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

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 \(B_{n,\lambda}(x)\).


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.)