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A NEW KIND OF LEGENDRE MATRIX POLYNOMIALS

Year 2016, Volume: 29 Issue: 2, 435 - 457, 21.06.2016

Abstract

The main aim of this paper is to introduce a new kind of Legendre matrix polynomials. Hypergeometric matrix representation of these matrix polynomials is given. The convergence properties and the integral form for the Legendre matrix polynomials are derived. The Legendre matrix differential equation of second order is established. Subsequently, Rodrigues formula, orthogonality property, matrix recurrence relation and types of generating matrix functions are then developed for the Legendre matrix polynomials. Furthermore, general families of bilinear and bilateral generating matrix functions for these matrix polynomials are obtained and their applications are presented. Finally, the composite Legendre matrix polynomials is introduced.

References

  • Aktaş, R., Çekim, B., and Çevik, A. Extended Jacobi matrix polynomials. Utilitas Mathematica, 92 (2013), 47-64.
  • Akta , R., Çekim, B., and Şahin, R. The matrix version for the multivariable Humbert polynomials. Miskolc Mathematical Notes, 13 (2) (2012), 197-208.
  • Altin, A., and Çekim, B. Generating matrix functions for Chebyshev matrix polynomials of the second kind. Hacettepe Journal of Mathematics and Statistics, 41 (1) (2012), 25–32.
  • Altin, A., and Çekim, B. Some properties associated with Hermite matrix polynomials. Utilitas Mathematica, 88 (2012), 171–181.
  • Altin, A., and Çekim, B. Some miscellaneous properties for Gegenbauer matrix polynomials. Utilitas Mathematica, 92 (2013), 377-387.
  • Altin, A., Aktaş, R., and Erkuş-Duman, E. On a multivariable extension for the extended Jacobi polynomials. Journal of Mathematical Analysis and Applications, 353 (2009), 121-133.
  • Çekim, B. New kinds of matrix polynomials. Miskolc Mathematical Notes, 14 (3) (2013), 817-826.
  • Çekim, B., Altin, A., and Aktaş, R. Some relations satisfied by orthogonal matrix polynomials. Hacettepe Journal of Mathematics and Statistics, 40 (2) (2011), 241-253.
  • Çekim, B., Altin, A., and Aktaş, R. Some new results for Jacobi matrix polynomials. Filomat, 27 (4) (2013), 713-719.
  • Çekim, B. and Altin, A. New matrix formulas for Laguerre matrix polynomials. Journal Classical Analysis, 3 (1), (2013), 59-67.
  • Çevik, A. Multivariable construction of extended Jacobi matrix polynomials. Journal of Inequalities and Special Functions, 4 (3) (2013), 6-12.
  • Defez, E., and Jódar, L. Some applications of the Hermite matrix polynomials series expansions. Journal of Computational and Applied Mathematics, 99 (1998), 105-117.
  • Defez, E., and Jódar, L. Chebyshev matrix polynomials and second order matrix differential equations.Utilitas Mathematica, 61 (2002), 107-123.
  • Defez, E., Jódar, L. and Law, A. Jacobi matrix differential equation, polynomial solutions, and their properties. Computers and Mathematics with Applications, 48 (2004), 789-803.
  • Defez, E., and Jódar, L., Law, A., and Ponsoda, E. Three-term recurrences and matrix orthogonal polynomials. Utilitas Mathematica, 57 (2000), 129-146.
  • Dunford, N., and Schwartz, J.T. Linear Operators, part I, General Theory. Interscience Publishers, INC. New York, 1957.
  • Golub, G., and Van-Loan, C.F. Matrix Computations, The Johns Hopkins Univ. Press, Baltimore, MD., (1991).
  • 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., Company R., and Ponsoda, E. Orthogonal matrix polynomials and systems of second order differential equations. Differential Equations and Dynamical Systems, 3 (1995), 269-288.
  • Jódar, L., and Cortés, J.C. Some properties of Gamma and Beta matrix functions. Applied Mathematics Letters, 11, (1998), 89-93.
  • Jódar, L., and Cortés, J.C. On the hypergeometric matrix function. Journal of Computational and Applied Mathematics, 99 (1998), 205-217.
  • Jódar, L., and Cortés, J.C. Closed form general solution of the hypergeometric matrix differential equation. Mathematical and Computer Modelling, 32 (2000), 1017-1028.
  • Jódar, L., and Defez, E. On Hermite matrix polynomials and Hermite matrix function. J. Approx. Theory Appl., 14 (1998), 36-48.
  • Kargin, L., and Kurt, V. Some relations on Hermite matrix polynomials. Mathematical and Computational Applications, 18 (3) (2013), 323-329.
  • Sayyed, K.A.M. Basic Sets of Polynomials of two Complex Variables and Convergence Propertiess. Ph. D. Thesis, Assiut University, 1975.
  • Sayyed, K.A.M., M.S. Metwally, M.S. and Mohammed, M.T. Certain hypergeometric matrix function. Scientiae Mathematicae Japonicae, 69 (2009), 315-321.
  • 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. Connections between Legendre with Hermite and Laguerre matrix polynomials. Gazi University Journal of Science, Vol. 28, No. 2 (2015), 221-230.
  • Shehata, A. On modified Laguerre matrix polynomials. Journal of Natural Sciences and Mathematics, Vol. 8, No. 2 (2015), 153-166.
  • Upadhyaya, L.M., and Shehata, A. On Legendre matrix polynomials and its applications. International Transactions in Mathematical Sciences and Computer, 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.
  • Taşdelen, F., Çekim, B., and Aktaş, R. On a multivariable extension of Jacobi matrix polynomials. Computers and Mathematics with Applications, 61 (9) (2011), 2412-2423.
  • Varma, S., Çekim, B., and Taşdelen F. On Konhauser matrix polynomials. Ars Combinatoria, 100 (2011), 193-204.
  • Varma, S., and Taşdelen, F. Biorthogonal matrix polynomials related to Jacobi matrix polynomials. Computers Math. Applications, 62(10) (2011), 3663-3668.
Year 2016, Volume: 29 Issue: 2, 435 - 457, 21.06.2016

