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

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.


41A10 Approximation by polynomials
33D45 Basic orthogonal polynomials and functions (Askey-Wilson polynomials, etc.)
Full Text: DOI arXiv


[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 · doi:10.1137/130921623
[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 · doi:10.1155/2013/757206
[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 · doi:10.1007/s00009-015-0542-2
[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 · doi:10.1016/j.amc.2016.06.023
[5] Carlitz, L., q-Bernoulli numbers and polynomials, Duke Mathematical Journal, 15, 987-1000, (1948) · Zbl 0032.00304 · doi:10.1215/S0012-7094-48-01588-9
[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 · doi:10.1155/2014/696454
[7] Kac, V.; Cheung, P., Quantum Calculus, (2002), New York, NY, USA: Springer, New York, NY, USA · Zbl 0986.05001 · doi:10.1007/978-1-4613-0071-7
[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 · doi:10.2991/jnmp.2006.13.1.2
[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 · doi:10.1007/s10958-009-9305-6
[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 · doi:10.1007/s10958-009-9314-5
[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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.