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Product of Sheffer sequences: properties and examples. (English) Zbl 1431.33014

Summary: This article is written with an objective to explore the product of two Sheffer sequences. This article is an attempt to explore such type of product which extends the possibility to consider hybrid type Sheffer polynomials. It is important to remark that although this product can be viewed as the umbral composition of two Sheffer sequences but the results related to this product cannot be deduced from the results of the Sheffer sequences. The set of this product of Sheffer sequences is also a non-abelian group and thus convoluting different members of Sheffer class allows us to consider a number of hybrid type special polynomials as its members. Certain results for this class including the quasi-monomiality and determinant form are established. The article concludes with the possibility of considering the product of \(n\)-Sheffer sequences.

MSC:

33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
11B83 Special sequences and polynomials
05A15 Exact enumeration problems, generating functions
12E10 Special polynomials in general fields
11C20 Matrices, determinants in number theory
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References:

[1] M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables, reprint of the 1972 edition, Dover Publications, Inc., New York, 1992. · Zbl 0643.33001
[2] C. M. Bender, Solution of operator equations of motion (Rigorous Results in Quantum Dynamics) Eds. J. Dittrich and P. Exner (World Scientific, Singapore, 1991) 99-112.
[3] R. P. Boas, R. C. Buck, Polynomial Expansions of Analytic Functions, Springer-Verlag, Berlin, Gottingen, Heidelberg, 1958. · Zbl 0082.05702
[4] G. Dattoli, Hermite-Bessel and Laguerre-Bessel functions: a by-product of the monomiality principle, advanced special functions and applications, (Melfi, 1999), 147-164, Proc. Melfi Sch. Adv. Top. Math. Phys., 1, Aracne, Rome, 2000. · Zbl 1022.33006
[5] C. Jordan, Calculus of Finite Differences, Third Edition, Chelsea Publishing Company, Bronx, New York, 1965. · Zbl 0154.33901
[6] M. Lahiri, On a generalisation of Hermite polynomials, Proc. Amer. Math. Soc. 27 (1971) 117-121. · Zbl 0219.33006
[7] E. D. Rainville, Special functions, Reprint of 1960 First Edition. Chelsea Publishing Company, Bronx, New York, 1971.
[8] S. Roman, The theory of the umbral calculus, I. J. Math. Anal. 87(1) (1982) 58-115. · Zbl 0499.05009
[9] S. Roman, The Umbral Calculus, Academic Press, New York, 1984. · Zbl 0536.33001
[10] L. W. Shapiro, S. Getu, W. J. Woan, L. C. Woodson, The Riordan group, Discrete Appl. Math. 34 (1-3) (1991) 229-239. · Zbl 0754.05010
[11] W. Wang, T. Wang, Generalized Riordan arrays, Discrete Math. 308(24) (2008) 6466-6500. · Zbl 1158.05008
[12] W. Wang, A determinantal approach to Sheffer sequences, Linear Algebra Appl. 463 (2014) 228-254. · Zbl 1301.05025
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