Umbral Form
In the classic umbral calculus one formally treats the indices j in a sequence Bj as if they were exponents, so that, in this case we can apply the binomial theorem and say
In the modern umbral calculus, one considers the linear functional T on the vector space of polynomials in a variable b given by
Then one can say
Read more about this topic: Faulhaber's Formula
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“Our ideas are the offspring of our senses; we are not more able to create the form of a being we have not seen, without retrospect to one we know, than we are able to create a new sense. He whose fancy has conceived an idea of the most beautiful form must have composed it from actual existence.”
—Henry Fuseli (17411825)