Kyungpook Mathematical Journal

Department of Mathematics, Kyungpook National University (경북대학교 자연과학대학 수학과)
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The p-deformed Generalized Humbert Polynomials and Their Properties

  • Savalia, Rajesh V. (Department of Mathematical Sciences, P. D. Patel Institute of Applied Sciences, Faculty of Applied Sciences, Charotar University of Science and Technology) ;
  • Dave, B.I. (Department of Mathematics, Faculty of Science, The Maharaja Sayajirao University of Baroda)
  • Received : 2019.09.10
  • Accepted : 2020.08.04
  • Published : 2020.12.31
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Abstract

We introduce the p-deformation of generalized Humbert polynomials. For these polynomials, we derive the differential equation, generating function relations, Fibonacci-type representations, and recurrence relations and state the companion matrix. These properties are illustrated for certain polynomials belonging to p-deformed generalized Humbert polynomials.

Keywords

Acknowledgement

The authors express their sincere thanks to the reviewers and the editor-in-chief for their valuable suggestions for the improvement of the manuscript.

References

  1. Yu. A. Brychkov, On some properties of the Nuttall function Qμ,ν(a, b), Integral Transforms Spec. Funct., 25(1)(2014), 33-43.
  2. Yu. A. Brychkov and N. V. Savischenko, A special function of communication theory, Integral Transforms Spec. Funct., 26(6)(2015), 470-484. https://doi.org/10.1080/10652469.2015.1020307
  3. V. V. Bytev, M. Yu. Kalmykov and S.-O. Moch, HYPERDIRE: HYPERgeometric functions DIfferential REduction: MATHEMATICA based packages for differential reduction of generalized hypergeometric functions: FD and FS Horn-type hypergeometric functions of three variables, Comput Phys. Commun., 185(2014), 3041-3058. https://doi.org/10.1016/j.cpc.2014.07.014
  4. G. B. Costa and L. E. Levine, Nth-order differential equations with finite polynomial solutions, Int. J. Math. Educ. Sci. Tech., 29(6)(1998), 911-914.
  5. P. Deligne, P. Etingof, D. S. Freed, L. C. Jeffrey, D. Kazhdan, J. W. Morgan, D. R. Morrison and E. Witten, Quantum fields and strings: a course for mathematicians, American Mathematical Society, 1999.
  6. R. Diaz and E. Pariguan, Quantum symmetric functions, Comm. Algebra, 33(6)(2005), 1947-1978. https://doi.org/10.1081/AGB-200063327
  7. R. Diaz and E. Pariguan, On hypergeometric functions and Pochhammer k-symbol, Divulg. Mat., 15(2)(2007), 179-192.
  8. R. Diaz and C. Teruel, q, k-generalized gamma and beta function, J. Nonlinear Math. Phys., 12(1)(2005), 118-134. https://doi.org/10.2991/jnmp.2005.12.1.10
  9. H. W. Gould, Inverse series relations and other expansions involving Humbert polynomials, Duke Math. J., 32(4)(1965), 697-711. https://doi.org/10.1215/S0012-7094-65-03275-8
  10. F. B. Hildebrand, Introduction to numerical analysis, Second edition, Dover Publications INC., New York, 1987.
  11. P. Humbert, Some extensions of Pincherle's polynomials, Proc. Edinburgh Math. Soc., 39(1)(1920), 21-24. https://doi.org/10.1017/S0013091500035756
  12. P. Humbert, The confluent hypergeometric functions of two variables, Proc. R. Soc. Edinb., 41(1922), 73-96. https://doi.org/10.1017/S0370164600009810
  13. B. A. Kniehl and O. V. Tarasov, Finding new relationships between hypergeometric functions by evaluating Feynman integrals, Nuclear Phys. B, 854(3)(2012), 841-852. https://doi.org/10.1016/j.nuclphysb.2011.09.015
  14. M. Mignotte and D. Stefanescu, Polynomials: an algorithmic approach, Springer-Verlag, New York, 1999.
  15. G. Ozdemir, Y. Simsek, and G. V. Milovanovic, Generating functions for special polynomials and numbers including Apostol-type and Humbert-type polynomials, Mediterr. J. Math., 14(3)(2017), Paper No. 117, 17 pp. https://doi.org/10.1007/s00009-016-0808-3
  16. G. Polya and G. Szego, Problems and theorems in analysis I, Springer-Verlag, New York-Heidelberg and Berlin, 1972.
  17. E. D. Rainville, Special functions, The Macmillan Company, New York, 1960.
  18. P. C. Sofotasios , T. A. Tsiftsis, Yu. A. Brychkov, et al., Analytic expressions and bounds for special functions and applications in communication theory, IEEE Trans. Inform. Theory, 60(12)(2014), 7798-7823. https://doi.org/10.1109/TIT.2014.2360388
  19. H. M. Srivastava and H. L. Manocha, A treatise on generating functions, Ellis Horwood Limited, John Willey and Sons, 1984.
  20. E. T. Whittaker and G. N. Watson, A course of modern analysis, 4th ed., Cambridge University Press, Cambridge, 1927.