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Certain properties of Konhauser matrix polynomials via Lie Algebra techniques

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Abstract

The principal purpose of this paper is devoted to an investigation of some interesting generating matrix functions for the second-kind Konhauser matrix polynomials (KMPs) using a Lie group theory. We derive many interesting properties such as Rodrigues formula, integral representations, matrix recurrence relations, matrix differential equation, finite sums and generating matrix functions for the second-kind KMPs.

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References

  1. Agarwal, R., Jain, S.: Certain properties of some special matrix functions via Lie algebra. Int. Bull. Math. Res. 2(1), 9–15 (2015)

    MATH  Google Scholar 

  2. Çekim, B.: New kinds of matrix polynomials. Miskolc Math. Notes 14(3), 817–826 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  3. Çekim, B., Altin, A.: New matrix formulas for Laguerre matrix polynomials. J. Class. Anal. 3(1), 59–67 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  4. Çekim, B., Altin, A., Aktaş, R.: Some relations satisfied by orthogonal matrix polynomials. Hacet. J. Math. Stat. 40(2), 241–253 (2011)

    MathSciNet  MATH  Google Scholar 

  5. Chongdar, A.K.: Group-theoretic study of certain generating functions. Bull. Calcutta Math. Soc. 77(77), 151–157 (1985)

    MathSciNet  MATH  Google Scholar 

  6. Dunford, N., Schwartz, J.T.: Linear Operators, Part I, General Theory. Interscience Publishers INC., New York (1957)

    MATH  Google Scholar 

  7. Erkuş-Duman, E., Çekim, B.: New generating functions for Konhauser matrix polynomials. Commun. Fac. Sci. Univ. Ankara Ser. A 1 Math. Stat. 63(1), 35–41 (2014)

    MathSciNet  MATH  Google Scholar 

  8. Jódar, L., Company, R., Navarro, E.: Laguerre matrix polynomials and systems of second order differential equations. Appl. Numer. Math. 15, 53–63 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  9. Jódar, L., Cortés, J.C.: Some properties of Gamma and Beta matrix functions. Appl. Math. Lett. 11, 89–93 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  10. Jódar, L., Cortés, J.C.: On the hypergeometric matrix function. J. Comput. Appl. Math. 99, 205–217 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  11. Jódar, L., Sastre, J.: On Laguerre matrix polynomials. Util. Math. 53, 37–48 (1998)

    MathSciNet  MATH  Google Scholar 

  12. Khan, S.: Harmonic oscillator group and Laguerre 2D-polynomials. Rep. Math. Phys. 52(2), 227–234 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  13. Khan, S., Ali, M.: Lie algebra representations and 1-parameter 2D-Hermite polynomials. Integral Transforms Spec. Funct. 28(4), 315–327 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  14. Khan, S., Hassan, N.A.M.: \(2\)-variable Laguerre matrix polynomials and Lie-algebraic techniques. J. Phys. A. Math. Theor. 43(23), 235204 (21pp) (2010)

    Article  MathSciNet  MATH  Google Scholar 

  15. Khan, S., Pathan, M.A., Yasmin, G.: Representation of a Lie algebra \(G(0,1)\) and three variables generalized Hermite polynomials, \(H_{n}(x, y, z)\). Integral Transforms Spec. Funct. 13, 59–64 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  16. Khan, S., Raza, N.: \(2\)-variable generalized Hermite matrix polynomials and Lie algebra representation. Rep. Math. Phys. 66(2), 159–174 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  17. Khan, S., Raza, N.: Hermite–Laguerre matrix polynomials and generating relations. Rep. Math. Phys. 73(2), 137–164 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  18. Konhauser, J.D.E.: Some properties of biorthogonal polynomials. J. Math. Anal. Appl. 11, 242–260 (1965)

