Approximation properties of \(q\)-Bernoulli polynomials. (English) Zbl 1470.41005

Momenzadeh, M.; Kakangi, I. Y.

Abstr. Appl. Anal. 2017, Article ID 9828065, 6 p. (2017).

Summary: We study the \(q\)-analogue of Euler-Maclaurin formula and by introducing a new \(q\)-operator we drive to this form. Moreover, approximation properties of \(q\)-Bernoulli polynomials are discussed. We estimate the suitable functions as a combination of truncated series of \(q\)-Bernoulli polynomials and the error is calculated. This paper can be helpful in two different branches: first we solve the differential equations by estimating functions and second we may apply these techniques for operator theory.

MSC:

41A10	Approximation by polynomials
33D45	Basic orthogonal polynomials and functions (Askey-Wilson polynomials, etc.)

References:

[1]	Goldman, R.; Simeonov, P.; Simsek, Y., Generating functions for the q-Bernstein bases, SIAM Journal on Discrete Mathematics, 28, 3, 1009-1025, (2014) · Zbl 1304.65264
[2]	Tohidi, E.; Kiliçman, A., A collocation method based on the bernoulli operational matrix for solving nonlinear BVPs which arise from the problems in calculus of variation, Mathematical Problems in Engineering, 2013, (2013) · Zbl 1299.49043
[3]	Tohidi, E.; Khorsand Zak, M., A new matrix approach for solving second-order linear matrix partial differential equations, Mediterranean Journal of Mathematics, 13, 3, 1353-1376, (2016) · Zbl 1350.35061
[4]	Zogheib, B.; Tohidi, E., A new matrix method for solving two-dimensional time-dependent diffusion equations with Dirichlet boundary conditions, Applied Mathematics and Computation, 291, 1-13, (2016) · Zbl 1410.65408
[5]	Carlitz, L., q-Bernoulli numbers and polynomials, Duke Mathematical Journal, 15, 987-1000, (1948) · Zbl 0032.00304
[6]	Mahmudov, N. I.; Momenzadeh, M., On a class of q -Bernoulli, q -Euler, and q -genocchi polynomials, Abstract and Applied Analysis, 2014, (2014) · Zbl 1474.11065
[7]	Kac, V.; Cheung, P., Quantum Calculus, (2002), New York, NY, USA: Springer, New York, NY, USA · Zbl 0986.05001
[8]	Ernst, T., The history of q-calculus and a new method
[9]	Hegazi, A. S.; Mansour, M., A note on q-Bernoulli numbers and polynomials, Journal of Nonlinear Mathematical Physics, 13, 1, 9-18, (2006) · Zbl 1109.33024
[10]	Carmichael, R., A treatise on the calculus of operations, Longman, brown, green, and longmans, 92-103, (1855)
[11]	Ricci, P. E.; Tavkhelidze, I., An introduction to operational techniques and special polynomials, Journal of Mathematical Sciences, 157, 1, 161-189, (2009) · Zbl 1184.33001
[12]	Cassisa, C.; Ricci, P. E.; Tavkhelidze, I., Operational methods and solutions of boundary-value problems with periodic data, Journal of Mathematical Sciences, 157, 1, 85-97, (2009) · Zbl 1184.35108
[13]	Davis, H., The Thory of Linear Operators, (1936), Elsah, Ill, USA: Principia, Elsah, Ill, USA
[14]	Boole, G., A treatise on the calculus of finite differences · Zbl 1202.65003
[15]	Chan, O.; Manna, D., A new q-analogue for Bernoulli numbers, Ams clas, 11, 68, 1-6, (2010)
[16]	Rudin, W., Principles of Mathematical Analysis, (1964), New York, NY, USA: McGraw-Hill Book Co., New York, NY, USA · Zbl 0148.02903
[17]	Annaby, M. H.; Mansour, Z. S., q-Fractional Calculus and Equation, (2012), NY, USA: Springer, NY, USA · Zbl 1267.26001