Connections between Legendre with Hermite and Laguerre Matrix Polynomials

Year 2015, Volume: 28 Issue: 2, 221 - 230, 22.06.2015

Ayman Shehata

Abstract

The aim of this paper is to develop a connection between Legendre and Hermite matrix polynomials recently introduced in \cite{Lasm1} is derived. We also obtain various new generalized forms of the Legendre and Hermite matrix polynomials by using the integral representation method. The expansion of Legendre matrix polynomials in a series of Laguerre matrix polynomials is established.

Keywords

Hermite , Laguerre and Legendre matrix polynomials; Integral representation; Finite summation.

Connections Between Legendre with Hermite and Laguerre Matrix Polynomials

Abstract

References

• Aktas, R. A note on multivariable Humbert matrix polynomials. Gazi University Journal of Science, 27 (2) (2014), 747-754.
• Altin, A., and Çekim, B. Some properties associated matrix with Mathematica, 88 (2012), 171-181. polynomials. Utilitas
• Altin, A., and Çekim, B. Some miscellaneous properties for Gegenbauer matrix polynomials. Utilitas Mathematica, 92 (2013) 377-387.
• Çekim, B., Altin, A., and Aktas, R. Some new results for Jacobi matrix polynomials. Filomat, 27 (4) (2013), 713-719
• Defez, E., and Jódar, L. Some applications of the Hermite matrix polynomials series expansions. J. Comp. Appl. Math., 99 (1998), 105-117.
• Dunford, N., and Schwartz, J.T. Linear Operators, part I, General Theory. Interscience Publishers, INC. New York, 1957.
• Jódar, L., and Cortés, J.C. Some properties of Gamma and Beta matrix functions. Applied Mathematics Letters, 11 (1998), 89-93.
• Jódar, L., and Company, R. Hermite matrix polynomials and second order matrix differential equations. J. Approx. Theory Appl., 12 (1996), 20-30.
• Jódar, L. Company, R., and Navarro, E. Laguerre matrix polynomials and system of second-order differential equations. Appl. Num. Math., 15 (1994), 53-63.
• Jódar, L., and Defez, E. A connection between Laguerre’s and Hermite’s matrix polynomials. Appl. Math. Lett., 11 (1998), 13-17.
• Jódar, L., and Defez, E. On Hermite matrix polynomials and Hermite matrix function. J. Approx. Theory Appl., 14 (1998), 36-48.
• Jódar, L., and Sastre, J. On Laguerre matrix polynomials. Utilitas Math., 53 (1998), 37-48.
• Metwally, M.S., Mohamed, M.T., and Shehata, A. On Hermite-Hermite Bohemica. 133 (4) (2008), 421-434. polynomials. Math.
• Metwally, M.S., Mohamed, M.T., and Shehata, A. Generalizations of two-index two-variable Hermite matrix polynomials. Demonstratio Mathematica, 42 (2009), 687-701.
• Rainville, E.D. Special Functions. The Macmillan Company, New York, 1960.
• Shehata, A. On Tricomi and Hermite-Tricomi matrix functions of complex variable. Communications Math. Applications, 2 (2-3) (2011), 97-109.
• Shehata, A. A new extension of Hermite-Hermite matrix polynomials and their properties. Thai Journal of Mathematics, 10 (2) (2012), 433-444.
• Shehata, A. A new extension of Gegenbauer matrix polynomials and their properties. Bulletin Inter. Math. Virtual Institute, 2 (2012), 29-42.
• Shehata, A. On Rice’s matrix polynomials. Afrika Matematika, 25 (3) (2014), 757-777.
• Shehata, A. On Rainville’s matrix polynomials. Sylwan Journal, 158 (9) (2014), 158-178.
• Shehata, A. Some relations on Humbert matrix polynomials. Mathematica Bohemica, (in press).
• Shehata, A. Some relations on Konhauser matrix polynomials. Miskolc Mathematical Notes, (in press).
• Shehata, A. New kinds of Hypergeometric matrix functions. British Journal of Mathematics and Computer Science, 5 (1) (2015), 92-102.
• Shehata, A. On a new family of the extended generalized Mitteilungen Klosterneuburg Journal, 65 (2) (2015), 100-121. matrix polynomials.
• Upadhyaya, L.M., and Shehata, A. On Legendre matrix polynomials and its applications. Inter. Trans. Mathematical Sci. Computer (ITMSC), 4 (2) (2011), 291-310.
• Upadhyaya, L.M., and Shehata, A. A new extension of generalized Hermite matrix polynomials. Bulletin Malaysian Mathematical Sci. Soc., 38 (1) (2015), 165–179.
• Varma, S., Çekim, B., and Tasdelen, F. On Konhauser matrix polynomials. Ars Combinatoria, 100 (2011), 193-204.