    Article  MathSciNet  MATH  Google Scholar 

  19. Konhauser, J.D.E.: Biorthogonal polynomials suggested by the Laguerre polynomials. Pac. J. Math. 21, 303–314 (1967)

    Article  MathSciNet  MATH  Google Scholar 

  20. Osler, T.J.: The fractional derivative of a composite function. SIAM J. Math. Anal. 1(2), 288–293 (1970)

    Article  MathSciNet  MATH  Google Scholar 

  21. Miller Jr., W.: Lie Theory and Special Functions. Academic Press, New York (1968)

    MATH  Google Scholar 

  22. Samanta, K.P., Samanta, B.: On bilateral generating functions of Konhauser biorthogonal polynomials. Univers. J. Appl. Math. 3(2), 18–23 (2015)

    Article  Google Scholar 

  23. Shahwan, M.J.S., Pathan, M.A.: Origin of certain generating relations of Hermite matrix functions from the view point of Lie Algebra. Integral Transform Spec. Funct. 17(10), 743–747 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  24. Shahwan, M.J.S., Pathan, M.A.: Generating relations of Hermite matrix polynomials by Lie Algebraic method. Ital. J. Pure Appl. Math. 25, 187–192 (2009)

    MathSciNet  MATH  Google Scholar 

  25. Shehata, A.: Some relations on Konhauser matrix polynomials. Miskolc Math. Notes 17(1), 605–633 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  26. Shehata, A.: Certain generating relations of Konhauser matrix polynomials from the view point of Lie algebra method. Univ. Politech. Buchar. Sci. Bull. Ser. A Appl. Math. Phys. 79(4), 123–136 (2017)

    MathSciNet  Google Scholar 

  27. Shehata, A.: Certain generating matrix relations of generalized Bessel matrix polynomials from the view point of Lie algebra method. Bull. Iran. Math. Soc. 44(4), 1025–1043 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  28. Shehata, A.: Certain generating matrix functions of Legendre matrix polynomials using Lie algebraic method. Kragujev. J. Math. 44(3), 353–368 (2020)

    Google Scholar 

  29. Shehata, A.: Certain properties of generalized Hermite-Type matrix polynomials using Weisner’s group theoretic techniques. In: Bulletin of the Brazilian Mathematical Society, New Series, (BBMS-D-17-00298) (in Press)

  30. Srivastava, H.M., Singhal, J.P.: A class of polynomials defined by generalized Rodrigues formula. Annali di Matematica Pura ed Applicata Series IV 90(4), 75–85 (1971)

    Article  MathSciNet  MATH  Google Scholar 

  31. Varma, S., Çekim, B., Taşdelen, F.: On Konhauser matrix polynomials. Ars Comb. 100, 193–204 (2011)

    MathSciNet  MATH  Google Scholar 

  32. Varma, S., Taşdelen, F.: Some properties of Konhauser matrix polynomials. Gazi Univ. J. Sci. 29(3), 703–709 (2016)

    Google Scholar 

  33. Weisner, L.: Group-theoretic origin of certain generating functions. Pac. J. Math. 5, 1033–1039 (1955)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

(a) The author expresses sincere appreciation to Dr. Mohamed Saleh Metwally [Department of Mathematics, Faculty of Science (Suez), Suez Canal University, Egypt], and Dr. Mahmoud Tawfik Mohamed [Department of Mathematics, Faculty of Science (New Valley), Assiut University, New Valley, EL-Kharga 72111, Egypt] for their kind interests, encouragement, help, suggestions, comments and the investigations for this series of papers. (b) The author would like to thank the anonymous reviewers for their valuable comments and suggestions, which improve the readability of the paper.

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Authors and Affiliations

  1. Department of Mathematics, Faculty of Science, Assiut University, Assiut, 71516, Egypt

    Ayman Shehata

  2. Department of Mathematics, College of Science and Arts, Unaizah, Qassim University, Qassim, Kingdom of Saudi Arabia

    Ayman Shehata

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  1. Ayman Shehata
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Correspondence to Ayman Shehata.

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Shehata, A. Certain properties of Konhauser matrix polynomials via Lie Algebra techniques. Bol. Soc. Mat. Mex. 26, 99–120 (2020). https://doi.org/10.1007/s40590-019-00232-8

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