Abstract

References

  • Aktaş, R., Çekim, B., and Çevik, A. Extended Jacobi matrix polynomials. Utilitas Mathematica, 92 (2013), 47-64.
  • Akta , R., Çekim, B., and Şahin, R. The matrix version for the multivariable Humbert polynomials. Miskolc Mathematical Notes, 13 (2) (2012), 197-208.
  • Altin, A., and Çekim, B. Generating matrix functions for Chebyshev matrix polynomials of the second kind. Hacettepe Journal of Mathematics and Statistics, 41 (1) (2012), 25–32.
  • Altin, A., and Çekim, B. Some properties associated with Hermite matrix polynomials. Utilitas Mathematica, 88 (2012), 171–181.
  • Altin, A., and Çekim, B. Some miscellaneous properties for Gegenbauer matrix polynomials. Utilitas Mathematica, 92 (2013), 377-387.
  • Altin, A., Aktaş, R., and Erkuş-Duman, E. On a multivariable extension for the extended Jacobi polynomials. Journal of Mathematical Analysis and Applications, 353 (2009), 121-133.
  • Çekim, B. New kinds of matrix polynomials. Miskolc Mathematical Notes, 14 (3) (2013), 817-826.
  • Çekim, B., Altin, A., and Aktaş, R. Some relations satisfied by orthogonal matrix polynomials. Hacettepe Journal of Mathematics and Statistics, 40 (2) (2011), 241-253.
  • Çekim, B., Altin, A., and Aktaş, R. Some new results for Jacobi matrix polynomials. Filomat, 27 (4) (2013), 713-719.
  • Çekim, B. and Altin, A. New matrix formulas for Laguerre matrix polynomials. Journal Classical Analysis, 3 (1), (2013), 59-67.
  • Çevik, A. Multivariable construction of extended Jacobi matrix polynomials. Journal of Inequalities and Special Functions, 4 (3) (2013), 6-12.
  • Defez, E., and Jódar, L. Some applications of the Hermite matrix polynomials series expansions. Journal of Computational and Applied Mathematics, 99 (1998), 105-117.
  • Defez, E., and Jódar, L. Chebyshev matrix polynomials and second order matrix differential equations.Utilitas Mathematica, 61 (2002), 107-123.
  • Defez, E., Jódar, L. and Law, A. Jacobi matrix differential equation, polynomial solutions, and their properties. Computers and Mathematics with Applications, 48 (2004), 789-803.
  • Defez, E., and Jódar, L., Law, A., and Ponsoda, E. Three-term recurrences and matrix orthogonal polynomials. Utilitas Mathematica, 57 (2000), 129-146.
  • Dunford, N., and Schwartz, J.T. Linear Operators, part I, General Theory. Interscience Publishers, INC. New York, 1957.
  • Golub, G., and Van-Loan, C.F. Matrix Computations, The Johns Hopkins Univ. Press, Baltimore, MD., (1991).
  • 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., Company R., and Ponsoda, E. Orthogonal matrix polynomials and systems of second order differential equations. Differential Equations and Dynamical Systems, 3 (1995), 269-288.
  • Jódar, L., and Cortés, J.C. Some properties of Gamma and Beta matrix functions. Applied Mathematics Letters, 11, (1998), 89-93.
  • Jódar, L., and Cortés, J.C. On the hypergeometric matrix function. Journal of Computational and Applied Mathematics, 99 (1998), 205-217.
  • Jódar, L., and Cortés, J.C. Closed form general solution of the hypergeometric matrix differential equation. Mathematical and Computer Modelling, 32 (2000), 1017-1028.
  • Jódar, L., and Defez, E. On Hermite matrix polynomials and Hermite matrix function. J. Approx. Theory Appl., 14 (1998), 36-48.
  • Kargin, L., and Kurt, V. Some relations on Hermite matrix polynomials. Mathematical and Computational Applications, 18 (3) (2013), 323-329.
  • Sayyed, K.A.M. Basic Sets of Polynomials of two Complex Variables and Convergence Propertiess. Ph. D. Thesis, Assiut University, 1975.
  • Sayyed, K.A.M., M.S. Metwally, M.S. and Mohammed, M.T. Certain hypergeometric matrix function. Scientiae Mathematicae Japonicae, 69 (2009), 315-321.
  • 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. Connections between Legendre with Hermite and Laguerre matrix polynomials. Gazi University Journal of Science, Vol. 28, No. 2 (2015), 221-230.
  • Shehata, A. On modified Laguerre matrix polynomials. Journal of Natural Sciences and Mathematics, Vol. 8, No. 2 (2015), 153-166.
  • Upadhyaya, L.M., and Shehata, A. On Legendre matrix polynomials and its applications. International Transactions in Mathematical Sciences and Computer, 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.
  • Taşdelen, F., Çekim, B., and Aktaş, R. On a multivariable extension of Jacobi matrix polynomials. Computers and Mathematics with Applications, 61 (9) (2011), 2412-2423.
  • Varma, S., Çekim, B., and Taşdelen F. On Konhauser matrix polynomials. Ars Combinatoria, 100 (2011), 193-204.
  • Varma, S., and Taşdelen, F. Biorthogonal matrix polynomials related to Jacobi matrix polynomials. Computers Math. Applications, 62(10) (2011), 3663-3668.
There are 37 citations in total.

