
doi: 10.1007/bf02896465
The author studies zeros of Genocchi polynomials. Genocchi polynomials \(G_n(x)\) are defined by the generating function as \[ {2t\over e^t+ 1} e^{xt}= \sum^\infty_{n=0} G_n(x){t^n\over n!}, \] and the Genocchi number \(G_n\) is defined by \(G_n= G_n(0)\). These are closely related to Euler polynomials and Euler numbers from their definitions. In Section 2, first the graphs of \(G_n(x)\) are given for \(1\leq n\leq 10\) and \(-3\leq x\leq 3\). And then zeros of \(G_n(x)\) for \(n=10\), \(20\), \(30\), \(40\) are plotted in the complex plane. These approximate numerical zeros are computed by Mathematica software. From that he observes the regular structure of the complex roots of \(G_n(x)\). In Section 3, he lists several open questions.
Genocchi polynomial, Euler number, Euler polynomial, Zeta functions and \(L\)-functions, Bernoulli and Euler numbers and polynomials, Other analytic theory (analogues of beta and gamma functions, \(p\)-adic integration, etc.), numerical experiments, Genocchi number
Genocchi polynomial, Euler number, Euler polynomial, Zeta functions and \(L\)-functions, Bernoulli and Euler numbers and polynomials, Other analytic theory (analogues of beta and gamma functions, \(p\)-adic integration, etc.), numerical experiments, Genocchi number
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