???? −Bernoulli Matrices and Their Some Properties

Öz In this study, we define ???? −Bernoulli matrix B( ) q and ???? −Bernoulli polynomial matrix B( , ) x q by using ???? −Bernoulli numbers, and polynomials respectively. We obtain some properties of B( ) q and B( , ) x q . We obtain factorizations ???? −Bernoulli polynomial matrix and shifted ???? −Bernoulli matrix using special matrices.

___

Bernoulli, J., “Ars conjectandi”, Published Posthumously Basel, Switzerland, 1-33, (1713).

Nörlund, N. E., “Vorlesungen uber dierenzenrechnung”, Chelsea Publishing Company, New York, (1924).

Carlitz, L., “Some theorems on Bernoulli Numbers of higher order received”, Pacic J. Math., 2: 127-139 (1952).

Carlitz, L., “ −Bernoulli numbers and polynomials”, Duke Math. J., 15 (4): 987-1000 (1948).

Carlitz, L., “Expansions of −Bernoulli numbers”, Duke Math. J., 25, 355-364 (1958).

Kac, V., Cheung P., “Quantum Calculus”, Springer, New York (2002)

Lalin, M. N., “Bernoulli numbers” Junior Number Theory Seminar-Universty of Texas at Austin September 6th (2005)

Zhang, Z.,Whang,J., “Bernoulli matrix and its algebraic properties” Discrete Appl. Math., 154: 1622-1632 (2006).

Ernst,T., “ −Pascal and −Bernoulli matrices and umbral approach”, Department of Mathematics Uppsala Universty D.M. Report 2008:23 (2008).

Hegazi, A. S. and Mansour, M., “A note on −Bernoulli numbers Phys.,13(1):9-18(2005). J. Nonlinear Math.

Song, S.-Z., Cheon, G.-S., Jun, Y.-B., and Beasley, L.-B., “A − analogue of the generalized factorial numbers”, J. Korean Math. Soc., 47, 645-657 (2010).