Abstract
We propose a matrix analogue of a general inverse series relation with an objective to introduce the generalized Humbert matrix polynomial, Wilson matrix polynomial, and the Rach matrix polynomial together with their inverse series representations. The matrix polynomials of Kiney, Pincherle, Gegenbauer, Hahn, Meixner-Pollaczek etc. occur as the special cases. It is also shown that the general inverse matrix pair provides the extension to several inverse pairs due to John Riordan [An Introduction to Combinatorial Identities, Wiley, 1968].
(Communicated by Marek Balcerzak)
Acknowledgement
Author is thankful to her guide Prof. B. I. Dave for his guidance during the preparation of the manuscript. Author also sincerely thanks the referees for their useful suggestions for the improvement of the manuscript.
References
[1] Askey, R.—Wilson, J.: Some basic hypergeometric orthogonal polynomials that generalize jacobi polynomials, Memoirs Amer. Math. Soc. 54(319) (1985), 1–55.10.1090/memo/0319Suche in Google Scholar
[2] Braumann, C. A.—Cortés, J. C.—Jódar, L.—Villafuerte, L.: On the random gamma function: Theory and computing, J. Comput. Appl. Math. 335 (2018), 142–155.10.1016/j.cam.2017.11.045Suche in Google Scholar
[3] Carlitz, L.: The inverse of certin formulas invovling bessel functions, Duke Math. J. 28 (1961), 431–438.10.1215/S0012-7094-61-02839-3Suche in Google Scholar
[4] Carlitz, L.: Some inverseformulas, Rend. Circ. Mat. Palermo (2) 12 (1963), 183–199.10.1007/BF02843964Suche in Google Scholar
[5] Carlitz, L.: Some inverse relations, Duke Math. J. 40 (1973), 893–901.10.1215/S0012-7094-73-04083-0Suche in Google Scholar
[6] Constantine, A. G.—Muirhead, R. J.: Partial differential equations for hypergeometric functions of two argument matrix, J. Multivariate Anal. 3 (1972), 332–338.10.1016/0047-259X(72)90020-6Suche in Google Scholar
[7] Cortés, J. C.—Jódar, L.—Soĺis, F. J.—Ku-Carrillo, R.: Infinite Matrix Products and the Representation of the Matrix Gamma Function, Abstr. Appl. Anal. 2015, Article ID 564287.10.1155/2015/564287Suche in Google Scholar
[8] Defez, E.—Hervás, A.—Law, A.—Villanueva-Oller, J.—Villanueva, R.: Progressive transmission of images: {PC-based computations, using orthogonal matrix polynomials, Math. Comput. Modelling 32 (2000), 1125–1140.10.1016/S0895-7177(00)00195-3Suche in Google Scholar
[9] Dunford, N.—Schwartz, J.: Linear Operators, Part I: General Theory, vol. I, Interscience Publishers, INC., New York, 1957.Suche in Google Scholar
[10] Dwivedi, R.—Sahai, V.: Models of lie algebra sl(2, c) and special matrix functions by means of a matrix integral transformation, J. Math. Anal. Appl 473 (2019), 786–802.10.1016/j.jmaa.2018.12.070Suche in Google Scholar
[11] Golub, G.—Loan, C. V.: Matrix Computations, Johns Hopkins University Press, Baltimore, 1989.Suche in Google Scholar
[12] Gould, H. W.: A new convolution formula and some new orthogonal relations for inversion of series, Duke Math. J. 29(3) (1962), 393–404.10.1215/S0012-7094-62-02938-1Suche in Google Scholar
[13] Gould, H. W.: A new series transform with applications to {Bessel, {Legendre, and {Tchebycheff polynomials, Duke Math. J. 31(2) (1964), 325–334.10.1215/S0012-7094-64-03131-XSuche in Google Scholar
[14] Gould, H. W.: Inverse series relations and other expansions involving {Humbert polynomials, Duke Math. J. 32(4) (1965), 697–711.10.1215/S0012-7094-65-03275-8Suche in Google Scholar
[15] Hildebrand, F. B.: Introduction to Numerical Analysis, Dover Publications, 1987.Suche in Google Scholar
[16] Hille, E.: Lectures on Ordinary Differential Equations, Addison-Wesley, New York, 1969.Suche in Google Scholar
[17] James, A. T.: Special Functions of Matrix And Single Argument in Statistics, in Theory and Applications of Special Functions, Academic Press, New york, 1975.10.1016/B978-0-12-064850-4.50016-1Suche in Google Scholar
[18] Jódar, L.—Sastre, J.: On Laguerre matrix polynomials, Utilitas Mathematics 53 (1998), 37–48.Suche in Google Scholar
[19] Jódar, L.—Cortés, J. C.: Some properties of {Gamma and {Beta matrix functions, Appl. Math. Lett. 11(1) (1998), 89–93.10.1016/S0893-9659(97)00139-0Suche in Google Scholar
[20] KARGIN, L.—KURT, V.: On generalized humbert matrix polynomials, Miskolc Math. Notes 15 (2014), 509–524.10.18514/MMN.2014.1155Suche in Google Scholar
[21] Khatri, C. G.: On the exact finite series distribution of the smallest or the largest root of matrices in three situations, J. Multivariate Anal. 12(2) (1972), 201–207.10.1016/0047-259X(72)90027-9Suche in Google Scholar
