Generalizations on poly-Eulerian numbers and polynomials.

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Generalizations on poly-Eulerian numbers and polynomials.

1[section]. Introduction

In the 17th century a topic of mathematical interest was finite sums of powers of integers such as the series 1 + 2 + 3 + ... + (n - 1) or the series [1.sup.2] + [2.sup.2] + [3.sup.2] + ... + [(n - 1).sup.2]. The closed form for these finite sums were known, but the sum of the more general case [1.sup.k] + [2.sup.k] + [3.sup.k] + ... + [(n - 1).sup.k] was not. It was the mathematician Jacob Bernoulli who solved this problem. After introducing the Bernoulli numbers, Euler introduced the Euler numbers to study the sum [T.sub.k] (n) = [[summation of].sup.n-1.sub.r=0] [(-1).sup.r] [r.sup.k]. The Bernoulli and Euler arise in Taylor series in the expansion...

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