Abstract
In this article, the recurrence relations and differential equation for the 3-variable Hermite–Sheffer polynomials are derived by using the properties of the Pascal functional and Wronskian matrices. The corresponding results for certain members belonging to the Hermite–Sheffer polynomials are also obtained.
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References
Aceto L, Caçāo I (2017) A matrix approach to Sheffer polynomials. J Math Anal Appl 446:87–100
Aceto L, Malonek HR, Tomaz G (2015) A unified matrix approach to the representation of Appell polynomials. Integral Transform Spec. Funct. 26(6):426–441
Jordan C (1965) Calculus of finite differences, 3rd edn. Chelsea, New York
Cesarano C (2017) Operational methods and new identities for hermite polynomials. Math Model Nat Phenom 12(3):44–50
Dattoli G (1999) Hermite–Bessel and Laguerre–Bessel functions: a by-product of the monomiality principle. Advanced special functions and applications (Melfi, 1999) In: Proc. Melfi Sch. Adv. Top. Math. Phys. Aracne, Rome, 2000. vol 1, pp 147–164
Dattoli G, Lorenzutta S, Cesarano C (2006) Bernstein polynomials and operational methods. J Comput Anal Appl 8(4):369–377
Dattoli G, Lorenzutta S, Ricci PE, Cesarano C (2004) On a family of hybrid polynomials. Integral Transform Spec Funct 15(6):485–490
Kim DS, Kim T (2015) A matrix approach to some identities involving Sheffer polynomial sequences. Appl Math Comput 253:83–101
Khan S, Raza N (2012) Monomiality principle, operational methods and family of Laguerre–Sheffer polynomials. J Math Anal Appl 387:90–102
Khan S, Al-Saad MWM, Yasmin G (2010) Some properties of Hermite-based Sheffer polynomials. Appl Math Comput 217(5):2169–2183
Khan S, Yasmin G, Khan R, Hassan NAM (2009) Hermite-based Appell polynomials: Properties and applications. J Math Anal Appl 351:756–764
Sheffer IM (1939) Some properties of polynomial sets of type zero. Duke Math J 5:590–622
Srivastava HM, Özarslan MA, Yilmaz B (2014) Some families of differential equations associated with the Hermite-based Appell polynomials and other classes of Hermite-based polynomials. Filomat 28(4):695–708
Rainville ED (1971) Special functions, 1st edn. Chelsea Publishig Co., Bronx (Reprint of 1960)
Roman S (1984) The umbral calculus. Academic Press, New York
Yang Y, Micek C (2007) Generalized Pascal functional matrix and its applications. Linear Algebra Appl 423(2–3):230–245
Yang Y, Youn H (2009) Appell polynomial sequences: a linear algebra approach. JP J Algebra Number Theory Appl 13(1):65–98
Youn H, Yang Y (2011) Differential equation and recursive formulas of Sheffer polynomial sequences. ISRN Discret Math 2011:1–16 (Article ID 476462)
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The authors are thankful to the reviewer(s) for several useful comments and suggestions towards the improvement of this paper. The second and third authors thank the first author for her helpful discussion and excellent suggestions.
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Khan, S., Naikoo, S.A. & Ali, M. Recurrence Relations and Differential Equations of the Hermite–Sheffer and Related Hybrid Polynomial Sequences. Iran J Sci Technol Trans Sci 43, 1607–1618 (2019). https://doi.org/10.1007/s40995-018-0550-8
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DOI: https://doi.org/10.1007/s40995-018-0550-8
Keywords
- Hermite–Sheffer polynomials
- Generalized Pascal functional matrix
- Wronskian matrix
- Recurrence relations
- Differential equations