A Class of Multivariable Polynomials Associated with Humbert Polynomials

Abstract

In this paper, we present a generalization (and unification) of a classof Humbert polynomials which include well known families of ChanChyan-Srivastava, Lagrange-Hermite and Erkus-Srivastava multivariable polynomials. We derive various families of multilateral and multilinear generating functions for these polynomials. We also obtain othermiscellaneous properties of these polynomials. Furthermore, for somespecial cases of these polynomials, we present hypergeometric representations and give expansions of these polynomials in series of someorthogonal polynomials.

Keywords

• Humbert polynomials,Chan-Chyan-Srivastava multivariable polynomials,Lagrange-Hermite multivariable polynomials,Erkus-Srivastava multivariable polynomials,Multilinear and multilateral generating function,Recurrence relation,Hypergeometric function,2000 AMS Classification: 33C45

References

1. Akta¸s, R., S ¸ahin, R. and Altın, A. On a multivariable extension of Humbert polynomials, Appl. Math. Comp. 218, 662–666, 2011.
2. Altın, A., Akta¸s, R. and C ¸ ekim, B. On a multivariable extension of the Hermite and related polynomials, Ars Combinatoria 110, 487–503, 2013.
3. Altın, A. and Erku¸s, E. On a multivariable extension of the Lagrange-Hermite polynomials, Integral Transforms Spec. Funct. 17, 239–244, 2006.
4. Chan, W.-C. C. , Chyan, C.-J. and Srivastava, H. M. The Lagrange polynomials in several variables, Integral Transforms Spec. Funct. 12, 139–148, 2001.
5. Dattoli, G., Ricci, P.E. and Cesarano, C. The Lagrange polynomials, the associated generalizations, and the umbral calculus, Integral Transforms Spec. Funct. 14 , 181–186, 2003. Dattoli, G., Ricci, P.E. and Cesarano, C. Operational and umbral methods for the solution of partial differential equations, J. Concr. Appl. Math. 2(3), 281–288, 2004.
6. Dattoli, G., Ricci, P. E., Cesarano, C. and Khomasuridze, I. Bilateral generating functions and operational methods, South East Asian J. Math. Math. Sci. 4(2), 1–6, 2006.
7. Erd´ elyi, A., Magnus, W., Oberhettinger, F. and Tricomi, F. G. Higher Transcendental Functions, III, McGraw-Hill Book Company, New York, Toronto, London, 1955.
8. Erku¸s, E. and Srivastava, H.M. A unified presentation of some families of multivariable polynomials, Integral Transforms Spec. Funct. 17, 267–273, 2006.

9. Gould, H. W. Inverse series relation and other expansions involving Humbert polynomials, Duke Math. J. 32, 697–711, 1965.
10. Horadam, A.F. Gegenbauer polynomials revisited, Fibonacci Quart. 23, 294–299, 1985.
11. Horadam, A. F. and Pethe, S. Polynomials associated with Gegenbauer polynomials, Fibonacci Quart. 19, 393–398, 1981.
12. Humbert, P. Some extensions of Pincherle’s polynomials, Proc. Edinburgh Math. Soc. 39, 21–24, 1921.
13. Milovanovi´ c, G. V. and Dordevi´ c, G. P. On some properties of Humbert’s polynomials, Fibonacci Quart. 25, 356–360, 1987.
14. Milovanovi´ c, G. V. and Dordevi´ c, G. P. On some properties of Humbert’s polynomials II, Facta Univ. Ser. Math. Inform. 6, 23–30, 1991.
15. Pathan, M. A. and Khan, M. A. On polynomials associated with Humbert’s polynomials, Publ. Inst. Math. (Beograd) (N.S.) 62(76), 53–62, 1997.
16. Rainville, E. D. Special Functions, The Macmillan Company, New York, 1960.
17. Sinha, S. K. On a polynomial associated with Gegenbauer polynomial, Proc. Nat. Acad. Sci. India Sect. A 59, 439–455, 1989.
18. Singhal, J. P. A note on generalized Humbert polynomials, Glasnik Mat. Ser III 5(25), 241–245, 1970.
19. Srivastava, H. M. and Manocha, H. L. A Treatise on Generating Functions, Halsted Press (Ellis Horwood Limited, Chichester), John Wiley and Sons, New York, 1984.
20. Zeitlin, D. On a class of polynomials obtained from generalized Humbert polynomials, Proc. Amer. Math. Soc. 18, 28–34, 1967.