A numerical investigation of the roots of the second kind \(\lambda\)-Bernoulli polynomials. (English) Zbl 1242.65093

Lee, H. Y.; Jung, N. S.; Ryoo, C. S.

Neural Parallel Sci. Comput. 19, No. 3-4, 295-306 (2011).

Summary: We consider a new type of the Apostol Bernoulli numbers and polynomials. We call them the second kind \(\lambda\)-Bernoulli numbers \(B_{n,\lambda}\) and polynomials \(B_{n,\lambda}(x)\). We also observe the behavior of complex roots of the second kind \(\lambda\)-Bernoulli polynomials \(B_{n,\lambda}(x)\), using numerical investigation. Finally, we give a table for the solutions of the second kind \(\lambda\)-Bernoulli polynomials \(B_{n,\lambda}(x)\).

MSC:

65H04	Numerical computation of roots of polynomial equations
11B68	Bernoulli and Euler numbers and polynomials
11S40	Zeta functions and \(L\)-functions
11S80	Other analytic theory (analogues of beta and gamma functions, \(p\)-adic integration, etc.)

Keywords:

numerical examples; Apostol Bernoulli numbers; complex roots; \(\lambda\)-Bernoulli polynomials