×

On generalisation of polynomials in complex plane. (English) Zbl 1205.33011

Summary: The generalised Bell and Laguerre polynomials of fractional-order in complex \(z\)-plane are defined. Some properties are studied. Moreover, we proved that these polynomials are univalent solutions for second order differential equations. Also, the Laguerre-type of some special functions are introduced.

MSC:

33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
33C05 Classical hypergeometric functions, \({}_2F_1\)
33E12 Mittag-Leffler functions and generalizations
26A33 Fractional derivatives and integrals
PDFBibTeX XMLCite
Full Text: DOI EuDML

References:

[1] T. Isoni, P. Natalini, and P. E. Ricci, “Symbolic computation of Newton sum rules for the zeros of polynomial eigenfunctions of linear differential operators,” Numerical Algorithms, vol. 28, no. 1-4, pp. 215-227, 2001. · Zbl 0997.65102
[2] J. Riordan, An Introduction to Combinatorial Analysis, Wiley Publications in Mathematical Statistics, John Wiley & Sons, New York, NY, USA, 1958. · Zbl 0078.00805
[3] M. G. Kendall and A. Stuart, The Advanced Theory of Statistics, Grin, London, UK, 1958.
[4] P. Natalini and P. E. Ricci, “Laguerre-type Bell polynomials,” International Journal of Mathematics and Mathematical Sciences, vol. 2006, Article ID 45423, 7 pages, 2006. · Zbl 1113.33012
[5] L. Carlitz, “Some reduction formulas for generalized hypergeometric functions,” SIAM Journal on Mathematical Analysis, vol. 1, pp. 243-245, 1970. · Zbl 0199.10803
[6] P. Natalini and P. E. Ricci, “An extension of the Bell polynomials,” Computers & Mathematics with Applications, vol. 47, no. 4-5, pp. 719-725, 2004. · Zbl 1080.11019
[7] P. N. Rai and S. N. Singh, “Generalization of Bell polynomials and related operational formula,” Vijnana Parishad Anusandhan Patrika, vol. 25, no. 3, pp. 251-258, 1982.
[8] W. Yang, H. Li, and S. Jing, “Deformed legendre polynomial and its application,” http://arxiv.org/abs/math-ph/0212008.
[9] S. Jing and W. Yang, “A new kind of deformed hermite polynomials and its applications,” http://arxiv.org/abs/math-ph/0212011.
[10] A. M. A. El-Sayed, “Laguerre polynomials of arbitrary (fractional) orders,” Applied Mathematics and Computation, vol. 109, no. 1, pp. 1-9, 2000. · Zbl 1049.33005
[11] E. T. Bell, “Exponential polynomials,” Annals of Mathematics. Second Series, vol. 35, no. 2, pp. 258-277, 1934. · Zbl 0009.21202
[12] H. Srivastava and S. Owa, Univalent Functions, Fractional Calculus, and Their Applications, Ellis Horwood Series: Mathematics and Its Applications, John Wiley & Sons, New York, NY, USA, 1989. · Zbl 0683.00012
[13] K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, A Wiley-Interscience Publication, John Wiley & Sons, New York, NY, USA, 1993. · Zbl 0789.26002
[14] R. W. Ibrahim and M. Darus, “Subordination and superordination for univalent solutions for fractional differential equations,” Journal of Mathematical Analysis and Applications, vol. 345, no. 2, pp. 871-879, 2008. · Zbl 1147.30009
[15] R. P. Agarwal, “A propos d’une note de M. Pierre Humbert,” Comptes Rendus de l’Académie des Sciences, vol. 236, pp. 2031-2032, 1953. · Zbl 0051.30801
[16] C. Fox, “The asymptotic expansion of the generalized hypergeometric function,” Journal of the London Mathematical Society, vol. 27, pp. 389-400, 1928. · JFM 54.0392.03
[17] E. M. Wright, “The asymptotic expansion of the generalized hypergeometric function,” Journal of the London Mathematical Society, vol. 10, pp. 286-293, 1935. · Zbl 0013.02104
[18] E. M. Wright, “The asymptotic expansion of the generalized hypergeometric function,” Proceedings of the London Mathematical Society. Second Series, vol. 46, pp. 389-408, 1940. · Zbl 0025.40402
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.