×

A note of the modified Bernoulli polynomials and it’s the location of the roots. (English) Zbl 1463.11070

Summary: This type of polynomial is a generating function that substitutes \(e^{\lambda t}\) for \(e^t\) in the denominator of the generating function for the Bernoulli polynomial, but polynomials by using this generating function have interesting properties involving the location of the roots. We define these generating functions and observe the properties of the generating functions.

MSC:

11B68 Bernoulli and Euler numbers and polynomials
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] T.M. Apostal, On the Lerch Zeta function, Pacific J. Math. 1 (1951), 161-167. · Zbl 0043.07103 · doi:10.2140/pjm.1951.1.161
[2] L. Carlitz, q-Bernoulli numbers and polynomials, Duke Mathematical Journal 15 (1948), 987-1000. · Zbl 0032.00304 · doi:10.1215/S0012-7094-48-01588-9
[3] J. Choi, P.J. Anderson, H.M. srivastava, Some q-extensions of the Apostol-Bernoulli and the Apostol-Euler polynomials of order n, and the multiple Hurwitz zeta function, Appl. Math. Comput. 199 (2007), 723-737. · Zbl 1146.33001
[4] J.E. Choi and A.H. Kim, Some properties of twisted q-Bernoulli numbers and polynomials of the second kind, J. Appl. & Pure Math. 2 (2020), 89-97.
[5] H.Y. Lee, J.S. Jung, C.S. Ryoo, A numerical investigation of the roots of the second kind \(\lambda \)-Bernoulli polynomials, Neural Parallel and Scientific Computaions 19 (2011). · Zbl 1242.65093
[6] A.H. Kim, Multiplication formula and \(( \lambda \), q)-alternating series of \(( \lambda \), q)-Genocchi polynomials of the second kind, J. Appl. & Pure Math. 1 (2019), 167-179.
[7] Dae San Kim, Taekyun Kim, Dmitry V. Dolgy, Some Identities on Laguerre Polynomials in Connection with Bernoulli and Euler Numbers, Discrete Dynamics in Nature and Society 2012 (2012), 10 pages, Article ID 619197. · Zbl 1254.33010
[8] B.A. Kupershmidt, Reflection symmetries of q-Bernoulli polynomials, J. Nonlinear Math. Phys 12 (2005), 412-422. · Zbl 1362.33021 · doi:10.2991/jnmp.2005.12.s1.34
[9] C.S. Ryoo, On Appell-type degenerate twisted (h, q)-tangent numbers and polynomials, J. Appl. & Pure Math. 1 (2019), 69-77.
[10] C.S. Ryoo, Distribution of the roots of the second kind Bernoulli polynomials, J. Comput. Anal. Appl. 13 (2011), 971-976. · Zbl 1258.11044
[11] Y. Simsek, Theorem on twisted L-function and twisted Bernoulli numbers, Advan. Stud. Contemp. Math. 12 (2016), 237-246.
[12] Y.
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.