Riemann Zeta Matrix Function

Year 2015, Volume: 28 Issue: 4, 683 - 688, 16.12.2015

Levent Kargın , Veli Kurt

Abstract

In this study, obtaining the matrix analog of the Euler's reflection formula for the classical gamma function we expand the domain of the gamma matrix function and give a infinite product expansion of sinπxP.  Furthermore we define Riemann zeta matrix function and evaluate some other matrix integrals. We prove a functional equation for Riemann zeta matrix function.

Keywords

Gamma matrix function, Riemann Zeta matrix function, Matrix integrals

References

• R. Aktaş, B. Çekim, and R. Şahin, The Matrix Version Polynomials, Miskolc Mathematical Notes 13 (2) (2012) 197-208. Humbert
• R. Aktaş, B. Çekim and A. Çevik, Extended Jacobi matrix polynomials, Utilitas Mathematica, 92, 47- 64, 2013.
• A. Altın and B. Çekim, Some properties associated with Mathematica, 88, 171-181, 2012. polynomials, Utilitas
• A. Altın and B. Çekim, Some miscellaneous properties for Gegenbauer matrix polynomials, Utilitas Mathematica, 92, 377-387, 2013.
• S. Varma, B. Çekim, and F. Taşdelen, On Konhauser matrix polynomials, Ars Combinatoria 100 (2011) 193-204.
• E. Defez and L. Jódar, Chebyshev matrix polynomials and second order matrix differential equations, Utilitas Math. 61 (2002) 107-123.
• E. Defez and L. Jódar, Jacobi Matrix Differential Equation, Polynomial Solutions, and Their Properties, Computers and Math. with Appl., 48 (2004) 789-803.
• E. Defez, A Rodrigues-type formula for Gegenbauer matrix polynomials, Appl. Math. Lett. 26 (2013) 899–903.
• N. Dunford and J. Schwartz, Linear Operators, Vol. I, Interscience, New York, 1963.
• L. Jódar, R. Company and E. Navarro, Laguerre matrix polynomials and systems of second order differential equations, Appl. Numer. Math. 15 (1994) 53-63.
• L. Jódar and R. Company, Hermite matrix polynomials and second order matrix differential equations, J. Approx. Theory Appl. 12 (2) (1996) 20-30.
• L. Jódar, J.C. Cortés, Some properties of gamma and beta functions, Appl. Math. Lett. 11 (1) (1998) 89-93.
• L. Jódar and J.C. Cortés, On the hypergeometric matrix function, J. Comput. Appl. Maths., 99 (1998) 205-217.
• L. Jódar and J. Sastre: The growth of Laguerre matrix polynomials on bounded intervals. Appl. Math. Lett. 13 (2000) 21-26.
• L. Kargin and V. Kurt, Some relations on Hermite matrix Computational Applications, Mathematical and Computational Applications, 18 (3) (2013) 323- 329. Mathematical and