Abstract
The main objective of this work is to present some important results and formulas in the theory of Humbert matrix functions by using the concepts of matrix functional calculus. We define Humbert matrix functions assuming that not all the matrices involved are commuting. We show that these two variable Humbert matrix functions follow naturally as confluent cases of Appell matrix functions. We determine their regions of convergence, integral representations, transformation formulas, summation formulas, contiguous relations and matrix differential equations satisfied by them.
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Abd-Elmageed, H., Abdalla, M., Abul-Ez, M., Saad, N.: Some results on the first Appell matrix function. Linear Multilinear Algebra (2018). https://doi.org/10.1080/03081087.2018.1502254
Altin, A., Cekim, B., Sahin, R.: On the matrix versions of Appell hypergeometric functions. Quaest. Math. 37(1), 31–38 (2014)
Batahan, R.S., Metwally, M.S.: Differential and integral operators on Appell’s matrix functions. Andal. Soci. Appl. Sci. 3, 7–25 (2009)
Brychkov, A.Y.: Reduction formulas for the Appell and Humbert functions. Integral Transforms Spec. Funct. 28(1), 22–38 (2017)
Brychkov, A.Y., Kim, Y.S., Rathie, A.K.: On new reduction formulas for the Humbert functions \(\psi _2\), \(\phi _2\) and \(\phi _3\). Integral Transforms Spec. Funct. 28(5), 350–360 (2017)
Choi, J., Rathie, A.K.: Certain summation formulas for Humbert’s double hypergeometric series \(\Psi _{2}\) and \(\Phi _{2}\). Commun. Korean Math. Soc. 30(4), 439–446 (2015)
Constantine, A.G., Muirhead, R.J.: Partial differential equations for hypergeometric functions of two argument matrices. J. Multivar. Anal. 2, 332–338 (1972)
Defez, E., Jodar, L.: Chebyshev matrix polynomials and second order matrix differential equations. Util. Math. 61, 107–123 (2002)
Dunford, N., Schwartz, J.: Linear Operators, Part-I. Addison-Wesley, New York (1957)
Dwivedi, R., Sahai, V.: On the hypergeometric matrix functions of two variables. Linear Multilinear Algebra 66(9), 1819–1837 (2018)
Dwivedi, R., Sahai, V.: On the hypergeometric matrix functions of several variables. J. Math. Phys. 59(2), 023505 (2018)
Dwivedi, R., Sahai, V.: A note on the Appell matrix functions. Quaest. Math. (2019). https://doi.org/10.2989/16073606.2019.1577309
Golub, G.H., Van Loan, C.F.: Matrix Computations. Johns Hopkins Univ. Press, Baltimore (1989)
Humbert, P.: The confluent hypergeometric functions of two variables. Proc. R. Soc. Edinburgh 41, 73–96 (1920)
James, A.T.: Special functions of matrix and single argument in statistics. Theory and application of special functions (Proc. Advanced Sem., Math. Res. Center, Univ. Wisconsin, Madison, Wis., 1975), pp. 497–520. Math. Res. Center, Univ. Wisconsin, Publ. No. 35, Academic Press, New York (1975)
Jodar, L., Cortes, J.C.: Some properties of gamma and beta matrix functions. Appl. Math. Lett. 11(1), 89–93 (1998a)
Jodar, L., Cortes, J.C.: On the hypergeometric matrix function. Proceedings of the VIIIth Symposium on orthogonal polynomials and their applications (Seville, 1997). J. Comput. Appl. Math. 99(1–2), 205–217 (1998b)
Jodar, L., Cortes, J.C.: Closed form general solution of the hypergeometric matrix differential equation. (2000)
Mathai, A.M.: Appell’s and Humbert’s functions of matrix arguments. Linear Algebra Appl. 183, 201–221 (1993)
Mathai, A.M.: Some results on functions of matrix argument. Math. Nachr. 84, 171–177 (1978)
Metwally, M.S., Mohamed, M.T., Shehata, A.: On Horn matrix function \(H_{2}\) of two complex variables under differential operator. Advances in Linear Algebra and Matrix Theory (ALAMT) 8(2), 96–110 (2018)
Mohamed, M.T., Shehata, A.: A study of Appell’s matrix functions of two complex variables and some properties. Adv. Appl. Math. Sci. 9(1), 23–33 (2011)
Rashwan, R.A., Metwally, M.S., Mohamed, M.T., Shehata, A.: Certain Kummer’s matrix function of two complex variables under certain differential and integral operators. Thai J. Math. 11(3), 725–740 (2013)
Rashwan, R.A., Metwally, M.S., Mohamed, M.T., Shehata, A.: On composite l(m, n)-Kummer’s matrix functions of two complex variables. Thai J. Math. 14(1), 69–81 (2016)
Rida, S.Z., Abul-Dahab, M., Saleem, M.A., Mohamed, M.T.: On Humbert matrix function \(\Psi _{1}(A, B; C, C^{\prime };z, w)\) of two complex variables under differential operator. Int. J. Ind. Math. 32, 167–179 (2010)
Shehata, A.: On \(p\) and \(q\)-Horn’s matrix function of two complex variables. Appl. Math. 2(12), 1437–1442 (2011)
Shehata, A.: Certain \(pl(m, n)\)-Kummer matrix function of two complex variables under differential operator. Appl. Math. 4(1), 91–96 (2013)
Shehata, A.: New kinds of hypergeometric matrix functions. Br. J. Math. Comput. Sci. 5(1), 92–103 (2015)
Srivastava, H.M., Karlsson, P.W.: Multiple Gaussian Hypergeometric Series. Ellis Horwood Limited, Chichester (1985)
Van Loan, C.: The sensitivity of the matrix exponential. SIAM J. Numer. Anal. 14(6), 971–981 (1977)
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The authors thank the referee for valuable suggestions that led to a better presentation of the paper. The financial assistance provided to the second author in the form of a Senior Research Fellowship from Council of Scientific and Industrial Research, India is gratefully acknowledged.
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Çekim, B., Dwivedi, R., Sahai, V. et al. Certain Integral Representations, Transformation Formulas and Summation Formulas Related to Humbert Matrix Functions. Bull Braz Math Soc, New Series 52, 213–239 (2021). https://doi.org/10.1007/s00574-020-00198-6
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DOI: https://doi.org/10.1007/s00574-020-00198-6