[22] Pathan, M. A.—Bin-Saad, M. G.—Al-Sarahi, F.: On matrix polynomials associated with Humbert polynomials, J. Korea Soc. Math. Educ. Ser. B: Pure Appl. Math. 21(3) (2014), 207–218.10.7468/jksmeb.2014.21.3.207Suche in Google Scholar
[23] Riordan, J.: Combinatorial Identities, John Willey and Sons. Inc., 1968.Suche in Google Scholar
[24] Rowell, D.: Computing the matrix exponential the Cayley-Hamilton method, Massachusetts Institute of Technology Department of Mechanical Engineering, 2.151 Advanced System Dynamics and Control,Suche in Google Scholar
[25] Sanjhira R.—Nathwani, B. V.—Dave, B. I.: Generalized Mittag-Leffler matrix function and associated matrix polynomials, J. Indian Math. Soc. (N.S.) 86(1–2) (2019), 161–178.Suche in Google Scholar
[26] Sayyed, K. A.—Metwally M. S.—Batahn, R. S.: Gegenbauer matrix polynomials and second order matrix differential equations, Divulg. Mat. 12 (2004), 101–115.Suche in Google Scholar
[27] SHEHATA, A: Some relation on Konhauser matrix polynomials, Miskolc Math. Notes 17(1) (2014), 605–633.10.18514/MMN.2016.1126Suche in Google Scholar
[28] Srivastava, H. M.—Khan, W. A.—Haroon, H.: Some expansions for a class of generalised Humbert Matrix Polynomial, RACSAM, (2019), https://doi.org/10.1007/s13398-019-00720-6.10.1007/s13398-019-00720-6Suche in Google Scholar
© 2021 Mathematical Institute Slovak Academy of Sciences
Artikel in diesem Heft
- Regular papers
- Prof. RNDr. Michal Fečkan, DrSc. – Sexagenarian?
- Tribonacci numbers with two blocks of repdigits
- Padovan numbers that are concatenations of two distinct repdigits
- On the 2-rank and 4-rank of the class group of some real pure quartic number fields
- A general inverse matrix series relation and associated polynomials – II
- Some hardy type inequalities with finsler norms
- Starlikeness and convexity of the product of certain multivalent functions with higher-order derivatives
- Block Hessenberg matrices and spectral transformations for matrix orthogonal polynomials on the unit circle
- How is the period of a simple pendulum growing with increasing amplitude?
- Fourier transforms of convolution operators on orlicz spaces
- Some characterizations of property of trans-Sasakian 3-manifolds
- P-Adic metric preserving functions and their analogues
- On statistical convergence of sequences of closed sets in metric spaces
- A characterization of the uniform convergence points set of some convergent sequence of functions
- A nonparametric estimation of the conditional ageing intensity function in censored data: A local linear approach
- Donsker’s fuzzy invariance principle under the Lindeberg condition
- Characterization of generalized Gamma-Lindley distribution using truncated moments of order statistics
- Matrix variate pareto distributions
- Global exponential periodicity and stability of neural network models with generalized piecewise constant delay
- Optimal inequalities for contact CR-submanifolds in almost contact metric manifolds
Artikel in diesem Heft
- Regular papers
- Prof. RNDr. Michal Fečkan, DrSc. – Sexagenarian?
- Tribonacci numbers with two blocks of repdigits
- Padovan numbers that are concatenations of two distinct repdigits
- On the 2-rank and 4-rank of the class group of some real pure quartic number fields
- A general inverse matrix series relation and associated polynomials – II
- Some hardy type inequalities with finsler norms
- Starlikeness and convexity of the product of certain multivalent functions with higher-order derivatives
- Block Hessenberg matrices and spectral transformations for matrix orthogonal polynomials on the unit circle
- How is the period of a simple pendulum growing with increasing amplitude?
- Fourier transforms of convolution operators on orlicz spaces
- Some characterizations of property of trans-Sasakian 3-manifolds
- P-Adic metric preserving functions and their analogues
- On statistical convergence of sequences of closed sets in metric spaces
- A characterization of the uniform convergence points set of some convergent sequence of functions
- A nonparametric estimation of the conditional ageing intensity function in censored data: A local linear approach
- Donsker’s fuzzy invariance principle under the Lindeberg condition
- Characterization of generalized Gamma-Lindley distribution using truncated moments of order statistics
- Matrix variate pareto distributions
- Global exponential periodicity and stability of neural network models with generalized piecewise constant delay
- Optimal inequalities for contact CR-submanifolds in almost contact metric manifolds