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

### 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.)