A note on Boole polynomials. (English) Zbl 1369.11020
Summary: Boole polynomials are defined by the generating function
\[ \sum_{n=0}^\infty \mathrm{Bl}_n(x\mid \lambda)\frac{t^n}{n!}=\frac {1}{(1+(1+t)^\lambda)} (1+t)^x \] (for \(\lambda=1\) we have the Changhee polynomials) and play an important role in the area of number theory, algebra and umbral calculus. In this paper, we investigate some properties of Boole polynomials and consider Witt-type formulas for the Boole numbers and polynomials. Finally, we derive some new identities of these polynomials from the Witt-type formulas which are related to Euler polynomials.
\[ \sum_{n=0}^\infty \mathrm{Bl}_n(x\mid \lambda)\frac{t^n}{n!}=\frac {1}{(1+(1+t)^\lambda)} (1+t)^x \] (for \(\lambda=1\) we have the Changhee polynomials) and play an important role in the area of number theory, algebra and umbral calculus. In this paper, we investigate some properties of Boole polynomials and consider Witt-type formulas for the Boole numbers and polynomials. Finally, we derive some new identities of these polynomials from the Witt-type formulas which are related to Euler polynomials.
MSC:
| 11B68 | Bernoulli and Euler numbers and polynomials |
| 11S80 | Other analytic theory (analogues of beta and gamma functions, \(p\)-adic integration, etc.) |
References:
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