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Connection problems and matrix representations for certain hybrid polynomials. (English) Zbl 1435.33025

Summary: In this paper, we deal with the connection and duplication problems associated with the hybrid Sheffer family. The hybrid Sheffer polynomials are also studied via matrix approach. The properties of these polynomials are established using simple matrix operations. Examples providing the corresponding results for certain members of the hybrid Sheffer family are considered. This article is first attempt in the direction of obtaining connection and duplication coefficients and matrix representations for the hybrid polynomials.

MSC:

33F10 Symbolic computation of special functions (Gosper and Zeilberger algorithms, etc.)
11B83 Special sequences and polynomials
15A16 Matrix exponential and similar functions of matrices
41A58 Series expansions (e.g., Taylor, Lidstone series, but not Fourier series)
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References:

[1] L. Aceto, H. R. Malonek, G. Tomaz, A unified matrix approach to the representation of Appell polynomials, Integral Transform Spec. Funct. 26 (2015) 426-441. · Zbl 1309.15015
[2] L. Aceto, I. Cação, A matrix approach to Sheffer polynomials, J. Math. Anal. Appl. 446 (2017) 87-100. · Zbl 1351.11020
[3] L. Aceto, D. Trigiante, The matrices of Pascal and other greats, Amer. Math. Monthly 108 (2001) 232-245. · Zbl 1002.15024
[4] P. Appell, Sur une classe de polynômes, Ann. Sci. \( \acute{E}\) cole. Norm. Sup. 9 (1880) 119-144. · JFM 12.0342.02
[5] L. C. Andrews, Special functions for engineers and applied mathematicians, Macmillan Publishing Company, New York, 1985.
[6] R. P. Boas, R. C. Buck, Polynomial expansions of analytic functions, Springer-Verlag, Berlin, Gottingen, Heidelberg, 1964. · Zbl 0116.28105
[7] Y. Ben Cheikh, H. Chaggara, Connection problems via lowering operators, J. Comput. Appl. Math. 178 (2005) 45-61. · Zbl 1061.33006
[8] F. A. Costabile, E. Longo, A determinantal approach to Appell polynomials, J. Comput. Appl. Math. 234(5) (2010) 1528-1542. · Zbl 1200.33020
[9] M. E. A. El-Mikkawy, An algorithm for solving Vandermonde systems, J. Inst. Math. Comp. Sci. 3(3) (1990) 293-297. · Zbl 0844.65015
[10] M. El-Mikkawy, B. El-Desouky, On a connection between symmetric polynomials, generalized Stirling numbers and the Newton general divided difference interpolation polynomials, Appl. Math. Comput. 138(2-3) (2003) 375-385. · Zbl 1043.41005
[11] C. Jordan, Calculus of finite differences, Third Edition, Chelsea Publishing Company, Bronx, New York, 1965.
[12] Subuhi Khan, M. Riyasat, A determinantal approach to Sheffer-Appell polynomials via monomiality principle, J. Math. Anal. Appl. 421 (2015) 806-829. · Zbl 1310.33012
[13] M. Lahiri, On a generalisation of Hermite polynomials, Proc. Amer. Math. Soc. 27(1) (1971) 117-121. · Zbl 0219.33006
[14] S. Roman, The umbral calculus, Academic Press, New York, 1984. · Zbl 0536.33001
[15] H.M. Srivastava, Some formulas for the Bernoulli and Euler polynomials at rational arguments, Math. Proc. Cambridge Phil. Soc. 129 (2000) 77-84. · Zbl 0978.11004
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