Details

Journal Section Mathematics
Authors

Ayman Shehata

Publication Date June 21, 2016
Published in Issue Year 2016 Volume: 29 Issue: 2

Cite

APA Shehata, A. (2016). A NEW KIND OF LEGENDRE MATRIX POLYNOMIALS. Gazi University Journal of Science, 29(2), 435-457.
AMA Shehata A. A NEW KIND OF LEGENDRE MATRIX POLYNOMIALS. Gazi University Journal of Science. June 2016;29(2):435-457.
Chicago Shehata, Ayman. “A NEW KIND OF LEGENDRE MATRIX POLYNOMIALS”. Gazi University Journal of Science 29, no. 2 (June 2016): 435-57.
EndNote Shehata A (June 1, 2016) A NEW KIND OF LEGENDRE MATRIX POLYNOMIALS. Gazi University Journal of Science 29 2 435–457.
IEEE A. Shehata, “A NEW KIND OF LEGENDRE MATRIX POLYNOMIALS”, Gazi University Journal of Science, vol. 29, no. 2, pp. 435–457, 2016.
ISNAD Shehata, Ayman. “A NEW KIND OF LEGENDRE MATRIX POLYNOMIALS”. Gazi University Journal of Science 29/2 (June 2016), 435-457.
JAMA Shehata A. A NEW KIND OF LEGENDRE MATRIX POLYNOMIALS. Gazi University Journal of Science. 2016;29:435–457.
MLA Shehata, Ayman. “A NEW KIND OF LEGENDRE MATRIX POLYNOMIALS”. Gazi University Journal of Science, vol. 29, no. 2, 2016, pp. 435-57.
Vancouver Shehata A. A NEW KIND OF LEGENDRE MATRIX POLYNOMIALS. Gazi University Journal of Science. 2016;29(2):435-57.