119-144, 1880.
• [2] D. Bedoya, M. Ortega, W. Ramírez and A. Urieles, New biparametric families of Apostol-Frobenius-Euler polynomials of level m, Mat. Stud. 55, 10-23, 2021.
• [3] G. Bretti, C. Cesarano and P. Ricci, Laguerre type exponentials and generalized Appell polynomials, Comput. Math. Appl. 48, 833-839, 2004.
• [4] L. Carlitz, A degenerate Staudt-Clausen theorem, Arch. Math. (Basel) 7, 28-33, 1956.
• [5] L. Carlitz, Degenerate Stirling, Bernoulli and Eulerian numbers, Util. Math. 15, 51- 88, 1979.
• [6] C. Cesarano and W. Ramírez, Some new classes of degenerated generalized Apostol- Bernoulli, Apostol-Euler and Apostol-Genocchi polynomials, Carpathian Math. Publ., 4 (2), 2022.
• [7] C. Cesarano, W. Ramírez and S. Diaz, New results for degenerated generalized Apostol-Bernoulli, Apostol-Euler and Apostol- Genocchi polynomials, WSEAS Transactions on Mathematics 21, 604-608, 2022.
• [8] C. Cesarano, W. Ramírez and S. Khan, A new class of degenerate Apostoltype Hermite polynomials and applications, Dolomites Res. Notes Approx. 15, 110, 2022.
• [9] F.A. Costabile, M.I. Gualtieri and A. Napoli, Bivariate general Appell interpolation problem, Numer. Algorithms 91, 531-556, 2022. doi: 10.1007/s11075-022-01272-4.
• [10] F.A. Costabile, M.I. Gualtieri and A. Napoli, General bivariate Appell polynomials via matrix calculus and related interpolation hints, Mathematics 964 (9), 2021.
• [11] F.A. Costabile and E. Longo, The Appell interpolation problem, J. Comput. Appl. Math. 236, 1024-1032, 2011.
• [12] F.A. Costabile and E. Longo, Δh - Appell sequences and related interpolation problem, Numer. Algorithms 63 (1), 165-186, 2013. doi: 10.1007/s11075-012-9619-1.
• [13] S. Khan and N. Raza, General-Appell polynomials within the context of monomiality principle, Int. J. Anal. 2013, Art. ID. 328032, 2013.
• [14] D. Kim, A class of Sheffer sequences of some complex polynomials and their degenerate types, Symmetry 7, 1064-1080, 2019.
• [15] D. Kim, A note on the degenerate type of complex Appell polynomials, Symmetry 11, 1339-1352, 2019.
• [16] W.A. Khan, Degenerate Hermite-Bernoulli numbers and polynomials of the second kind, Prespacetime Journal 7 (9), 1200-1208, 2016.
• [17] M. Riyasat, Generalized 3D extension of degenerate Fubini polynomials and their applications, submitted for publication.
• [18] M. Riyasat, T. Nahid and S. Khan, An algebraic approach to degenerate Appell polynomials and their hybrid forms via determinants, Acta Math. Sci. 43 (2), 719-735, 2023.